1.2.3: Talk Concerning the First Beginning (Zuni)
Yes, indeed. In this world there was no one at all. Always the sun came up; always he went in. No one in the morning gave him sacred meal; no one gave him prayer sticks; it was very lonely. He said to his two children: “You will go into the fourth womb.  Thus he said to them. They said, “But how shall we go in?” “That will be all right.” Laying their lightning arrow across their rainbow bow, they drew it. Drawing it and shooting down, they entered.
When they entered the fourth womb it was dark inside. They could not distinguish anything. They said, “Which way will it be best to go?” They went toward the west. They met someone face to face. They said, “Whence come you?” “I come from over this way to the west.” “What are you doing going around?” “I am going around to look at my crops. Where do you live?” “No, we do not live any place. There above our father the Sun, priest, made us come in. We have come in,” they said. “Indeed,” the younger brother said. “Come, let us see,” he said. They laid down their bow.  When they had kindled the fire, light came out from the coals. As it came out, they blew on it and it caught fire. Aglow! It is growing light. “Ouch! What have you there?” he said. He fell down crouching. He had a slimy horn, slimy tail, he was slimy allover, with webbed hands. The elder brother said, “Poor thing! Put out the light.” Saying thus, he put out the light. The youth said, “Oh dear, what have you there?” “Why, we have fire,” they said. “Well, what (crops) do you have coming up?” “Yes, here are our things coming up.” Thus he said. He was going around looking after wild grasses.
He said to them, “Well, now, let us go.” They went toward the west, the two leading. There the people were sitting close together. They questioned one another. Thus they said, “Well, now, you two, speak. I think there is something to say. It will not be too long a talk. If you lotus know that we shall always remember it.” “That is so, that is so,” they said. “Yes, indeed, it is true. There above is our father, Sun.  Because it is thus we have come to you, in order that you may go out standing yonder into the daylight of your sun father. Now you will say which way (you decide).” Thus the two said. “Hayi! Yes, indeed. Because it is thus you have passed us on our roads. Now that you have passed us on our roads here where we stay miserably, far be it from us to speak against it. We can not see one another. Here inside where we just trample on one another, where we just spit on one another, where we just urinate on one another, where we just befoul one another, where we just follow one another about, you have passed us on our roads. None of us can speak against it. But rather, as the priest of the north says, so let it be. Now you two call him.” Thus they said to the two, and they came up close toward the north side.
They met the north priest on his road. “You have come,” he said. “Yes, we have come. How have you lived these many days?” “Here where I live happily you have passed me on my road. Sit down.” When they were seated he questioned them. “Now speak. I think there is something to say. It will not be too long a talk. So now, that you will let me know.” “Yes, indeed, it is so. In order that you may go out standing there into the daylight of your sun father we have passed you on your road. However you say, so shall it be.” “Yes, indeed, now that you have passed us on our road here where we live thus wretchedly, far be it from me to talk against it. Now that you have come to us here inside where, we just trample on one another, where we just spit on one another, where we just urinate on one another, where we just befoul one another, where we just follow one another about, how should I speak against it?” so he said. Then they arose. They came back. Coming to the village where they were sitting in the middle place, there they questioned one another. “Yes, even now we have met on our roads. Indeed there is something to say; it will not be too long a talk. When you let me know that, I shall always remember it,” thus they said to one another. When they had spoken thus, “Yes, indeed. In order that you may go out standing into the daylight of your sun father, we have passed you on your road,” thus they said. “Hayi! Yes, indeed. Now that you have passed us on our road here where we cannot see one another, where we just trample on one another, where we just urinate on one another, where we just befoul one another, where we just follow one another around, far be it from me to speak against it. But rather let it be as my younger brother, the priest of the west shall say. When he says, ‘Let it be thus,’ that way it shall be. So now, you two call him.” Thus said the priest of the north and they went and stood close against the west side.
“Well, perhaps by means of the thoughts of someone somewhere it may be that we shall go out standing into the daylight of our sun father.” Thus he said. The two thought. “Come, let us go over there to talk with eagle priest.” They went. They came to where eagle was staying. “You have come.” “Yes.” “Sit down.” They sat down. “Speak!” “We want you.” “Where?” “Near by, to where our fathers, käeto·‑we, tcu‑eto·we, stay quietly, we summon you.” “Haiyi!” So they went. They came to where käeto·‑we stayed. “Well, even now when you summoned me, I have passed you on your roads. Surely there is something to say; it will not be too long a talk. So now if you let me know that I shall always remember it,” thus he said. “Yes, indeed, it is so. Our fathers, käeto·‑we, tcu‑eto·we, mu‑eto·we, łhe·‑eto‑we, all the society priests shall go out standing into the daylight of their sun father. You will look for their road.” “Very well,” he said, “I am going,” he said. He went around. Coming back to his starting place he went a little farther out. Coming back to his starting place again he went still farther out. Coming back to his starting place he went way far out. Coming back to his starting place, nothing was visible. He came. To where käeto·‑we stayed he came. After he sat down they questioned him. “Now you went yonder looking for the road going out. What did you see in the world?” “Nothing was visible.” “Haiyi!” “Very well, I am going now.” So he went.
When he had gone the two thought. “Come, let us summon our grandson, cokäpiso,” thus they said. They went. They came to where cokäpiso stayed. “Our grandson, how have you lived these days?” “Where I live happily you have passed me on my road. I think perhaps there is something to say; it will not be too long a talk. So now when you let me know that, I shall always remember it,” thus he said. “Yes, indeed, it is so. Our fathers, käeto·‑we, tcu‑eto·we, mu‑eto·we, łhe·‑eto‑we, all the society priests are about to come outstanding into the daylight of their sun father. We summon you that you may be the one to look for their road.” “Indeed?” Thus he said. They went. When they got there, they questioned them where they were sitting. “Even now you have summoned me. Surely there is something to say; it will not be too long a talk. So now when you let me know that, I shall always remember it.” “Yes, indeed, it is so. When our fathers, our mothers, käeto·‑we, tcu‑eto·we, mu‑eto·we, łhe·‑eto‑we, the society priests, go forth standing into the daylight of their sun father, you will look for their road.” Thus the two said. He went out to the south. He went around. Coming back to the same place, nothing was visible. A second time he went, farther out. Coming back to the same place, nothing was visible. A third time, still farther out he went. Nothing was visible. A fourth time he went, way far, but nothing was visible. When he came to where käeto·‑we were staying, the two questioned him.  What did you see in the world?” Thus the two asked him. “Well, nothing was visible.” “Well indeed?” the two said. “Very well, I am going now.” Saying this, he went.
When cokäpiso had gone the two thought. “Come, let us go and talk to our grandson chicken hawk.” Thus they said. They went. They reached where chicken hawk stayed. “You have come.” “Yes.” “Sit down.” “How have you lived these days?” “Happily. Well now, speak. I think there is something to say; it will not be too long a talk. So now, when you let me know it, I shall always remember that.” “Yes, indeed, it is so. When our fathers, käeto·‑we, tcu‑eto·we, mu‑eto·we, łhe·‑eto‑we, the society priests, go out standing into the sunlight of their sun father, you will look for their road.” So they went. When they got there they sat down. There he questioned them. “Yes, even now you summoned me. Perhaps there is something to say; it will not be too long a talk. When you let me know that, I shall always remember it.” Thus he said. “Yes, indeed, it is so. When our fathers, käeto·‑we, tcu‑eto·we, mu‑eto·we, łhe·‑eto‑we, the society priests, go out standing into the daylight of their sun father, you will look for their road.” “Is that so?” Saying this, he went out. He went to the south. He went where cokäpiso had been. Coming back to his starting place, nothing was visible. A second time he went, farther out. He came back to his starting place, nothing was visible. He went a third time, along the shore of the encircling ocean. A fourth time farther out he went. He came back to his starting place. Nothing was visible. To where käeto·‑we stayed he came. “Nothing is visible.” “Haiyi!” Yes, so I am going.” “Well, go.” So he went.
Then the two thought. “Come on, let us summon our grandson,” thus they said. They went. They came to where humming bird was staying. “You have come?” “Yes, how have you lived these days?” “Where I live happily these days you have passed me on my road. Sit down.” When they had sat down: “Well, now, speak. I think there is something to say; it will not be too long a talk. So now if you let me know that, I shall always remember it.” “Yes, indeed, it is so. Is that so?” Saying this, they went. When they got there, he questioned them. “Well, even now you summoned me. Surely there is something to say. It will not be too long a talk. So now when you let me know that I shall always remember it.” Thus he said. “Yes, indeed, it is so. When our fathers, käeto·‑we, tcu‑eto·we, mu‑eto·we, łhe·‑eto‑we, the society priests, go out into the daylight of their sun father, that you shall be the one to look for their road, for that we have summoned you.” Thus the two said. He went out toward the south. He went on. Coming back to his starting place, nothing was visible. Farther out he went. Coming back to the same place, nothing was visible. Then for the third time he went. Coming back to the same place, nothing was visible. For the fourth time he went close along the edge of the sky. Coming back to the same place, nothing was visible. He came. Coming where käeto·‑we were staying, “Nothing is visible.” “Hayi!” “Yes. Well, I am going now.” “Very well, go.” He went.
The two said, “What had we better do now? That many different kinds of feathered creatures, the ones who go about without ever touching the ground, have failed.” Thus the two said. “Come, let us talk with our grandson, locust. Perhaps that one will have a strong spirit because he is like water.” Thus they said. They went. Their grandson, locust, they met. “You have come.” “Yes, we have come.” “Sit down. How have you lived these days?” “Happily.” “Well, even now you have passed me, on my road. Surely there is something to say; it will not be too long a talk. So now when you let me know that, that I shall always remember.” Thus he said. “Yes, indeed, it is so. In order that our fathers, käeto·‑we, tcu‑eto·we, mu‑eto·we, łhe·‑eto‑we, the society priests, may go out standing into the daylight of their sun father, we have come to you.” “Is that so?” Saying this, they went. When they arrived they sat down. Where they were sitting, he questioned them. “Well, just now you came to me. Surely there is something to say; it will not be too long a talk. So now if you let me know that, that I shall always remember.” “Yes, indeed. In order that our fathers, käeto·‑we, tcu‑eto·we, mu‑eto·we, łhe·‑eto‑we, the society priests, may go out standing into the daylight of their sun father, we have summoned you.” “Indeed?” Saying this, locust rose right up. He goes up. He went through into another world. And again he goes right up. He went through into another world. And again he goes right up. Again he went through into another world. He goes right up. When he had just gone a little way his strength gave out, he came back to where käeto·‑we were staying and said, “Three times I went through and the fourth time my strength gave out.” “Hayi! Indeed?” Saying this, he went.
When he had gone the two thought. “Come, let us speak with our grandson, Reed Youth. For perhaps that one with his strong point will be all right.” Saying this, they went. They came to where Reed Youth stayed. “You have come?” “Yes; how have you lived these days.” “Where I stay happily you have passed me on my road. Sit down.” Thus he said. They sat down. Then he questioned them. “Yes. Well, even now you have passed me on my road. I think there is something to say; it will not be too long a talk. When you let me know that, that I shall always remember.” Thus he said. “Yes, indeed, in order that our fathers, käeto·‑we, tcu‑eto·we, mu‑eto·we, łhe·‑eto‑we, the society priests, may go out standing into the daylight of their sun father, we have come to you.” “Hayi! Is that so?” Having spoken thus, they went. When they arrived they sat down. There he questioned them. “Yes, even now that you have summoned me I have passed you on your roads. Surely there is something to say; it will not be too long a talk. When you let me know that, that I shall always remember . . . . Yes, indeed, it is so. In order that our fathers, käeto·‑we, tcu‑eto·we, mu‑eto·we, łhe·‑eto‑we, the society priests, may go forth standing into the daylight of their sun father, we have summoned you.” Thus they said. “Hayi! Is that so?” Saying this, he went out. Where Locust had gone out he went out. The first time he passed through, the second time he passed through, the third time he passed through. Having passed through the fourth time and come forth standing into the daylight of his sun father, he went back in. Coming back in he came to where käeto·‑we were staying. “You have come?” Thus they said. “Yes,” he said. “Far off to see what road there may be you have gone. How may it be there now?” Thus they said. “Yes, indeed, it is so. There it is as you wanted it. As you wished of me, I went forth standing into the daylight of my sun father now.” Thus he said. “Halihi! Thank you!” “Now I am going.” “Go.” Saying this, he went.
After he had gone they were sitting around. Now as they were sitting around, there the two set up a pine tree for a ladder. They stayed there. For four days they stayed there. Four days, they say, but it was four years. There all the different society priests sang their song sequences for one another. The ones sitting in the first row listened carefully. Those sitting next on the second row heard all but a little. Those sitting on the third row heard here and there. Those sitting last on the fourth row heard just a little bit now and then. It was thus because of the rustling of the dry weeds.
When their days there were at an end, gathering together their sacred things they arose. “Now what shall be the name of this place?” “Well, here it shall be sulphur‑smell‑inside‑world; and furthermore, it shall be raw‑dust world.” Thus they said. “Very well. Perhaps if we call it thus it will be all right.” Saying this, they came forth.
After they had come forth, setting down their sacred things in a row at another place, they stayed there quietly. There the two set up a spruce tree as a ladder. When the ladder was up they stayed there for four days. And there again the society priests sang their song sequences for one another. Those sitting on the first row listened carefully. Those sitting there on the second row heard all but a little. Those sitting there on the third row heard here and there. Those sitting last distinguished a single word now and then. It was thus because of the rustling of some plants. When their days there were at an end, gathering together their sacred things there they arose. “Now what shall it be called here?” “Well, here it shall be called soot‑inside‑world, because we still can not recognize one another.” “Yes, perhaps if it is called thus it will be all right.” Saying this to one another, they arose.
Passing through to another place, and putting down their sacred things in a row, they stayed there quietly. There the two set up a piñon tree as a ladder. When the piñon tree was put up, there all the society priests and all the priests went through their song sequences for one another.  Those sitting on the second row heard all but a little. Those sitting behind on the third row heard here and there. Those sitting on the fourth row distinguished only a single word now and then. This was because of the rustling of the weeds.
When their days there were at an end, gathering together their sacred things they arose. Having arisen, “Now what shall it be called here?” “Well, here it shall be fog‑inside‑world, because here just a little bit is visible.” “Very well, perhaps if it is called thus it will be all right.” Saying this, rising, they came forth.
Passing through to another place, there the two set down their sacred things in a row, and there they sat down. Having sat down, the two set up a cottonwood tree as a ladder. Then all the society priests and all the priests went through their song sequences for one another. Those sitting first heard everything clearly. Those sitting on the second row heard all but a little. Those sitting on the third row heard here and there. Those sitting last on the fourth row distinguished a single word now and then. It was thus because of the rustling of some plants.
When their days there were at an end, after they had been there, when their four days were passed, gathering together their sacred possessions, they arose. When they arose, “Now what shall it be called here?” “Well, here it shall be wing‑inner‑world, because we see our sun father’s wings.” Thus they said. They came forth.
Into the daylight of their sun father they came forth standing. Just at early dawn they came forth. After they had come forth there they set down their sacred possessions in a row. The two said, “Now after a little while when your sun father comes forth standing to his sacred place you will see him face to face. Do not close your eyes.” Thus he said to them. After a little while the sun came out. When he came out they looked at him. From their eyes the tears rolled down. After they had looked at him, in a little while their eyes became strong. “Alas!” Thus they said. They were covered all over with slime. With slimy tails and slimy horns, with webbed fingers, they saw one another. “Oh dear! is this what we look like?” Thus they said.
Then they could not tell which was which of their sacred possessions. Meanwhile, near by an old man of the Dogwood clan lived alone. Spider said to him, “Put on water. When it gets hot, wash your hair.” “Why?” “Our father, our mothers, käeto·‑we, tcu‑eto·we, mu‑eto·we, łhe·‑eto‑we, all the society priests, into the daylight of their sun father have come forth standing. They can not tell which is which. You will make this plain to them.” Thus she said. “Indeed? Impossible. From afar no one can see them. Where they stay quietly no one can recognize them.” Thus he said. “Do not say that. Nevertheless it will be all right. You will not be alone. Now we shall go.” Thus she said. When the water was warm he washed his hair.
Meanwhile, while he was washing his hair, the two said, “Come let us go to meet our father, the old man of the Dogwood clan. I think he knows in his thoughts; because among our fathers, käeto·‑we, tcu‑eto·we, mu‑eto·we, łhe·‑eto‑we, we can not tell which is which.” Thus they said. They went. They got there. As they were climbing tip, “Now indeed! They are coming.” Thus Spider said to him. She climbed up his body from his toe. She clung behind his ear. The two entered. “You have come,” thus he said. “Yes. Our father, how have you lived these days?” “As I live happily you pass me on my road. Sit down.” They sat down. “Well, now, speak. I think some word that is not too long, your word will be. Now, if you let me know that, remembering it, I shall live.” “Indeed it is so. Our fathers, käeto·‑we, tcu‑eto·we, mu‑eto·we, łhe·‑eto‑we, all the society priests, into the daylight of their sun father have risen and come out. It is not plain which is which. Therefore we have passed you on your road.” “Haiyi, is that so? Impossible! From afar no one can see them. Where they stay quietly no one can recognize them.” Thus he said. “Yes, but we have chosen you.” Thus the two said. They went. When they came there, “My fathers, my mothers, how have you lived these days?” “Happily, our father, our child. Be seated.” Thus they said. He sat down. Then he questioned them. “Yes, now indeed, since you have sent for me, I have passed you on your road. I think some word that is not too long your word will be. Now if you let me know that, remembering it, I shall always live.”
Thus he said. “Indeed, it is so. Even though our fathers, our mothers, käeto·‑we, tcu‑eto·we, mu‑eto·we, łhe·‑eto‑we, have come out standing into the daylight of their sun father, it is not plain which of these is which. Therefore we have sent for you.” Thus they said. “Haiyi. Well, let me try.” “Impossible. From afar no one can see them. Where they stay quietly no one can tell which is which.” “Well, let me try.” Thus he said. Where they lay in a row he stood beside them. Spider said to him, “Here, the one that lies here at the end is käeto·‑we and these next ones touching it are tcu‑eto·we, and this next one is łhe·‑eto‑we, and these next ones touching it are mu‑eto·we.” Thus she said. He said, “Now this is käeto·‑we, and these all touching it are tcu‑eto·we, and this one is łhe·‑eto‑we, and all these touching it are mu‑eto·we.” Thus he said. “Halihi! Thank you. How shall be the cycle of the months for them?” Thus he said: “This one Branches‑broken‑down. This one No‑snow‑on‑the‑road. This one Little‑sand‑storms. This one Great‑sand‑storms. This the Month‑without‑a‑name. This one Turn‑about. This one Branches‑broken‑down. This one No‑snow‑on‑the‑road. This one Little‑sand‑storms. This one Great‑sand‑storms. This the Month‑without‑a‑name. This one Turn‑about. Thus shall be all the cycle of the months.” “Halihi! Thank you. Our father, you shall not be poor. Even though you have no sacred possessions toward which your thoughts bend, whenever Itiwana is revealed to us, because of your thought, the ceremonies of all these shall come around in order. You shall not be a slave.” This they said. They gave him the sun. “This shall be your sacred possession.” Thus they said. When this had happened thus they lived.
Four days—four days they say, but it was four years—there they stayed. When their days were at an end, the earth rumbled. The two said, “Who was left behind?” “I do not know, but it seems we are all here.” Thus they said. Again the earth rumbled. “Well, does it not seem that some one is still left behind?” Thus the two said. They went. Coming to the place where they had come out, there they stood. To the mischief‑maker and the Mexicans they said, “Haiyi! Are you still left behind?” “Yes.” “Now what are you still good for?” Thus they said. “Well, it is this way. Even though käeto·‑we have issued forth into the daylight, the people do not live on the living waters of good corn; on wild grasses only they live. Whenever you come to the middle you will do well to have me. When the people are many and the land is all used up, it will not be well. Because this is so I have come out.” Thus he said. “Haiyi! Is that so? So that’s what you are. Now what are you good for?” Thus they said. “Indeed, it is so. When you come to the middle, it will be well to have my seeds. Because käeto·‑we do not live on the good seeds of the corn, but on wild grasses only. Mine are the seeds of the corn and all the clans of beans.” Thus he said. The two took him with them. They came to where käeto·‑we were staying. They sat down. Then they questioned him. “Now let us see what you are good for.” “Well, this is my seed of the yellow corn.” Thus he said. He showed an ear of yellow corn. “Now give me one of your people.” Thus he said. They gave him a baby. When they gave him the baby it seems he did something to her. She became sick. After a short time she died. When she had died he said, “Now bury her.”  After four days he said to the two, “Come now. Go and see her.” The two went to where they had come out. When they got there the little one was playing in the dirt. When they came, she laughed. She was happy. They saw her and went back. They came to where the people were staying. “Listen! Perhaps it will be all right for you to come. She is still alive. She has not really died.” “Well, thus it shall always be.” Thus he said.
Gathering together all their sacred possessions, they came hither. To the place called since the first beginning, Moss Spring, they came. There they set down their sacred possessions in a row. There they stayed. Four days they say, but it was four years. There the two washed them. They took from all of them their slimy tails, their slimy horns. “Now, behold! Thus you will be sweet.” There they stayed.
When their days were at an end they came hither. Gathering together all their sacred possessions, seeking Itiwana, yonder their roads went. To the place called since the first beginning Massed‑cloud Spring, they came. There they set down their sacred possessions in a row. There they stayed quietly. Four days they stayed. Four days they say, but it was four years. There they stayed. There they counted up the days. For käeto·‑we, four nights and four days. With fine rain caressing the earth, they passed their days. The days were made for łhe·‑eto‑we, mu‑eto·we. For four days and four nights it snowed. When their days were at an end there they stayed.
When their days were at an end they arose. Gathering together all their sacred possessions, hither their roads went. To the place called since the first beginning Mist Spring their road came. There they sat down quietly. Setting out their sacred possessions in a row, they sat down quietly. There they counted up the days for one another. They watched the world for one another’s waters. For käeto·‑we, four days and four nights, with heavy rain caressing the earth they passed their days. When their days were at an end the days were made for łhe·‑eto‑we and mu‑eto·we. Four days and four nights with falling snow the world was filled. When their days were at an end, there they stayed.
When all their days were passed, gathering together all their sacred possessions, hither their road went. To Standing‑wood Spring they came. There they sat down quietly. Setting out their sacred possessions in a row, they stayed quietly. There they watched one another’s days. For käeto·‑we, four days and four nights with fine rain caressing the earth, they passed their days. When all their days were at an end, the days were made for lhe‑eto:we and mu‑eto·we. For four days and four nights, with falling snow, the world was filled. When all their days were at an end, there they stayed.
When all their days were passed, gathering together their sacred possessions, and arising, hither they came. To the place called since the first beginning Upuilima they came. When they came there, setting down their sacred possessions in a row, they stayed quietly. There they strove to outdo one another. There they planted all their seeds. There they watched one another’s days for rain. For käeto·‑we, four days with heavy rain caressing the earth. There their corn matured. It was not palatable, it was bitter. Then the two said, “Now by whose will will our corn become fit to eat?” Thus they said. They summoned raven. He came and pecked at their corn, and it became good to eat. “It is fortunate that you have come.” With this then, they lived.
When their days were at an end they arose. Gathering together their sacred possessions, they came hither. To the place called since the first beginning, Cornstalk‑place they came. There they set down their sacred possessions in a row. There they stayed four days. Four days they say, but it was four years. There they planted all their seeds. There they watched one another’s days for rain. During käeto·‑we’s four days and four nights, heavy rain fell. During lhe‑eto:we’s and mu‑eto·we’s four days and four nights, the world was filled with falling snow. Their days were at an end. Their corn matured. When it was mature it was hard. Then the two said, “By whose will will our corn become soft? Well, owl.” Thus they said. They summoned owl. Owl came. When he came he pecked at their corn and it became soft.
Then, when they were about to arise, the two said, “Come, let us go talk to the corn priest.” Thus they said. They went. They came to where the corn priest stayed. “How have you lived these days?” “As we are living happily you have passed us on our road. Sit down.” They sat down. There they questioned one another. “Well, speak. I think some word that is not too long, your word will be. Now, if you let me know that, remembering it, I shall always live.” “Indeed, it is so. To‑morrow, when we arise, we shall set out to seek Itiwana. Nowhere have we found the middle. Our children, our women, are tired. They are crying. Therefore we have come to you. Tomorrow your two children will look ahead. Perhaps if they find the middle when our fathers, our mothers, käeto·‑we, tcu‑eto·we, mu‑eto·we, lhe‑eto:we, all the society priests, come to rest, there our children will rest themselves. Because we have failed to find the middle.” “Haiyi! Is that so? With plain words you have passed us on our road. Very well, then, thus it shall be.” Thus he said. The two went.
Next morning when they were about to set out they put down a split ear of corn and eggs. They made the corn priest stand up. They said, “Now, my children, some of you will go yonder to the south. You will take these.” Thus he said (indicating) the tip of the ear and the macaw egg. And then the ones that were to come this way took the base of the ear and the raven egg. Those that were to go to the south took the tip of the ear and the macaw egg. “Now, my children, yonder to the south you will go. If at any time you come to Itiwana, then some time we shall meet one another.” Thus they said. They came hither.
They came to the place that was to be Katcina village. The girl got tired. Her brother said, “Wait, sit down for a while. Let me climb up and look about to see what kind of a place we are going to.” Thus he said. His sister sat down. Her brother climbed the hill. When he had climbed up, he stood looking this way. “Eha! Maybe the place where we are going lies in this direction. Maybe it is this kind of a place.” Thus he said and came down. Meanwhile his sister had scooped out the sand. She rested against the side of the hill. As she lay sleeping the wind came and raised her apron of grass. It blew up and she lay with her vulva exposed. As he came down he saw her. He desired her. He lay down upon his sister and copulated with her. His sister awoke. “Oh, dear, oh, dear,” she was about to say (but she said,) “Watsela, watsela.” Her brother said, “Ah!” He sat up. With his foot he drew a line. It became a stream of water. The two went about talking. The brother talked like Koyemci. His sister talked like Komakatsik. The people came.
“Oh alas, alas! Our children have become different beings.” Thus they said. The brother speaking: “Now it will be all right for you to cross here.” Thus he said. They came and went in. They entered the river. Some of their children turned into water snakes.
Some of them turned into turtles. Some of them turned into frogs. Some of them turned into lizards. They bit their mothers. Their mothers cried out and dropped them. They fell into the river. Only the old people reached the other side. They sat down on the bank. They were half of the people. The two said, “Now wait. Rest here.” Thus they said. Some of them sat down to rest. The two said (to the others), “Now you go in. Your children will turn into some kind of dangerous animals and will bite you. But even though you cry out, do not let them go.  Thus they said to them. They entered the water. Their children became different creatures and bit them. Even though they cried out, they crossed over. Then their children once more became the kind of creatures they had been. “Alas! Perhaps had we done that it would have been all right.” Now all had crossed over.
There setting down their sacred possessions in a row, they stayed quietly. They stayed there quietly for four days. Thus they say but they stayed for four years. There each night they lived gaily with loud singing. When all their time was passed, the two said “Come, let us go and talk to Ne‑we‑kwe.” Thus they said. They went to where the Ne‑we‑kwe were staying. They came there. “How have you passed these days?’ “Happily. You have come? Be seated.” They sat down. Then they questioned them. “Now speak. I think some word that is not too long your word will be. If you let me know that, remembering it I shall always live.” “Indeed it is so. To‑morrow we shall arise. Our fathers, our mothers, käeto·‑we, tcu‑eto·we, mu‑eto·we, lhe‑eto:we, all the society priests, are going to seek the middle. But nowhere have we come to the middle. Our children and our women are tired. They are crying now. Therefore we have passed you on your road. To‑morrow you will look ahead. If perhaps somewhere you come to Itiwana there our children will rest.” Thus they said. “Alas! but we are just foolish people. If we make some mistake it will not be right.” Thus he said. “Well, that is of no importance. It can’t be helped. We have chosen you.” Thus they said. “Well indeed?’ “Yes. Now we are going.” “Go ahead.” The two went out.
They came (to where the people were staying). “Come, let us go and speak to our children.” Thus they said. They went. They entered the lake.  “Now stand still a moment. Our two fathers have come.” Thus they said. The katcinas suddenly stopped dancing. When they stopped dancing they said to the two, “Now our two fathers, now indeed you have passed us on our road. I think some word that is not too long your word will be. If you will let us know that we shall always remember it.” Thus he said. “Indeed it is so. Tomorrow we shall arise. Therefore we have come to speak to you.” “Well indeed? May you go happily. You will tell our parents, ‘Do not worry.’ We have not perished. In order to remain thus forever we stay here. To Itiwana but one day’s travel remains. Therefore we stay nearby. When our world grows old and the waters are exhausted and the seeds are exhausted, none of you will go back to the place of your first beginning. Whenever the waters are exhausted and the seeds are exhausted you will send us prayer sticks. Yonder at the place of our first beginning with them we shall bend over to speak to them. Thus there will not fail to be waters. Therefore we shall stay quietly near by.” Thus they said to them. “Well indeed?” “Yes. You will tell my father, my mother, ‘Do not worry.’ We have not perished.” Thus they said. They sent strong words to their parents. “Now we are going. Our children, may you always live happily.” “Even thus may you also go.” Thus they said to the two. They went out. They arrived. They told them. “Now our children, here your children have stopped. ‘They have perished,’ you have said. But no. The male children have become youths, and the females have become maidens. They are happy. They live joyously. They have sent you strong words. ‘Do not worry,’ they said.” “Haiyi! Perhaps it is so.”
They stayed overnight. Next morning they arose. Gathering together all their sacred possessions, they came hither. They came to Hanlhipingka. Meanwhile the two Ne‑we‑kwe looked ahead. They came to Rock‑in‑the‑river. There two girls were washing a woolen dress. They killed them. After they had killed them they scalped them. Then someone found them out. When they were found out, because they were raw people, they wrapped themselves in mist. There to where käeto·‑we were staying they came. “Alack, alas! We have done wrong!” Thus they said. Then they set the days for the enemy. There they watched one another’s days for rain. käeto·‑we’s four days and four nights passed with the falling of heavy rain. There where a waterfall issued from a cave the foam arose. There the two Ahaiyute appeared. They came to where käeto·‑we were staying. Meanwhile, from the fourth inner world, Unasinte, Uhepololo, Kailuhtsawaki, Hattungka, Oloma, Catunka, came out to sit down in the daylight. There they gave them the comatowe Song cycle. Meanwhile, right there, Coyote was going about hunting. He gave them their pottery drum. They sang comatowe.
After this had happened, the two said, “Now, my younger brother, Itiwana is less than one day distant. We shall gather together our children, all the beast priests, and the winged creatures, this night.” They went. They came yonder to Comk?äkwe. There they gathered together all the beasts, mountain lion, bear, wolf, wild cat, badger, coyote, fox, squirrel; eagle, buzzard, cokapiso, chicken hawk, baldheaded eagle, raven, owl. All these they gathered together. Now squirrel was among the winged creatures, and owl was among the beasts. “Now my children, you will contest together for your sun father’s daylight. Whichever side has the ball, when the sun rises, they shall win their sun father’s daylight.” Thus the two said. “Indeed?” They went there. They threw up the ball. It fell on the side of the beasts. They hid it. After they had hidden it, the birds came one by one but they could not take it. Each time they paid four straws. They could not take it.
At this time it was early dawn. Meanwhile Squirrel was lying by the fireplace. Thus they came one by one but they could not take it. Eagle said, “Let that one lying there by the fireplace go.” They came to him and said, “Are you asleep?” “No. I am not asleep.” “Oh dear! Now you go!” Thus they said. “Oh no, I don’t want to go,” he said. He came back. “The lazy one does not wish to.” Thus they said. Someone else went. Again they could not take it. Now it was growing light. “Let that one lying by the fireplace go.” Thus they said. Again Buzzard went. “Alas, my boy, you go.” “Oh, no, I don’t feel like it.” Thus he said. Again he went back. “He does not want to,” he said. Again some one else went. Again they did not take it. Now it was growing light. Spider said to him, “Next time they come agree to go.” Thus she said. Then again they said, “Let that one lying by the fireplace go.” Thus they said; and again someone went. When he came there he said, “Alas, my boy, you go.” “All right, I shall go.” Thus he said and arose. As he arose Spider said to him, “Take that stick.” He took up a stick, so short. Taking it, he went. Now the sun was about to rise. They came there. Spider said to him, “Hit those two sitting on the farther side.” Thus she said. Bang! He knocked them down. He laid them down. Then, mountain lion, who was standing right there, said, “Hurry up, go after it. See whether you can take it.” Thus he said. Spider said to him, “Say to him, ‘Oh, no, I don’t want to take it.’ So she said.” “Oh, no, I don’t want to take it. Perhaps there is nothing inside. How should I take it? There is nothing in there.” “That is right. There is nothing in there. All my children are gathered together. One of them is holding it. If you touch the right one, you will take it.” “All right.” Now Spider is speaking: “No one who is sitting here has it. That one who goes about dancing, he is holding it.” Thus she said. He went. He hit Owl on the hand. The white ball came out. He went. He took up the hollow sticks and took them away with him. Now the birds hid the ball. Spider came down. Over all the sticks she spun her web. She fastened the ball with her web. Now the animals came one by one. Whenever they touched a stick, she pulled (the ball) away. Each time they paid ten straws. The sun rose. After sunrise, he was sitting high in the sky. Then the two came. They said, “Now, all my children, you have won your sun father’s daylight, and you, beasts, have lost your sun father’s daylight. All day you will sleep. After sunset, at night, you will go about hunting. But you, owl, you have not stayed among the winged creatures. Therefore you have lost your sun father’s daylight. You have made a mistake. If by daylight, you go about hunting, the one who has his home above will find you out. He will come down on you.  Then thinking, ‘Let it be here,’ you will come to the end of your life. This kind of creature you shall be.” Thus they said. They stayed there overnight. The animals all scattered.
The two went. They came to where käeto·‑we were staying. Then they arose. Gathering together all their sacred possessions, they arose. Lhe‑eto:we said, “Now, my younger brothers, hither to the north I shall take my road. Whenever I think that Itiwana has been revealed to you, then I shall come to you.” Thus he said, and went to the north. Now some woman, seeing them, said, “Oh dear! Whither are these going?” Thus she said:
Naiye heni aiye
Naiye heni aiye.
In white stripes of hail they went.
Meanwhile käeto·‑we came hither. They came to House Mountain. When they came there they would not let them pass through. They fought together. A giant went back and forth before them. Thus they fought together. Thus evening came. In the evening they came back to Hanlhipingka. Next day they went again. In heavy rain they fought together. In the evening they went back again. Next morning they went again for the third time. Again they fought together. The giant went back and forth in front. Even though she had arrows sticking in her body she did not die. At sunset they went back again. Next morning they went. They came there, and they fought together. Still they would not surrender. The giant went back and forth in front. Although she was wounded with arrows, she would not surrender. Ahaiyute said, “Alas, why is it that these people will not let us pass? Wherever may her heart be, that one that goes back and forth? Where her heart should be we have struck her, yet she does not surrender. It seems we can not overcome her.  Without doubt he knows.” Thus he said. His younger brother climbed up to where the sun was.
It was nearly noon when he arrived. “You have come?” “Yes, I have come.” “Very well, speak. I think some word that is not too long your word will be. So if you let me know that, I shall always remember it.” Thus he said. “Indeed, it is so. Our fathers, our mothers, käeto·‑we, tcu‑eto·we, mu‑eto·we, lhe‑eto:we, all the society priests, have issued forth into the daylight. Here they go about seeking Itiwana. These people will not let them pass. Where does she have her heart, that one who goes back and forth before them? In vain have we struck her where her heart should be. Even though the arrows stick in her body, she does not surrender.” “Haiyi! For nothing are you men! She does not have her heart in her body. In vain have you struck her there. Her heart is in her rattle.” Thus he said. “This is for you and this is for your elder brother.” Thus he said, and gave him two turquoise rabbit sticks. “Now, when you let these go with my wisdom I shall take back my weapons.’ “Haiyi! Is that so? Very well, I am going now.” “Go ahead. May you go happily.” Thus he said. He came down. His elder brother said to him, “Now, what did he tell you?” “Indeed, it is so. In vain do we shoot at her body. Not there is her heart; but in her rattle is her heart. With these shall we destroy her.” Thus he said, and gave his brother one of the rabbit sticks. When he had given his brother the rabbit stick, “Now go ahead, you.” Thus he said. The younger brother went about to the right. He threw it and missed. Whiz! The rabbit stick went up to the sun. As the rabbit stick came up the sun took it. “Now go ahead, you try.” Thus he said. The elder brother went around to the left. He threw it. As he threw it, zip! His rabbit stick struck his rattle. Tu ‑‑‑ n! They ran away. As they started to run away, their giant died. Then they all ran away. The others ran after them. They came to a village. They went into the houses. “This is my house;” “This is my house;” and “This is mine.” Thus they said. They went shooting arrows into the roof. Wherever they first came, they went in.
In the center of their room was standing a jar of urine. They stuffed their nostrils with känaite flowers and with cotton wool. Then they thrust their noses into the jar. The people could see them. “Oh, dear! These are ghosts!” Thus they said. Then the two said to them, “Do not harm them, for I think they know something. So even though it is dangerous they are still alive.” Thus they said. The two entered. As they came in they questioned them. “And now do you know something? Therefore, even though it is dangerous, you have not perished.” “Well, we have a sacred object.” “Indeed! Very well, take them. We shall go.  If your days are the same as theirs you will not be slaves. It does not matter that he is only a little boy. Even so, he will be our father. It does not matter that she is a little girl, she will be our mother.” Thus he said. Taking their sacred object they went. They came to where käeto·‑we were staying. There they said to them, “Now make your days.” “Oh, no! We shall not be first. When all your days are at an end, then we shall add on our days.” Thus they said. Then they worked for käeto·‑we. käeto·‑we’s days were made. Four days and four nights, with fine rain falling, were the days of käeto·‑we. When their days were at an end, the two children and their grandmother worked. Their days were made. Four days and four nights, with heavy rain falling, were their days. Then they removed the evil smell. They made flowing canyons. Then they said, “Halihi! Thank you! Just the same is your ceremony. What may your clan be?” “Well, we are of the Yellow Corn clan.” Thus they said. “Haiyi! Even though your eton:e is of the Yellow Corn clan, because of your bad smell, you have become black.  Thus they said to them.
Then they arose. Gathering together all their sacred possessions, they came hither, to the place called, since the first beginning, Halona‑Itiwana, their road came. There they saw the Navaho helper, little red bug. “Here! Wait! All this time we have been searching in vain for Itiwana. Nowhere have we seen anything like this.” Thus they said. They summoned their grandchild, water bug. He came. “How have you lived these many days?” “Where we have been living happily you have passed us on our road. Be Seated.” Thus they said. He sat down. Then he questioned them. “Now, indeed, even now, you have sent for me. I think some word that is not too long your word will be. So now, if you will let me know that, I shall always remember it.” “Indeed, it is so. Our fathers, our mothers, käeto·‑we, tcu‑eto·we, mu‑eto·we, lhe‑eto:we, all the society priests, having issued forth into the daylight, go about seeking the middle. You will look for the middle for them. This is well.  Following after those, toward whom our thoughts bend, we shall pass our days.” Thus they said. He sat down facing the east. To the left he stretched out his arm. To the right he stretched out his arm, but it was a little bent. He sat down facing the north. He stretched out his arms on both sides. They were just the same. Both arms touched the horizon. “Come, let us cross over to the north. For on this side my right arm is a little bent.” Thus he said. They crossed (the river). They rested. He sat down. To all directions he stretched out his arms. Everywhere it was the same. “Right here is the middle.” Thus he said. There his fathers, his mothers, käeto·‑we, tcu‑eto·we, mu‑eto·we, lhe‑eto:we, all the society priests, the society p̂ekwins, the society bow priests, and all their children came to rest. Thus it happened long ago.


11.5: Warranties
A warranty is a contractual assurance that goods will meet a certain standard. An express warranty is a guarantee, created by the words or actions of the seller, that goods will meet certain standards. Under the UCC, a seller may create an express warranty in three ways:
- With an affirmation of a fact or a promise;
- With a description of the goods; or
- With a sample or model.
To constitute a warranty, the seller’s words or actions must be part of the basis of the bargain. For example, if a salesperson says that a car’s timing belt was just replaced and will not need to be replaced for another 100,000 miles, and the buyer relied on that statement when deciding to buy the car, then the salesperson’s statement becomes an express warranty.
An implied warranty is a guarantee created by the UCC and imposed on the seller of goods. There are two main implied warranties: merchantability and fitness for a particular purpose.
The implied warranty of merchantability is a warranty that the goods are fit for the ordinary purposes for which they are used. To disclaim this warranty, a merchant must use the term “merchantability.” The novel or unusual use of goods is not protected by this warranty. If a buyer uses goods for something other than their intended purpose, the warranty does not apply.
The implied warranty of fitness for a particular purpose is a warranty that the property is suitable for the buyer’s special purpose. For the warranty to apply, the seller must know what the buyer’s purpose is and the buyer must rely on the seller’s judgment that the goods meet the buyer’s needs.


5.5: Decimals and Fractions (Part 1)
- Convert fractions to decimals
- Order decimals and fractions
- Simplify expressions using the order of operations
- Find the circumference and area of circles
Before you get started, take this readiness quiz.
- Divide: 0.24 ÷ 8. If you missed this problem, review Example 5.4.9 .
- Order 0.64__0.6 using < or >. If you missed this problem, review Example 5.2.7 .
- Order −0.2__−0.1 using < or >. If you missed this problem, review Example 5.2.8 .
Convert Fractions to Decimals
In Decimals , we learned to convert decimals to fractions. Now we will do the reverse—convert fractions to decimals. Remember that the fraction bar indicates division. So \(\dfrac{4}{5}\) can be written 4 ÷ 5 or \(5 \overline{)4}\). This means that we can convert a fraction to a decimal by treating it as a division problem.
To convert a fraction to a decimal, divide the numerator of the fraction by the denominator of the fraction.
Write the fraction \(\dfrac{3}{4}\) as a decimal.
Solution
| A fraction bar means division, so we can write the fraction 3 4 using division. | |
| Divide. |
So the fraction \(\dfrac{3}{4}\) is equal to 0.75.
Write the fraction as a decimal: \(\dfrac{1}{4}\).
- Answer
-
\(0.25\)
Write the fraction as a decimal: \(\dfrac{3}{8}\).
- Answer
-
\(0.375\)
Write the fraction \(− \dfrac{7}{2}\) as a decimal.
Solution
| The value of this fraction is negative. After dividing, the value of the decimal will be negative. We do the division ignoring the sign, and then write the negative sign in the answer. | $$- \dfrac{7}{2}$$ |
| Divide 7 by 2. |
So, \(− \dfrac{7}{2}\) = −3.5.
Write the fraction as a decimal: \(− \dfrac{9}{4}\).
- Answer
-
\(-2.25\)
Write the fraction as a decimal: \(− \dfrac{11}{2}\).
- Answer
-
\(-5.5\)
Repeating Decimals
So far, in all the examples converting fractions to decimals the division resulted in a remainder of zero. This is not always the case. Let’s see what happens when we convert the fraction \(\dfrac{4}{3}\) to a decimal. First, notice that \(\dfrac{4}{3}\) is an improper fraction. Its value is greater than 1. The equivalent decimal will also be greater than 1.
We divide 4 by 3.
No matter how many more zeros we write, there will always be a remainder of 1, and the threes in the quotient will go on forever. The number 1.333… is called a repeating decimal. Remember that the “…” means that the pattern repeats.
A repeating decimal is a decimal in which the last digit or group of digits repeats endlessly.
How do you know how many ‘repeats’ to write? Instead of writing 1.333 … we use a shorthand notation by placing a line over the digits that repeat. The repeating decimal 1.333 … is written 1.\(\overline{3}\). The line above the 3 tells you that the 3 repeats endlessly. So 1.333… = 1.\(\overline{3}\). For other decimals, two or more digits might repeat. Table \(\PageIndex{1}\) shows some more examples of repeating decimals.
| 1.333… = 1.\(\overline{3}\) | 3 is the repeating digit |
| 4.1666… = 4.1\(\overline{6}\) | 6 is the repeating digit |
| 4.161616… = 4.\(\overline{16}\) | 16 is the repeating block |
| 0.271271271… = 0.\(\overline{271}\) | 271 is the repeating block |
Write \(\dfrac{43}{22}\) as a decimal.
Solution
Divide 43 by 22.
Notice that the differences of 120 and 100 repeat, so there is a repeat in the digits of the quotient; 54 will repeat endlessly. The first decimal place in the quotient, 9, is not part of the pattern. So,
\[\dfrac{43}{22} = 1.9 \overline{54}\]
Write as a decimal: \(\dfrac{27}{11}\).
- Answer
-
\(2. \overline{45}\)
Write as a decimal: \(\dfrac{51}{22}\).
- Answer
-
\(2.3 \overline{18}\)
It is useful to convert between fractions and decimals when we need to add or subtract numbers in different forms. To add a fraction and a decimal, for example, we would need to either convert the fraction to a decimal or the decimal to a fraction.
Simplify: \(\dfrac{7}{8}\) + 6.4.
Solution
| Change \(\dfrac{7}{8}\) to a decimal. | 0.875 + 6.4 | |
| Add. | 7.275 |
Simplify: \(\dfrac{3}{8}\) + 4.9.
- Answer
-
\(5.275\)
Simplify: 5.7 + \(\dfrac{13}{20}\).
- Answer
-
\(6.35\)
Order Decimals and Fractions
In Decimals , we compared two decimals and determined which was larger. To compare a decimal to a fraction, we will first convert the fraction to a decimal and then compare the decimals.
Order \(\dfrac{3}{8}\)__0.4 using < or >.
Solution
| Convert \(\dfrac{3}{8}\) to a decimal. | 0.375__0.4 |
| Compare 0.375 to 0.4 | 0.375 < 0.4 |
| Rewrite with the original fraction. | \(\dfrac{3}{8}\) < 0.4 |
Order each of the following pairs of numbers, using < or >.
\[\dfrac{17}{20} \_ \_ \; 0.82\]
- Answer
-
\(>\)
Order each of the following pairs of numbers, using < or >.
\[\dfrac{3}{4} \_ \_ \; 0.785\]
- Answer
-
\(<\)
When ordering negative numbers, remember that larger numbers are to the right on the number line and any positive number is greater than any negative number.
Order −0.5___\(− \dfrac{3}{4}\) using < or >.
Solution
| Convert \(− \dfrac{3}{4}\) to a decimal. | −0.5___−0.75 |
| Compare −0.5 to −0.75. | −0.5 > −0.75 |
| Rewrite the inequality with the original fraction. | −0.5 > \(− \dfrac{3}{4}\) |
Order each of the following pairs of numbers, using < or >:
\[− \dfrac{5}{8} \_ \_ −0.58\]
- Answer
-
\(<\)
Order each of the following pairs of numbers, using < or >:
\[−0.53 \_ \_ − \dfrac{11}{20}\]
- Answer
-
\(>\)
Write the numbers \(\dfrac{13}{20}\), 0.61, \(\dfrac{11}{16}\) in order from smallest to largest.
Solution
| Convert the fractions to decimals. | 0.65, 0.61, 0.6875 |
| Write the smallest decimal number first. | 0.61, ____, _____ |
| Write the next larger decimal number in the middle place. | 0.61, 0.65, _____ |
| Write the last decimal number (the larger) in the third place. | 0.61, 0.65, 0.6875 |
| Rewrite the list with the original fractions. | 0.61, \(\dfrac{13}{20}, \dfrac{11}{16}\) |
Write each set of numbers in order from smallest to largest: \(\dfrac{7}{8}, \dfrac{4}{5}\), 0.82.
- Answer
-
\(\frac{4}{5}\), \(0.82\), \(\frac{7}{8}\)
Write each set of numbers in order from smallest to largest: 0.835, \(\dfrac{13}{16}, \dfrac{3}{4}\).
- Answer
-
\(\frac{3}{4}\), \(\frac{13}{16}\), \(0.835\)
Simplify Expressions Using the Order of Operations
The order of operations introduced in Use the Language of Algebra also applies to decimals. Do you remember what the phrase “Please excuse my dear Aunt Sally” stands for?
Simplify the expressions: (a) 7(18.3 − 21.7) (b) \(\dfrac{2}{3}\) (8.3 − 3.8)
Solution
(a) 7(18.3 − 21.7)
| Simplify inside parentheses. | 7(−3.4) |
| Multiply. | −23.8 |
(b) \(\dfrac{2}{3}\) (8.3 − 3.8)
| Simplify inside parentheses. | $$\dfrac{2}{3} (4.5)$$ |
| Write 4.5 as a fraction. | $$\dfrac{2}{3} \left(\dfrac{4.5}{1}\right)$$ |
| Multiply. | $$\dfrac{9}{3}$$ |
| Simplify. | $$3$$ |
Simplify: (a) 8(14.6 − 37.5) (b) \(\dfrac{3}{5}\) (9.6 − 2.1).
- Answer a
-
\(-183.2\)
- Answer b
-
\(4.5\)
Simplify: (a) 25(25.69 − 56.74) (b) \(\dfrac{2}{7}\) (11.9 − 4.2).
- Answer a
-
\(-776.25\)
- Answer b
-
\(2.2\)
Simplify each expression: (a) 6 ÷ 0.6 + (0.2)4 − (0.1) 2 (b) \(\left(\dfrac{1}{10}\right)^{2}\) + (3.5)(0.9)
Solution
(a) 6 ÷ 0.6 + (0.2)4 − (0.1) 2
| Simplify exponents. | 6 ÷ 0.6 + (0.2)4 − 0.01 |
| Divide. | 10 + (0.2)4 − 0.01 |
| Multiply. | 10 + 0.8 − 0.01 |
| Add. | 10.8 − 0.01 |
| Subtract. | 10.79 |
(b) \(\left(\dfrac{1}{10}\right)^{2}\) + (3.5)(0.9)
| Simplify exponents. | \(\dfrac{1}{100}\) + (3.5)(0.9) |
| Multiply. | \(\dfrac{1}{100}\) + 3.15 |
| Convert \(\dfrac{1}{100}\) to a decimal. | 0.01 + 3.15 |
| Add. | 3.16 |
Simplify: 9 ÷ 0.9 + (0.4)3 − (0.2) 2 .
- Answer
-
\(11.16\)
Simplify: \(\left(\dfrac{1}{2}\right)^{2}\) + (0.3)(4.2).
- Answer
-
\(1.51\)
Contributors and Attributions
-
Lynn Marecek (Santa Ana College) and MaryAnne Anthony-Smith (Formerly of Santa Ana College). This content is licensed under Creative Commons Attribution License v4.0 "Download for free at<EMAIL_ADDRESS>."


3.13: Cauchy Sequences. Completeness
This page is a draft and is under active development.
A convergent sequence is characterized by the fact that its terms \(x_{m}\) become (and stay) arbitrarily close to its limit, as \(m \rightarrow+\infty .\) Due to this, however, they also get close to each other; in fact, \(\rho\left(x_{m}, x_{n}\right)\) can be made arbitrarily small for sufficiently large \(m\) and \(n .\) It is natural to ask whether the latter property, in turn, implies the existence of a limit. This problem was first studied by Augustin-Louis Cauchy \((1789-1857) .\) Thus we shall call sequences Cauchy sequences. More precisely, we formulate the following.
A sequence \(\left\{x_{m}\right\} \subseteq(S, \rho)\) is called a Cauchy sequence (we briefly say that \("\left\{x_{m}\right\}\) is Cauchy") iff, given any \(\varepsilon>0\) (no matter how small), we have \(\rho\left(x_{m}, x_{n}\right)<\varepsilon\) for all but finitely many \(m\) and \(n .\) In symbols,
\[(\forall \varepsilon>0)(\exists k)(\forall m, n>k) \quad \rho\left(x_{m}, x_{n}\right)<\varepsilon.\]
Observe that here we only deal with terms \(x_{m}, x_{n},\) not with any other point. The limit (if any) is not involved, and we do not have to know it in advance. We shall now study the relationship between property \((1)\) and convergence.
Every convergent sequence \(\left\{x_{m}\right\} \subseteq(S, \rho)\) is Cauchy.
- Proof
-
Let \(x_{m} \rightarrow p .\) Then given \(\varepsilon>0,\) there is a \(k\) such that
\[(\forall m>k) \quad \rho\left(x_{m}, p\right)<\frac{\varepsilon}{2}.\]
As this holds for any \(m>k,\) it also holds for any other term \(x_{n}\) with \(n>k\).
Thus
\[(\forall m, n>k) \quad \rho\left(x_{m}, p\right)<\frac{\varepsilon}{2} \text{ and } \rho\left(p, x_{n}\right)<\frac{\varepsilon}{2}.\]
Adding and using the triangle inequality, we get
\[\rho\left(x_{m}, x_{n}\right) \leq \rho\left(x_{m}, p\right)+\rho\left(p, x_{n}\right)<\varepsilon,\]
and \((1)\) is proved. \(\square\)
Every Cauchy sequence \(\left\{x_{m}\right\} \subseteq(S, \rho)\) is bounded.
- Proof
-
We must show that all \(x_{m}\) are in some globe. First we try an arbitrary radius \(\varepsilon\). Then by \((1),\) there is \(k\) such that \(\rho\left(x_{m}, x_{n}\right)<\varepsilon\) for \(m, n>k .\) Fix some \(n>k .\) Then
\[(\forall m>k) \rho\left(x_{m}, x_{n}\right)<\varepsilon, \text{ i.e., } x_{m} \in G_{x_{n}}(\varepsilon).\]
Thus the globe \(G_{x_{n}}(\varepsilon)\) contains all \(x_{m}\) except possibly the \(k\) terms \(x_{1}, \ldots, x_{k}\). To include them as well, we only have to take a larger radius \(r,\) greater than \(\rho\left(x_{m}, x_{n}\right), m=1, \ldots, k .\) Then all \(x_{m}\) are in the enlarged globe \(G_{x_{n}}(r) .\) \(\square\)
Note 1. In \(E^{1},\) under the standard metric, only sequences with finite limits are regarded as convergent. If \(x_{n} \rightarrow \pm \infty,\) then \(\left\{x_{n}\right\}\) is not even a Cauchy sequence in \(E^{1}(\text { in view of Theorem } 2) ;\) but in \(E^{*},\) under a suitable metric (cf. Problem 5 in §11, it is convergent (hence also Cauchy and bounded).
If a Cauchy sequence \(\left\{x_{m}\right\}\) clusters at a point \(p,\) then \(x_{m} \rightarrow p\).
- Proof
-
We want to show that \(x_{m} \rightarrow p,\) i.e., that
\[(\forall \varepsilon>0)(\exists k)(\forall m>k) \quad \rho\left(x_{m}, p\right)<\varepsilon.\]
Thus we fix \(\varepsilon>0\) and look for a suitable \(k .\) Now as \(\left\{x_{m}\right\}\) is Cauchy, there is a \(k\) such that
\[(\forall m, n>k) \quad \rho\left(x_{m}, x_{n}\right)<\frac{\varepsilon}{2}.\]
Also, as \(p\) is a cluster point, the globe \(G_{p}\left(\frac{\varepsilon}{2}\right)\) contains infinitely many \(x_{n},\) so we can fix one with \(n>k\left(k \text { as above). Then } \rho\left(x_{n}, p\right)<\frac{\varepsilon}{2} \text { and, as noted above, }\right.\) also \(\rho\left(x_{m}, x_{n}\right)<\frac{\varepsilon}{2}\) for \(m>k .\) Hence
\[(\forall m>k) \quad \rho\left(x_{m}, x_{n}\right)+\rho\left(x_{n}, p\right)<\varepsilon,\]
implying \(\rho\left(x_{m}, p\right) \leq \rho\left(x_{m}, x_{n}\right)+\rho\left(x_{n}, p\right)<\varepsilon,\) as required. \(\square\)
Note 2. It follows that a Cauchy sequence can have at most one cluster point \(p,\) for \(p\) is also its limit and hence unique; see §14, Corollary 1.
These theorems show that Cauchy sequences behave very much like convergent ones. Indeed, our next theorem (a famous result by Cauchy) shows that, in \(E^{n}\left(^{*} \text { and } C^{n}\right)\) the two kinds of sequences coincide.
(Cauchy's convergence criterion). A sequence \(\left\{\overline{x}_{m}\right\}\) in \(E^{n}\) (*or \(C^{n}\) ) converges if and only if it is a Cauchy sequence.
- Proof
-
Conversely, let \(\left\{x_{m}\right\}\) be a Cauchy sequence. Then by Theorem \(2,\) it is bounded. Hence by the Bolzano-Weierstrass theorem (Theorem 2 of §16, it has a cluster point \(\overline{p} .\) Thus by Theorem 3 above, it converges to \(\overline{p},\) and all is proved. \(\square\)
Unfortunately, this theorem (along with the Bolzano-Weierstrass theorem used in its proof) does not hold in all metric spaces. It even fails in some subspaces of \(E^{1} .\) For example, we have
\[x_{m}=\frac{1}{m} \rightarrow 0 \text{ in } E^{1}.\]
By Theorem 1 , this sequence, being convergent, is also a Cauchy sequence. Moreover, it still preserves \((1)\) even if we remove the point 0 from \(E^{1}\) since the distances \(\rho\left(x_{m}, x_{n}\right)\) remain the same. However, in the resulting subspace \(S=E^{1}-\{0\},\) the sequence no longer converges because its limit (and unique cluster point) 0 has disappeared, leaving a "gap" in its place. Thus we have a Cauchy sequence in \(S,\) without a limit or cluster points, so Theorem 4 fails in \(S\) (along with the Bolzano-Weierstrass theorem).
Quite similarly, both theorems fail in \((0,1)\) (but not in \([0,1] )\) as a subspace of \(E^{1}\). By analogy to incomplete ordered fields, it is natural to say that \(S\) is "incomplete" because of the missing cluster point \(0,\) and call a space (or subspace) "complete" if it has no such "gaps," i.e., if Theorem 4 holds in it.
Thus we define as follows.
A metric space (or subspace) \((S, \rho)\) is said to be complete iff every Cauchy sequence in \(S\) converges to some point \(p\) in \(S .\)
Similarly, a set \(A \subseteq(S, \rho)\) is called complete iff each Cauchy sequence \(\left\{x_{m}\right\} \subseteq A\) converges to some point \(p\) in \(A,\) i.e., iff \((A, \rho)\) is complete as a metric subspace of \((S, \rho) .\)
In particular, \(E^{n}\left(^{*} \text { and } C^{n}\right)\) are complete by Theorem \(4 .\) The sets \((0,1)\) and \(E^{1}-\{0\}\) are incomplete in \(E^{1},\) but \([0,1]\) is complete. Indeed, we have the following theorem.

(ii) Every complete set \(A \subseteq(S, \rho)\) is necessarily closed.
- Proof
-
(i) Let \(A\) be a closed set in a complete space \((S, \rho) .\) We have to show that Theorem 4 holds in \(A(\text { as it does in } S) .\) Thus we fix any Cauchy sequence \(\left\{x_{m}\right\} \subseteq A\) and prove that it converges to some \(p\) in \(A .\)
Now, since \(S\) is complete, the Cauchy sequence \(\left\{x_{m}\right\}\) has a limit \(p\) in \(S .\) As \(A\) is closed, however, that limit must be in \(A\) by Theorem 4 in §16. Thus (i) is proved.
(ii) Now let \(A\) be complete in a metric space \((S, \rho) .\) To prove that \(A\) is closed, we again use Theorem 4 of §16. Thus we fix any convergent sequence \(\left\{x_{m}\right\} \subseteq A, x_{m} \rightarrow p \in S,\) and show that \(p\) must be \(i n A .\)
Now, since \(\left\{x_{m}\right\}\) converges in \(S,\) it is a Cauchy sequence, in \(S\) as well as in A. Thus by the assumed completeness of \(A,\) it has a limit \(q\) in \(A .\) Then, however, the uniqueness of \(\lim _{x_{m}} x_{m}(\text { in } S)\) implies that \(p=q \in A,\) so that \(p\) is in \(A,\) indeed. \(\square\)


16.7: Complex Circuits
What do we do if we come across a circuit more complex than the simple series configurations we’ve seen so far? Take this circuit as an example:
The simple time constant formula (τ=RC) is based on a simple series resistance connected to the capacitor. For that matter, the time constant formula for an inductive circuit (τ=L/R) is also based on the assumption of a simple series resistance. So, what can we do in a situation like this, where resistors are connected in a series-parallel fashion with the capacitor (or inductor)?
The answer comes from our studies in network analysis. Thevenin’s Theorem tells us that we can reduce any linear circuit to an equivalent of one voltage source, one series resistance, and a load component through a couple of simple steps. To apply Thevenin’s Theorem to our scenario here, we’ll regard the reactive component (in the above example circuit, the capacitor) as the load and remove it temporarily from the circuit to find the Thevenin voltage and Thevenin resistance. Then, once we’ve determined the Thevenin equivalent circuit values, we’ll re-connect the capacitor and solve for values of voltage or current over time as we’ve been doing so far.
After identifying the capacitor as the “load,” we remove it from the circuit and solve for voltage across the load terminals (assuming, of course, that the switch is closed):
This step of the analysis tells us that the voltage across the load terminals (same as that across resistor R 2 ) will be 1.8182 volts with no load connected. With a little reflection, it should be clear that this will be our final voltage across the capacitor, seeing as how a fully-charged capacitor acts like an open circuit, drawing zero current. We will use this voltage value for our Thevenin equivalent circuit source voltage.
Now, to solve for our Thevenin resistance, we need to eliminate all power sources in the original circuit and calculate resistance as seen from the load terminals:
Re-drawing our circuit as a Thevenin equivalent, we get this:
Our time constant for this circuit will be equal to the Thevenin resistance times the capacitance (τ=RC). With the above values, we calculate:
Now, we can solve for voltage across the capacitor directly with our universal time constant formula. Let’s calculate for a value of 60 milliseconds. Because this is a capacitive formula, we’ll set our calculations up for voltage:
Again, because our starting value for capacitor voltage was assumed to be zero, the actual voltage across the capacitor at 60 milliseconds is equal to the amount of voltage change from zero, or 1.3325 volts.
We could go a step further and demonstrate the equivalence of the Thevenin RC circuit and the original circuit through computer analysis. I will use the SPICE analysis program to demonstrate this:
Comparison RC analysis * first, the netlist for the original circuit: v1 1 0 dc 20 r1 1 2 2k r2 2 3 500 r3 3 0 3k c1 2 3 100u ic=0 * then, the netlist for the thevenin equivalent: v2 4 0 dc 1.818182 r4 4 5 454.545 c2 5 0 100u ic=0 * now, we analyze for a transient, sampling every .005 seconds * over a time period of .37 seconds total, printing a list of * values for voltage across the capacitor in the original * circuit (between modes 2 and 3) and across the capacitor in * the thevenin equivalent circuit (between nodes 5 and 0) .tran .005 0.37 uic .print tran v(2,3) v(5,0) .end time v(2,3) v(5) 0.000E+00 4.803E-06 4.803E-06 5.000E-03 1.890E-01 1.890E-01 1.000E-02 3.580E-01 3.580E-01 1.500E-02 5.082E-01 5.082E-01 2.000E-02 6.442E-01 6.442E-01 2.500E-02 7.689E-01 7.689E-01 3.000E-02 8.772E-01 8.772E-01 3.500E-02 9.747E-01 9.747E-01 4.000E-02 1.064E+00 1.064E+00 4.500E-02 1.142E+00 1.142E+00 5.000E-02 1.212E+00 1.212E+00 5.500E-02 1.276E+00 1.276E+00 6.000E-02 1.333E+00 1.333E+00 6.500E-02 1.383E+00 1.383E+00 7.000E-02 1.429E+00 1.429E+00 7.500E-02 1.470E+00 1.470E+00 8.000E-02 1.505E+00 1.505E+00 8.500E-02 1.538E+00 1.538E+00 9.000E-02 1.568E+00 1.568E+00 9.500E-02 1.594E+00 1.594E+00 1.000E-01 1.617E+00 1.617E+00 1.050E-01 1.638E+00 1.638E+00 1.100E-01 1.657E+00 1.657E+00 1.150E-01 1.674E+00 1.674E+00 1.200E-01 1.689E+00 1.689E+00 1.250E-01 1.702E+00 1.702E+00 1.300E-01 1.714E+00 1.714E+00 1.350E-01 1.725E+00 1.725E+00 1.400E-01 1.735E+00 1.735E+00 1.450E-01 1.744E+00 1.744E+00 1.500E-01 1.752E+00 1.752E+00 1.550E-01 1.758E+00 1.758E+00 1.600E-01 1.765E+00 1.765E+00 1.650E-01 1.770E+00 1.770E+00 1.700E-01 1.775E+00 1.775E+00 1.750E-01 1.780E+00 1.780E+00 1.800E-01 1.784E+00 1.784E+00 1.850E-01 1.787E+00 1.787E+00 1.900E-01 1.791E+00 1.791E+00 1.950E-01 1.793E+00 1.793E+00 2.000E-01 1.796E+00 1.796E+00 2.050E-01 1.798E+00 1.798E+00 2.100E-01 1.800E+00 1.800E+00 2.150E-01 1.802E+00 1.802E+00 2.200E-01 1.804E+00 1.804E+00 2.250E-01 1.805E+00 1.805E+00 2.300E-01 1.807E+00 1.807E+00 2.350E-01 1.808E+00 1.808E+00 2.400E-01 1.809E+00 1.809E+00 2.450E-01 1.810E+00 1.810E+00 2.500E-01 1.811E+00 1.811E+00 2.550E-01 1.812E+00 1.812E+00 2.600E-01 1.812E+00 1.812E+00 2.650E-01 1.813E+00 1.813E+00 2.700E-01 1.813E+00 1.813E+00 2.750E-01 1.814E+00 1.814E+00 2.800E-01 1.814E+00 1.814E+00 2.850E-01 1.815E+00 1.815E+00 2.900E-01 1.815E+00 1.815E+00 2.950E-01 1.815E+00 1.815E+00 3.000E-01 1.816E+00 1.816E+00 3.050E-01 1.816E+00 1.816E+00 3.100E-01 1.816E+00 1.816E+00 3.150E-01 1.816E+00 1.816E+00 3.200E-01 1.817E+00 1.817E+00 3.250E-01 1.817E+00 1.817E+00 3.300E-01 1.817E+00 1.817E+00 3.350E-01 1.817E+00 1.817E+00 3.400E-01 1.817E+00 1.817E+00 3.450E-01 1.817E+00 1.817E+00 3.500E-01 1.817E+00 1.817E+00 3.550E-01 1.817E+00 1.817E+00 3.600E-01 1.818E+00 1.818E+00 3.650E-01 1.818E+00 1.818E+00 3.700E-01 1.818E+00 1.818E+00
At every step along the way of the analysis, the capacitors in the two circuits (original circuit versus Thevenin equivalent circuit) are at equal voltage, thus demonstrating the equivalence of the two circuits.
Review
- To analyze an RC or L/R circuit more complex than simple series, convert the circuit into a Thevenin equivalent by treating the reactive component (capacitor or inductor) as the “load” and reducing everything else to an equivalent circuit of one voltage source and one series resistor. Then, analyze what happens over time with the universal time constant formula.


5.1: Polynomial Interpolation
The \(n+1\) points \(\left(x_{0}, y_{0}\right),\left(x_{1}, y_{1}\right), \ldots,\left(x_{n}, y_{n}\right)\) can be interpolated by a unique polynomial of degree \(n .\) When \(n=1\) , the polynomial is a linear function; when \(n=2\) , the polynomial is a quadratic function. There are three standard algorithms that can be used to construct this unique interpolating polynomial, and we will present all three here, not so much because they are all useful, but because it is interesting to learn how these three algorithms are constructed.
When discussing each algorithm, we define \(P_{n}(x)\) to be the unique \(n\) th degree polynomial that passes through the given \(n+1\) data points.
5.1.1. Vandermonde polynomial
This Vandermonde polynomial is the most straightforward construction of the interpolating polynomial \(P_{n}(x)\) . We simply write
\[P_{n}(x)=c_{0} x^{n}+c_{1} x^{n-1}+\cdots+c_{n} . \nonumber \]
Then we can immediately form \(n+1\) linear equations for the \(n+1\) unknown coefficients \(c_{0}, c_{1}, \ldots, c_{n}\) using the \(n+1\) known points:
\[\begin{gathered} y_{0}=c_{0} x_{0}^{n}+c_{1} x_{0}^{n-1}+\cdots+c_{n-1} x_{0}+c_{n} \\ y_{2}=c_{0} x_{1}^{n}+c_{1} x_{1}^{n-1}+\cdots+c_{n-1} x_{1}+c_{n} \\ \vdots \\ y_{n}=c_{0} x_{n}^{n}+c_{1} x_{n}^{n-1}+\cdots+c_{n-1} x_{n}+c_{n} \end{gathered} \nonumber \]
The system of equations in matrix form is
\[\left(\begin{array}{ccccc} x_{0}^{n} & x_{0}^{n-1} & \cdots & x_{0} & 1 \\ x_{1}^{n} & x_{1}^{n-1} & \cdots & x_{1} & 1 \\ \vdots & \vdots & \ddots & \vdots & \\ x_{n}^{n} & x_{n}^{n-1} & \cdots & x_{n} & 1 \end{array}\right)\left(\begin{array}{c} c_{0} \\ c_{1} \\ \vdots \\ c_{n} \end{array}\right)=\left(\begin{array}{c} y_{0} \\ y_{1} \\ \vdots \\ y_{n} \end{array}\right) \nonumber \]
The matrix is called the Vandermonde matrix , and can be constructed using the MATLAB function vander.m. The system of linear equations can be solved in MATLAB using the \operator, and the MATLAB function polyval.m can used to interpolate using the \(c\) coefficients. I will illustrate this in class and place the code on the website.
5.1.2. Lagrange polynomial
The Lagrange polynomial is the most clever construction of the interpolating polynomial \(P_{n}(x)\) , and leads directly to an analytical formula. The Lagrange polynomial is the sum of \(n+1\) terms and each term is itself a polynomial of degree \(n\) . The full polynomial is therefore of degree \(n\) . Counting from 0 , the \(i\) th term of the Lagrange polynomial is constructed by requiring it to be zero at \(x_{j}\) with \(j \neq i\) , and equal to \(y_{i}\) when \(j=i\) . The polynomial can be written as
\[\begin{array}{r} P_{n}(x)=\dfrac{\left(x-x_{1}\right)\left(x-x_{2}\right) \cdots\left(x-x_{n}\right) y_{0}}{\left(x_{0}-x_{1}\right)\left(x_{0}-x_{2}\right) \cdots\left(x_{0}-x_{n}\right)}+\dfrac{\left(x-x_{0}\right)\left(x-x_{2}\right) \cdots\left(x-x_{n}\right) y_{1}}{\left(x_{1}-x_{0}\right)\left(x_{1}-x_{2}\right) \cdots\left(x_{1}-x_{n}\right)} \\ +\cdots+\dfrac{\left(x-x_{0}\right)\left(x-x_{1}\right) \cdots\left(x-x_{n-1}\right) y_{n}}{\left(x_{n}-x_{0}\right)\left(x_{n}-x_{1}\right) \cdots\left(x_{n}-x_{n-1}\right)} . \end{array} \nonumber \]
It can be clearly seen that the first term is equal to zero when \(x=x_{1}, x_{2}, \ldots, x_{n}\) and equal to \(y_{0}\) when \(x=x_{0}\) ; the second term is equal to zero when \(x=x_{0}, x_{2}, \ldots x_{n}\) and equal to \(y_{1}\) when \(x=x_{1}\) ; and the last term is equal to zero when \(x=x_{0}, x_{1}, \ldots x_{n-1}\) and equal to \(y_{n}\) when \(x=x_{n}\) . The uniqueness of the interpolating polynomial implies that the Lagrange polynomial must be the interpolating polynomial.
5.1.3. Newton polynomial
The Newton polynomial is somewhat more clever than the Vandermonde polynomial because it results in a system of linear equations that is lower triangular, and therefore can be solved by forward substitution. The interpolating polynomial is written in the form
\[P_{n}(x)=c_{0}+c_{1}\left(x-x_{0}\right)+c_{2}\left(x-x_{0}\right)\left(x-x_{1}\right)+\cdots+c_{n}\left(x-x_{0}\right) \cdots\left(x-x_{n-1}\right) \nonumber \]
which is clearly a polynomial of degree \(n\) . The \(n+1\) unknown coefficients given by the \(c^{\prime}\) s can be found by substituting the points \(\left(x_{i}, y_{i}\right)\) for \(i=0, \ldots, n\) :
\[\begin{aligned} y_{0} &=c_{0} \\ y_{1} &=c_{0}+c_{1}\left(x_{1}-x_{0}\right) \\ y_{2} &=c_{0}+c_{1}\left(x_{2}-x_{0}\right)+c_{2}\left(x_{2}-x_{0}\right)\left(x_{2}-x_{1}\right) \\ \vdots & \vdots \\ y_{n} &=c_{0}+c_{1}\left(x_{n}-x_{0}\right)+c_{2}\left(x_{n}-x_{0}\right)\left(x_{n}-x_{1}\right)+\cdots+c_{n}\left(x_{n}-x_{0}\right) \cdots\left(x_{n}-x_{n-1}\right) \end{aligned} \nonumber \]
This system of linear equations is lower triangular as can be seen from the matrix form
\[\left(\begin{array}{ccccc} 1 & 0 & 0 & \cdots & 0 \\ 1 & \left(x_{1}-x_{0}\right) & 0 & \cdots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 1 & \left(x_{n}-x_{0}\right) & \left(x_{n}-x_{0}\right)\left(x_{n}-x_{1}\right) & \cdots & \left(x_{n}-x_{0}\right) \cdots\left(x_{n}-x_{n-1}\right) \end{array}\right)\left(\begin{array}{c} c_{0} \\ c_{1} \\ \vdots \\ c_{n} \end{array}\right) \nonumber \]
and so theoretically can be solved faster than the Vandermonde polynomial. In practice, however, there is little difference because polynomial interpolation is only useful when the number of points to be interpolated is small.


3.3: Order of Operations
After completing this module, you should be able to:
- Simplify expressions using order of operations.
- Simplify expressions using order of operations involving grouping symbols.
Calculates else sure someone be rules expect explicit we what that needs to need make, we to that them what be calculate to calculated first.
You probably read that sentence and couldn't make heads or tails of it. Seems like it might concern calculations, but maybe it concerns needs? You may even be attempting to unscramble the sentence as you read it, placing words in the order you might expect them to appear in. The reason that the sentence makes no sense is that the words don't follow the order you expect them to follow. Unscrambled, the sentence was intended to be “To be sure that someone else calculates when we expect them to calculate, we need rules that make explicit what needs to be calculated first.”
Similarly, when working with math expressions and equations, if we don't follow the rules for order of operations , arithmetic expressions make no sense. Just a simple expression would be problematic if we didn't have some rules to tell us what to calculate first. For instance, can be calculated in many ways. You could get 5,184. Or, you could get 80. Or, 96. The issue is that without following a set of rules for calculation, the same expression will give various results. In case you are curious, using the appropriate order of operations, we find .
Simplify Expressions Using Order of Operations
The order in which mathematical operations is performed is a convention that makes it easier for anyone to correctly calculate. They follow the acronym EMDAS:
| E | Exponents |
| M/D | Multiplication and division |
| A/S | Addition and Subtraction |
So, what does EMDAS tell us to do? In an equation, moving left to right, we begin by calculating all the exponents first. Once the exponents have been calculated, we again move left to right, calculating the multiplications and divisions, one at a time. Multiplication and division hold the same position in the ordering, so when you encounter one or the other at this step, do it. Once the multiplications and divisions have been calculated, we again move left to right, calculating the additions and subtractions, one at a time. Additions and subtractions hold the same position in the ordering, so when you encounter one or the other at this step, do it. (You may have previously learned the order of operations as PEMDAS, with parentheses first; we will add that aspect later on.) We’ll explore this as we work an example.
Calculate .
- Answer
-
There are no exponents in this expression, so the next operations to check are multiplication and division.
Step 1: Moving left to right, the first multiplication encountered is 4 multiplied by 13. We perform that operation first.
Step 2: The only operation remaining is the subtraction.
.
So, .
1. Calculate \(43+18 \times 15\).
Calculate .
- Answer
-
Step 1: Moving left to right, we see there is an exponent. We calculate the exponent first.
Step 2: The only operation remaining is the multiplication.
So, .
1. Calculate \(60^2 / 50\).
Calculate .
- Answer
-
Step 1: To calculate this, move left to right, and compute all the exponents first. The only exponent we see is the squaring of the 3, so that is calculated first.
Step 2: Since the exponents are all calculated, now calculate all the multiplications and divisions moving left to right. The only multiplication or division present is 9 times 4.
Step 3: Moving left to right, perform the additions and subtractions. There is only one such operation, 2 plus 36.
So, .
1. Calculate \(3 \times 6^3-18\).
Even if the expression being calculated gets more complicated, we perform the operations in the order: EMDAS.
Order of Operations 1
Correctly apply the order of operations to compute the following:
.
- Answer
-
Step 1: To do so, calculate the exponents first, moving left to right. There are two occurrences of exponents in the expression, 3 squared and 2 cubed.
Step 2: Now that the exponents are calculated, perform the multiplication and division, moving left to right. The first is the product of 25 and 6.
Step 3: Next is the 150 divided by 10.
Step 4: Next is 15 multiplied by 9.
Step 5: Finally, multiply the 7 and 8.
As all the multiplications and divisions have been calculated, the additions and subtractions are performed, moving left to right.
The computed value is −75.
Correctly apply the order of operations to compute the following:
\[
13+3^4 \times 8 / 6 \times 5^2-3 \times 4
\nonumber \]
Order of Operations 2
Correctly apply the rules for the order of operations to accurately compute the following:
.
- Answer
-
Step 1: Calculate exponents first, moving left to right:
Step 2: Multiply and divide, moving left to right:
Step 3: Add and subtract, moving left to right:
Correctly apply the rules for the order of operations to accurately compute the following:
\[
12 /(-4)+8 \times 9 / 12 \times 2^3-24 \times 25 / 10
\nonumber \]
Correctly apply the rules for the order of operations to accurately compute the following: .
- Answer
-
Step 1: Calculate the exponents first, moving left to right:
Step 2: Multiply and divide, moving left to right:
Step 3: Add and subtract, moving left to right:
Correctly apply the rules for the order of operations to accurately compute the following:
\(3\times{4^3}\times7+24/6\times{7^2}-9/3\times8\)
Using the Order of Operations Involving Grouping Symbols
We have examined how to use the order of operations, denoted by EMDAS, to correctly calculate expressions. However, there may be expressions where a multiplication should happen before an exponent, or a subtraction before a division. To indicate an operation should be performed out of order, the operation is placed inside parentheses. When parentheses are present, the operations inside the parentheses are performed first. Adding the parentheses to our list, we now have PEMDAS, as shown below.
| P | Parentheses | |
| E | Exponents | |
| M/D | Multiplication and division | (division is just the multiplication by the reciprocal) |
| A/S | Addition and subtraction | (subtraction is just the addition of the negative) |
As said previously, parentheses indicate that some operation or operations will be performed outside the standard order of operation rules. For instance, perhaps you want to multiply 4 and 7 before squaring. To indicate that the multiplication happens before the exponent, the multiplication is placed inside parentheses: .
This means operations inside the parentheses take precedence, or happen before other operations. Now, the first step in calculating arithmetic expressions using the order of operations is to perform operations inside parentheses first. Inside the parentheses, you follow the order of operation rules EMDAS.
Correctly apply the rules for the order of operations to accurately compute the following:
.
- Answer
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Step 1: Perform all calculations within the parentheses before all other operations.
Step 2: Since all parentheses have been cleared, move left to right, and compute all the exponents next.
Step 3: Perform all multiplications and divisions moving left to right.
Correctly apply the rules for the order of operations to accurately compute the following:
\(8-(25-{2^2})/7\).
Be aware that there can be more than one set of parentheses, and parentheses within parentheses. When one set of parentheses is inside another set, do the innermost set first, and then work outward.
Order of Operations 3
Correctly apply the rules for order of operations to accurately compute the following:
.
- Answer
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Step 1: Perform all calculations within the parentheses before other operations. Evaluate the innermost parentheses first. We can work separate parentheses expressions at the same time. The innermost set of parentheses has the 2 + 5 inside. The 3 + 8 is in a separate set of parentheses, so that addition can occur at the same time as the 2 + 5.
Step 2: Now that those parentheses have been handled, move on to the next set of parentheses. Applying the order of operation rules inside that set of parentheses, the exponent is evaluated first, then the multiplication, and then the addition.
Step 3: Since all parentheses have been cleared, apply the EMDAS rules to finish the calculation.
Correctly apply the rules for the order of operations to accurately compute the following:
\({(8-6)^2}\times100-({(48/6-3)^2}-4\times7)\)
Order of Operations 4
Check Your Understanding
- Which operation has highest precedence?
- Which is performed first, exponents or addition?
- Calculate \(2 \times 3^2-5 \times 8\).
- What is used to indicate operations that should be performed out of order?
- Calculate \((4-3)^2+27 \times 8^2 \div 6^2\).