# Room temperature cavity electromechanics in the sideband-resolved regime

Anh Tuan Le  Department of Physics, University of Konstanz, 78457 Konstanz,
Germany  Department of Electrical & Computer Engineering, Technical University
of Munich, 80333 Munich, Germany  A. Brieussel  Department of Physics,
University of Konstanz, 78457 Konstanz, Germany  E. M. Weig<EMAIL_ADDRESS>Department of Physics, University of Konstanz, 78457 Konstanz, Germany
Department of Electrical & Computer Engineering, Technical University of
Munich, 80333 Munich, Germany  Munich Center for Quantum Science and
Technology (MCQST), 80799 Munich, Germany

###### Abstract

We demonstrate a sideband-resolved cavity electromechanical system operating
at room temperature. It consists of a nanomechanical resonator, a strongly
pre-stressed silicon nitride string, dielectrically coupled to a three-
dimensional microwave cavity made of copper. The electromechanical coupling is
characterized by two measurements, the cavity-induced eigenfrequency shift of
the mechanical resonator and the optomechanically induced transparency. While
the former is dominated by dielectric effects, the latter reveals a clear
signature of the dynamical backaction of the cavity field on the resonator.
This unlocks the field of cavity electromechanics for room temperature
applications.

††preprint: AIP/123-QED

##  I Introduction

In recent years, cavity (or circuit) electromechanics has been established as
a powerful implementation of cavity optomechanics. Aspelmeyer, Kippenberg, and
Marquardt (2014) Rather than relying on a light field circulating inside an
optical cavity, an electromagnetic circuit is employed to realize a cavity
mode in the microwave regime which parametrically couples to a mechanical
resonator. Regal, Teufel, and Lehnert (2008); Teufel  _et al._ (2008, 2011a);
Pernpeintner  _et al._ (2014) This seemingly simple modification has enabled a
series of breakthroughs, including quantum ground state cooling, Teufel  _et
al._ (2011b) entanglement generation, Palomaki  _et al._ (2013a); Barzanjeh
_et al._ (2019) long-lived quantum storage, Palomaki  _et al._ (2013b); Zhou
_et al._ (2013); Pechal, Arrangoiz-Arriola, and Safavi-Naeini (2018)
microwave-to-optical conversion, Andrews  _et al._ (2014); Forsch  _et al._
(2019) non-reciprocal signal transduction, Barzanjeh  _et al._ (2017) and
ultrastrong coupling Peterson  _et al._ , just to name a few. Following recent
developments in the field of superconducing qubits Paik  _et al._ (2011);
Reagor  _et al._ (2013) where three-dimensional microwave cavities have
replaced coplanar waveguide architectures for their large mode volumes and
remarkably high quality factors, three-dimensional superconducting microwave
cavities have been adapted for cavity electromechanics by capacitive coupling
to a mechanical resonator. Yuan  _et al._ (2015); Noguchi  _et al._ (2016);
Carvalho  _et al._ (2019) However, to date, the field of cavity
electromechanics is limited to millikelvin temperatures, since it relies on
superconducting circuits. Room temperature cavity electromechanics is impeded
by the non-zero resistance of normal conducting circuits which gives rise to
dissipation. Non-superconducting microwave cavities such as copper microstrip
resonators have been successfully employed for cavity-assisted displacement
sensing of nanomechanical resonators at room temperature. Faust  _et al._
(2012); Rieger  _et al._ (2012) Their use for cavity electromechanics Faust
_et al._ (2012) is limited by a cavity quality factor of about $100$ due to
dielectric and conductor losses, which not only constrains the displacement
sensitivity but also keeps the system deeply in the unresolved sideband, so-
called bad cavity regime where the linewidth of the cavity $\kappa/2\pi$
exceeds the frequency of the mechanical mode $\Omega_{m}/2\pi$.

Here we present a room temperature cavity electromechanical system capable of
sideband-resolved operation ($\kappa<\Omega_{m}$). Inspired by the
developments in the field of superconducing qubits Paik  _et al._ (2011);
Reagor  _et al._ (2013) and cavity electromechanics Yuan  _et al._ (2015);
Noguchi  _et al._ (2016); Carvalho  _et al._ (2019) we employ a three-
dimensional, cylindrically shaped cavity rather than a microstrip to
dielectrically probe the displacement of a strongly pre-stressed silicon
nitride nanostring resonator.

##  II Results and discussion

Figure 1(a) depicts a photograph of the cylindrical cavity. It consists of two
parts which have been machined from bulk copper (Cu > 99.90 %) and which can
be closed using screws. The cavity has a radius of $35$ mm and a height of
$70$ mm, it supports both transverse electric (TE) and transverse magnetic
(TM) modes. Pozar (2011) As the modes reside in the hollow cylinder, they are
only weakly affected by dielectric and conductor losses, enabling high quality
factor even at room temperature. Reagor  _et al._ (2013) The coupling to the
cavity is realized by injecting the microwave signal from a coaxial line
through a loop coupler via a $3.6$ mm sized hole in the cavity top. In all our
measurements, we use a circulator to physically separate the input and output
signals of the single port reflection cavity. Figure 1(b) (left) displays the
reflection coefficient $\left|S_{11}\right|$ of the $\textrm{TM}_{110}$ mode
which is found at $\omega_{c}/(2\pi)\approx 5.147$ GHz with a linewidth of
$\kappa/(2\pi)=2.824$ MHz for the empty cavity, in good agreement with finite
element simulations. This corresponds to a cavity quality factor of $1,800$,
exceeding the state of the art in microwave-cavity-assisted nanomechanical
displacement sensing at room temperature by more than an order of magnitude.
Faust  _et al._ (2012) The nanomechanical resonator under investigation is a
strongly pre-stressed nanostring fabricated from LPCVD silicon nitride on a
fused silica wafer, which is flanked by two gold electrodes for dielectric
transduction. Faust  _et al._ (2012); Rieger  _et al._ (2012) The reflection
coefficient of the cavity including the resonator chip is shown in Figure 1(b)
(right). Upon insertion of the resonator, the eigenfrequency of the
$\textrm{TM}_{110}$ mode shifts to $\omega_{c}/(2\pi)=5.226$ GHz. The
linewidth increases by a factor of $2$ to $\kappa/(2\pi)=5.572$ MHz.

Figure 1: Experimental setup of the cavity electromechanical system at room
temperature. (a), Photograph of the (open) cylindrical microwave 3D cavity
machined from bulk copper. The microwave signal is inductively injected into
the cavity via a loop coupler. The sample holder including the antenna is
visible on the cavity floor. (b), Reflection coefficient $\left|S_{11}\right|$
of the $\textrm{TM}_{110}$ mode (black) along with fit to the data (red),
showing the response of both the empty cavity (left) and the cavity including
the sample (right). Upon insertion of the sample the frequency of the
$\textrm{TM}_{110}$ mode shifts by $\sim 80$ MHz. Furthermore, an increase of
the linewidth by a factor of $2$ is observed. (c), Schematic visualizing the
physical implementation of the cavity electromechanical system. The doubly
clamped silicon nitride string resonator (blue) is situated between two
elevated gold electrodes (yellow). The resonator chip (light green) is glued
to a printed circuit board (green) and placed inside the cavity (beige). The
circuit board also hosts a looped dipole antenna which is bonded to one of the
electrodes to inductively couple to the $\textrm{TM}_{110}$ mode. The other
electrode is connected to a single layer capacitor (brown) which serves as a
capacitive ground for frequencies in the microwave frequency range, and to a
wire which is fed through a hole in the cavity wall. This allows to apply DC
voltages and RF signals to dielectrically drive the resonator inside the
cavity. (d), Scanning electron micrograph of a doubly clamped pre-stressed
nanomechanical resonator between two gold electrodes for dielectric actuation
as well as coupling to the three-dimensional microwave cavity.

In addition, the fit of the amplitude and phase of the cavity’s reflection
coefficient reveals an external dissipation rate of $\kappa_{ex}/(2\pi)=1.361$
MHz. This gives rise to a coupling efficiency $\eta=\kappa_{ex}/\kappa=0.244\
$, indicating that the cavity is undercoupled. Figure 1(c) shows the physical
realization of the cavity electromechanical system. The sample holder with the
resonator chip is placed inside the cavity. One of the electrodes is connected
to an antenna to inductively couple to the $\textrm{TM}_{110}$ of the cavity.
It consists of a looped silver wire on top of a coaxial cable which places the
loop into the electromagnetic field near the center of the hollow cylinder.
The other electrode is bonded across a single layer capacitor acting as
capacitive ground Rieger  _et al._ (2012) and connected to an RF signal
generator through a $1.5$ mm wide hole in the cavity wall for dielectric
actuation of the resonator. Chen  _et al._ (2011); de Graaf  _et al._ (2014);
Hao, Rouxinol, and LaHaye (2014); Kong  _et al._ (2015); Cohen  _et al._
(2017); Stammeier, Garcia, and Wallraff (2018) The nanostring under
investigation is $w=250$ nm wide, $t=100$ nm thick and $L=57\,\mu$m long,
similar to the one depicted in Fig. 1(d). While the cavity characterization is
done under ambient conditions, the experiments discussed in the following
involve the vibrational excitation of the nanostring and air damping needs to
be excluded. Hence, the entire cavity electromechanical system consisting of
the string resonator as well as the microwave cavity is placed inside a vacuum
chamber below $5\times 10^{-4}$ mbar. As indicated above, all measurements are
performed at room temperature. The nanomechanical resonator is characterized
by dielectrically driving its fundamental out-of-plane mode with a vector
network analyzer (VNA). The response of the nanomechanical resonator is
characterized using heterodyne cavity-assisted displacement detection Faust
_et al._ (2012) using the three-dimensional microwave cavity. The microwave
cavity is driven on resonance $\omega_{d}=\omega_{c}$, such that the periodic
modulation of the capacitance induced by the dielectric nanostring vibrating
between the two electrodes induces sidebands at $\omega_{c}\pm\Omega_{m}$ on
the cavity response. The reflected signal of the cavity is demodulated by an
in-phase quadrature (IQ) mixer, filtered and amplified as previously
described. Faust  _et al._ (2012) The resulting signal is fed back into the
VNA. The response of the fundamental out-of-plane mode of the resonator is
displayed in Fig. 2. Figure 2(a) shows the Lorentzian resonance curve observed
in the linear response regime for a drive power $P_{m}=-36$ dBm. Fitting
allows to extract the resonance frequency $\Omega_{m}/(2\pi)=6.4948$ MHz, as
well as a linewidth $\Gamma/(2\pi)=42$ Hz which gives rise to a quality factor
of $Q\approx 150.000$. Given the cavity linewidth $\kappa/(2\pi)=5.572$ MHz
the condition for sideband resolution, $\Omega_{m}>\kappa$, is fulfilled for
the $\textrm{TM}_{110}$ mode. The response of the resonator for increasing
drive power between $-36$ dBm to $-14$ dBm is depicted in Fig. 2(b). Clearly,
the transition from the linear, Lorentzian response to an asymmetric response
curve, which is well described by the cubic nonlinearity $\alpha x^{3}$ of the
Duffing model with a stiffening $\alpha>0$, is observed. Nayfeh and Mook
(1995) This demonstrates that the cavity-assisted displacement detection is
not impeded even under strong driving of the nanomechanical resonator inside
the 3D cavity.

Figure 2:  Characterization of the nanostring resonator. (a), Linear response
of the fundamental out-of-plane mode of the nanostring (black) and Lorentzian
fit (red) for a drive power of $P_{m}=-36$ dBm. (b), Response of the
nanostring by increasing drive power from $P_{m}=-36$ dBm to $-14$ dBm showing
the transition to the nonlinear Duffing regime.

Following the characterization of the microwave cavity and the resonator, we
discuss the electromechanical coupling between the two systems. First, we
explore how the cavity detuning affects the mechanical eigenfrequency of the
resonator. The radiation pressure of the power circulating in the microwave
cavity acts back on the mechanical resonator, causing a mechanical
eigenfrequency shift which depends on the microwave drive power $P_{d}$ and
detuning $\Delta=\omega_{d}-\omega_{c}$. This so-called optical spring effect
leads to a detuning-dependent softening or hardening of the resonator. In case
of a high-Q mechanical oscillator with small linewidth, where
$\Gamma\ll\kappa$, the frequency shift of the mechanical oscillator can be
defined as Aspelmeyer, Kippenberg, and Marquardt (2014)

$\displaystyle\delta\Omega_{m,\textrm{opt}}=g^{2}\left(\frac{\Delta-\Omega_{m}}{\frac{\kappa^{2}}{4}+\left(\Delta-\Omega_{m}\right)^{2}}+\frac{\Delta+\Omega_{m}}{\frac{\kappa^{2}}{4}+\left(\Delta+\Omega_{m}\right)^{2}}\right).$
(1)

The electromechanical coupling strength $g=g_{0}\cdot\sqrt{n_{d}}$ depends on
the single photon coupling rate $g_{0}$ and the number of photons circulating
in the cavity $n_{d}$. Note that this quantity strongly depends on the
detuning of the cavity. Hence, for a given power $P_{d}$ the intra-cavity
photon number
$n_{d}=P_{d}\kappa_{ex}/\left[\hbar\omega_{d}\left(\frac{\kappa^{2}}{4}+\Delta^{2}\right)\right]$
of the single-sided cavity can be significantly reduced for a finite cavity
detuning. As the static mechanical displacement of the resonator arising from
the radiation pressure force Aspelmeyer, Kippenberg, and Marquardt (2014) is
negligibly small, the detuning $\Delta$ rather than the effective detuning
$\bar{\Delta}$ is employed in eq. (1)

In addition to the optomechanical backacktion we also expect a quasi-static
dielectric force acting on the mechanical resonator. This force results from
the root-mean-square (RMS) electrical field which builds up inside the
microwave cavity and contributes to the dielectric frequency tuning of the
nanostring Rieger  _et al._ (2012). As the RMS cavity field exhibits the same
detuning-dependence as the intracavity photon number $n_{d}$, this translates
into a change in eigenfrequency of the mechanical oscillator as the cavity is
detuned from its resonance

$\displaystyle\delta\Omega_{m,\textrm{dielectric}}=c_{\mu w}\cdot\left(U_{\mu
w}^{rms}\left(\Delta\right)\right)^{2}=c_{\mu
w}R\frac{\kappa\kappa_{ex}}{\frac{\kappa^{2}}{4}+\Delta^{2}}P_{d}.$ (2)

The calibration factor $c_{\mu w}$ converts the change of the effective RMS
voltage $U_{\mu w}^{rms}$ inside the cavity into a frequency change, and $R$
is the impedance of the circuit allowing to express the RMS voltage in terms
of the power circulating in the cavity. In total, the detuning dependence of
the mechanical resonance is

$\displaystyle\Omega^{\prime}_{m}=\Omega_{m}+\delta\Omega_{m,\textrm{dielectric}}+\delta\Omega_{m,\textrm{opt}}$
(3)

where $\Omega_{m}$ is the unperturbed mechanical eigenfrequency.

Figure 3:  Cavity-induced eigenfrequency shift of the resonator for a cavity
drive power of $P_{d}=15$ dBm. (a), Experimental data is shown as black
circles, the fit of Eq. (3) is indicated as a red solid line. The observed
detuning dependence of the mechanical eigenfrequency is dominated by the
dielectric shift, which overwhelms the contribution of the optomechanical
backaction. Inset: Theoretical prediction of the eigenfrequency shift
according to eq. (3) for $g_{0}=0$ Hz (grey), $200\,\mu$Hz (light green) and
$20$ mHz (dark green). (b) Residual of the data shown in (a).

In Fig. 3(a) the nanomechanical resonator’s eigenfrequency is plotted as a
function of the detuning (black circles) for a cavity drive power of
$P_{d}=15$ dBm. The data is well described by eq. (3), which is apparent from
a fit to the data (red line). However, the fit yields an overwhelming
contribution of the dielectric frequency shift and only negligible
optomechanical backaction such that the optomechanical coupling strength $g$
can not be determined. The inset of Fig. 3(a) shows the theoretical prediction
of the eigenfrequency according to eq. (3) using the parameters of the
experiment for three different values of $g_{0}$: The grey solid line
illustrates the eigenfrequency shift for zero optomechanical coupling
$g_{0}=0$ Hz. As eq. (2) does not depend on the sign of $\Delta$, it is mirror
symmetric with respect to the $y$-axis. The light green dashed line
corresponds to $g_{0}=200\,\mu$Hz, which is the value of the single photon
coupling rate estimated from the data in Fig. 4. The line completely coincides
with the grey line, confirming that the optomechanical coupling has no
measurable effect on the eigenfrequency shift. Minute deviations start
becoming apparent from $g_{0}=2$ mHz which we thus estimate as an upper bound
of the single photon coupling rate. The dark green dashed line for $g_{0}=20$
mHz already shows a sizable deviation of the eigenfrequency shift resulting
from optomechanical coupling. Our observation of a negligible optomechanical
eigenfrequency shift is supported by the nearly mirror-symmetric shape of the
measured eigenfrequency shift of the resonator around zero detuning in Fig.
3(a). The slight deviation shifting the maximum to a small positive detuning
is likely caused by a slight drift of the mechanical eigenfrequency from slow
polarization effects within the nanostring during the cavity frequency sweep,
the opposite effect would be expected from radiation pressure effects. Figure
Fig. 3(b) shows the residual of the data vs. the fit. We attribute the
observed pattern to the same slow polarization effects. The small value of
$g_{0}$ is consistent with our observation that the mechanical linewidth does
not yield any measurable detuning dependence in our experiment (not shown).

In order to obtain a better estimation of the electromechanical coupling
strength, we apply a second technique to characterize the parametric coupling
between the nanostring and the three-dimensional cavity. It is based on the
optomechanically induced transparency (OMIT), Agarwal and Huang (2010); Weis
_et al._ (2010); Teufel  _et al._ (2011a) which arises from the coherent
interaction of two microwave tones with the mechanical resonator in the
resolved sideband regime. Singh  _et al._ (2014) In an OMIT experiment, the
cavity is strongly pumped by a drive tone $\omega_{d}=\omega_{c}-\Omega_{m}$
which is red detuned from the cavity resonance by the frequency of the
mechanical resonator. The cavity response is measured by a second, weak probe
tone $\omega_{p}$ that is scanned across the cavity resonance. The beating
between the two microwave tones induces a radiation pressure force coherently
driving the mechanical resonator. In turn, the resonator imprints sidebands on
the drive which interfere constructively with the probe. This opens a
transparency window in the cavity transmission (or, as in our case, the cavity
reflection which strictly speaking leads to optomechanically induced
reflection (OMIR) Singh  _et al._ (2014)). According to the standard theory of
OMIT, the height of the transparency peak allows to directly extract the
cooperativity $C=4g^{2}/(\kappa\Gamma_{m})$, rendering OMIT an important tool
in cavity opto- and electromechanics.

In Fig. 4(a) we plot the response of the microwave cavity as a function of the
probe frequency $\omega_{p}$ for a drive applied at optimal detuning
$\Delta=\omega_{d}-\omega_{c}=-\Omega_{m}$. The drive and probe power are $15$
dBm and $-25$ dBm, respectively. A transparency feature at $\omega_{c}$ is
clearly apparent in the center of the cavity resonance. Note that the peak is
not symmetric. Its asymmetry reflects the nonlinear response of the mechanical
resonator to the radiation pressure drive exerted by the two microwave tones.
Zhou  _et al._ (2013); Shevchuk  _et al._ (2015); Singh  _et al._ (2016) In
agreement with the standard model for OMIR for the case of an undercoupled
cavity driven on the red sideband, the shape of the nonlinear transparency
peak directly follows the response of the Duffing resonator. Zhou  _et al._
(2013); Shevchuk  _et al._ (2015) This is also reflected in the hysteretic
behavior of the OMIR feature displayed in Fig. 4(b) for a forward (black) as
well as a reverse (red) sweep of the cavity probe at a somewhat lower probe
power to maximize the magnitude of the OMIR feature. Figure 4(c) illustrates
the dependence of OMIR peak on the probe power. The curves (from left to
right) correspond to an increasing probe power from $-46$ dBm to $-39$ dBm. As
expected, the width of the nonlinear feature broadens with increasing probe
power while its amplitude slightly decreases.Shevchuk  _et al._ (2015)
Finally, the effect of a detuning $\delta$ of the cavity drive from the red
sideband condition, $\Delta=-\Omega_{m}+\delta$, is explored in Fig. 4(d) for
a constant probe power of $-25$ dBm. For a drive tone red- or blue-shifted
from the red-sideband condition ($\delta=\pm 180$ kHz, red and blue trace,
respectively), the OMIR peak moves away from the minimum of the cavity
response at $\omega_{c}$. Shevchuk  _et al._ (2015) The data for the drive
matched to the red sideband is also included ($\delta=0$ Hz, black trace). For
clarity, the red and blue trace are vertically offset from the black trace.

Figure 4:  Optomechanically induced reflection. (a), Cavity reflection
coefficient $\left|S_{11}\right|$ showing an OMIR peak of the undercoupled
cavity in response to a weak probe tone $\omega_{p}\approx\omega_{c}$ in the
presence of an additional strong, red-detuned drive
$\omega_{d}=\omega_{c}-\Omega_{m}$. The asymmetric shape of the OMIR feature
reveals the nonlinearity of the mechanical resonator. The inset displays the
full cavity resonance (red box illustrates area shown in main panel). (b),
Forward (blue) and reverse (red) sweeps of the probe tone reveal a hysteresis
of the OMIR feature reflecting the bistability of the nonlinear mechanical
system. (c) OMIR feature as a function of the probe power. For increasing
probe power, the OMIR feature broadens, while its amplitude slightly
decreases. (d) OMIR feature as a function of the drive detuning. For a drive
tone red- or blue-shifted from the red sideband condition (red and blue trace,
respectively) the OMIR feature shifts to the left or right of the cavity
resonance. The situation for a drive on the red sideband is also included
(black trace). Red and blue trace are vertically offset for clarity.

According to the standard theory of OMIR, the height of the transparency peak
$S_{11}^{0}$ allows to quantify the cooperativity. For the case of an
undercoupled cavity driven on the red sideband, $C=(2\eta)/(1-S_{11}^{0})-1$.
Neither the nonlinear regime nor a poor sideband resolution greatly affect the
magnitude of the OMIR feature. Shevchuk  _et al._ (2015); Bodiya  _et al._
(2019) Using the yellow trace in Fig. 4(c), extract an approximate value of
$C\approx 0.025$. This translates into an optomechanical coupling strength of
$g/(2\pi)=1.2$ kHz, and, given a cavity photon number of $4.0\cdot 10^{13}$ at
a drive power of $15$ dBm on the red sideband, to a single photon coupling
rate of $g_{0}/(2\pi)=200\,\mu$Hz.

It is noteworthy to mention that such a feeble single photon coupling can
produce observable features in the OMIT experiment at all. This is enabled by
the large number of photons supported by the three-dimensional microwave
cavity, which exceeds photon numbers achieved in planar microwave resonators
at low temperatures Singh  _et al._ (2014); Teufel  _et al._ (2011a) by at
least four orders of magnitude and is at present only limited by the maximum
output power of our microwave generator. The observed weak single photon
coupling strength is attributed to our cavity design, as the loop antenna
could not be precisely positioned in the cavity in our experiment and
presumably only weakly couples to the $\textrm{TM}_{110}$ cavity mode. At the
same time, the electromechanical coupling of the nanostring to the electrodes
is limited by a relatively large electrode-electrode separation of approx.
$600$ nm for the sample under investigation. For future work on the room
temperature cavity electromechanics platform, an improved control of the
antenna position as well as a smaller electrode gap will enable to
significantly increase $g_{0}$.

##  III Conclusion

In conclusion, we have demonstrated a cavity electromechanical system
operating in the sideband resolved regime at room temperature. This was
accomplished by introducing a three-dimensional, non-superconducting microwave
cavity made of copper which replaces the previously employed copper microstrip
cavity, the quality factor of which is outperformed by more than an order of
magnitude. In our experiment a non-metallized silicon nitride nanostring
resonator was dielectrically coupled to the $\textrm{TM}_{110}$ mode of the
cavity which offers almost perfect sideband resolution $\kappa\leq\Omega_{m}$.
Electromechanical coupling was observed and characterized in one- and two-
microwave tone experiments. While the mechanical eigenfrequency shift is
dominated by dielectric frequency tuning, the optomechanically induced
transparency (in reflection geometry) establishes a clear proof of dynamical
backaction. Despite the minute single photon coupling rate of our first
implementation of the room temperature cavity electromechanical plattform in
the sub-mHz regime, a measureable coupling is enabled by the large number of
photons circulating in the three-dimensional microwave cavity. As a result of
the required strong red-detuned cavity drive, the response of the mechanical
resonator is nonlinear in our proof-of-principle experiment.

Our results translate the thriving field of cavity electromechanics from the
millikelvin realm to room temperature. For future exploitation, the
electromechanical vacuum coupling rate $g_{0}/(2\pi)$ will need to be
increased. This can be accomplished by an improved positioning of the loop
antenna providing the coupling between the cavity mode and the control
electrodes. Furthermore, the coupling can be enhanced by increasing the
dielectric transduction efficiency, i.e. by reducing the lateral gap between
the electrodes. Following these technical improvements, we expect to reveal
the electromechanical cooling or pumping of the mechanical mode. Finally, the
quality factor or the $\textrm{TE}_{011}$ mode exceeds that of the
$\textrm{TM}_{110}$ mode by another order of magnitude, which offers the
prospect of entering the deep-sideband-resolved regime of cavity
electromechanics at room temperature.

###### Acknowledgements.

Financial support from the Deutsche Forschungsgemeinschaft (DFG, German
Research Foundation) via Project-ID 425217212 (Collaborative Research Center
SFB 1432)); the European Unions Horizon 2020 programme for Research and
Innovation under grant agreement No. 732894 (FET Proactive HOT), as well as
the German Federal Ministry of Education and Research through contract no.
13N14777 funded within the European QuantERA cofund project QuaSeRT is
gratefully acknowledged.

## Data Availability

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

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# The paradoxes and the infinite dazzled ancient mathematics and continue to
do so today

Fairouz Kamareddine and Jonathan Seldin

###### Abstract

This paper looks at how ancient mathematicians (and especially the Pythagorean
school) were faced by problems/paradoxes associated with the infinite which
led them to juggle two systems of numbers: the discrete whole/rationals which
were handled arithmetically and the continuous magnitude quantities which were
handled geometrically. We look at how approximations and mixed numbers (whole
numbers with fractions) helped develop the arithmetization of geoemtry and the
development of mathematical analysis and real numbers.

##  1 Why did it take so long to develop Real Numbers and Analysis?

God made the integers; all else is the work of man.  Kronecker

The concepts and language of mathematics have been under development slowly
but surely since ancient times. Despite the obstacles, this development
uncovered fascinating results, which include as late as the 20th century, a
sound foundation of the theory of the infinitesimal (which is in essence the
foundation of mathematics) and the theory of the computable. Well before then,
Leibniz (1646–1717) conceived of automated deduction where he wanted to find a
language $L$ and a method that could carry out proof checking/finding to
determine the correctness of statements in $L$.111Now we know, due to later
results by Gödel, Church and Turing, that such a method can not work for every
statement. Leibniz was frustrated by the limitations in expressing thoughts:

If we could find characters or signs appropriate for expressing all our
thoughts as definitely and as exactly as arithmetic expresses numbers or
geometric analysis expresses lines, we could in all subjects in so far as they
are amenable to reasoning accomplish what is done in Arithmetic/Geometry.
Leibniz

But at the time of Leibniz, expressibility in Arithmetic was far from complete
and the real numbers were still not developed. The later development of real
analysis222Thanks to Euler who converted the calculus of Newton and Leibniz
from a geometrical field to a field where mathematical formulae are analysed.
would be based on the real numbers and the arithmetisation of geometry.

###  1.1 From naturals to intergers and rationals

Natural numbers were long understood, but it may come as a surprise that as
late as the 14th century, negative numbers were not known in Europe. In Italy,
a double entry bookkeeping system compensated for their absence. Accounts in
which debits may be greater than credits were compared without using negative
integers. If $c$ and $d$ are in $\mathbb{N}^{+}$, then _account_ $c\ominus d$
has credit $c$ and debit $d$. Define $\mbox{accounts}=\\{m\ominus
n\>|\>m,n,p,q\in\mathbb{N}^{+}\\}$. Just like the arithmetic
$(\mathbb{N}^{+},=,+,\cdot,1)$ on natural numbers
$\mathbb{N}^{+}=\\{1,2,\cdots\\}$ is defined with equality =, addition $+$,
multiplication $\cdot$, and identity element 1 for $\cdot$, we define
$(\mbox{accounts},\cong,+_{c},\cdot_{c})$ by:

  * •

$m\ominus n\cong p\ominus q\mbox{ iff }m+q=n+p.$

  * •

$(m\ominus n)+_{c}(p\ominus q)=(m+p)\ominus(n+q)$.

  * •

$(m\ominus n)\cdot_{c}(p\ominus q)=(mp+nq)\ominus(mq+np).$

The integers $(\mathbb{Z},+_{i},\cdot_{i},0_{i},1_{i},-\alpha)$ are then
defined from the equivalence classes: $\left[m\ominus
n\right]=\left\\{p\ominus q:p\ominus q\cong m\ominus n\right\\}$ by:

  * •

$\mathbb{Z}=\\{[m\ominus n]\>|\>m,n\in\mathbb{N}^{+}\\}$.

  * •

$[(m\ominus n)]+_{i}[(p\ominus q)]=[(m\ominus n)+_{c}(p\ominus q)]$.

  * •

$[(m\ominus n)]\cdot_{i}[(p\ominus q)]=[(m\ominus n)\cdot_{c}(p\ominus q)]$.

  * •

Identity $0_{i}$ for $+_{i}$: for any $m$, $n$ in $\mathbb{N}^{+}$, $[m\ominus
m]=[n\ominus n]$.

  * •

Identity $1_{i}$ for $\cdot_{i}$: take $1_{i}=[(p+1)\ominus p]$ for any
$p\in\mathbb{N}^{+}$.

  * •

Inverse for $+_{i}$: if $\alpha=[m\ominus n]$, then $-\alpha=[n\ominus m]$.

Like we defined $(\mathbb{Z},+_{i},\cdot_{i},0_{i},1_{i},-\alpha)$ from
$(\mbox{accounts},\cong,+_{c},\cdot_{c})$ which were defined from
$(\mathbb{N}^{+},=,+,\cdot,1)$, we can define positive rational numbers
$(\mathbb{Q}^{+},+_{r},\cdot_{r},1_{r},\mathbf{a}^{-1})$ from
$\mbox{fractions}=\\{\frac{\displaystyle{m}}{\displaystyle{n}}\>|\>m,n\in\mathbb{N}^{+}\\}$
where the arithmetic of $(\mbox{fractions},\mbox{$\asymp$},+_{f},\cdot_{f})$
is defined from $(\mathbb{N}^{+},=,+,\cdot,1)$ by:

  * •

$\frac{\displaystyle{m}}{\displaystyle{n}}\mbox{$\asymp$}\frac{\displaystyle{p}}{\displaystyle{q}}\mbox{
if and only if }mq=np$,

  * •

$\frac{\displaystyle{m}}{\displaystyle{n}}+_{f}\frac{\displaystyle{p}}{\displaystyle{q}}=\frac{\displaystyle{mq+np}}{\displaystyle{nq}}$
and
$\frac{\displaystyle{m}}{\displaystyle{n}}\cdot_{f}\frac{\displaystyle{p}}{\displaystyle{q}}=\frac{\displaystyle{mp}}{\displaystyle{nq}}$.

Then, we define $(\mathbb{Q}^{+},+_{r},\cdot_{r},1_{r},\mathbf{a}^{-1})$ from
equivalence classes
$\left[\frac{\displaystyle{m}}{\displaystyle{n}}\right]=\left\\{\frac{\displaystyle{p}}{\displaystyle{q}}\>|\>\frac{\displaystyle{p}}{\displaystyle{q}}\mbox{$\asymp$}\frac{\displaystyle{m}}{\displaystyle{n}}\right\\}$
as follows:

  * •

$\mathbb{Q}^{+}=\\{\left[\frac{\displaystyle{m}}{\displaystyle{n}}\right]\>|\>m,n\in\mathbb{N}^{+}\\}$.

  * •

$\left[\frac{\displaystyle{m}}{\displaystyle{n}}\right]+_{r}\left[\frac{\displaystyle{p}}{\displaystyle{q}}\right]=\left[\frac{\displaystyle{m}}{\displaystyle{n}}+_{f}\frac{\displaystyle{p}}{\displaystyle{q}}\right]$
and
$\left[\frac{\displaystyle{m}}{\displaystyle{n}}\right]\cdot_{r}\left[\frac{\displaystyle{p}}{\displaystyle{q}}\right]=\left[\frac{\displaystyle{m}}{\displaystyle{n}}\cdot_{f}\frac{\displaystyle{p}}{\displaystyle{q}}\right]$.

  * •

Identity $1_{r}$ for $\cdot_{r}$: take
$1_{r}=\left[\frac{\displaystyle{1}}{\displaystyle{1}}\right]$.

  * •

Inverse for $+_{r}$:
$\left[\frac{\displaystyle{m}}{\displaystyle{n}}\right]^{-1}=\left[\frac{\displaystyle{n}}{\displaystyle{m}}\right]$.

The steps to build $(\mathbb{Z},+_{i},\cdot_{i},0_{i},1_{i},-\alpha)$ and
$(\mathbb{Q}^{+},+_{r},\cdot_{r},1_{r},\mathbf{a}^{-1})$ lead to a
generalisation as follows:

###### Definition 1

If $(S,\circ)$ is a Commutative Cancellation Semigroup333I.e., $\circ$
satisfies closure, commutativity, associativity and cancellation law on $S$
where cancellation means that $a\circ b=a\circ c$ implies $b=c$. (CCS), then
build $(S\times S,\approx,*)$ as follows:

  * •

Define congruence $\approx$ on $S\times S$ based on $(S,\circ)$ by:
$(x,y)\approx(u,v)$ iff $x\circ v=y\circ u$.

  * •

The operation $*$ on $S\times S$ inherited from $\circ$ is defined by
$(x,y)*(u,v)=(x\circ u,y\circ v)$.

Then, define $[(x,y)]=\\{(u,v):(u,v)\approx(x,y)\\}$ and
$S_{d}=\\{[(x,y)]:x,y\in S\\}$, and build
$(s_{d},\circ_{d},e_{d},\mathfrak{a}^{-1})$ as follows:

  * •

Define $[(x,y)]e_{d}[(u,v)]=[(x,y)*(u,v)]=[(x\circ u,y\circ v)]$. Note that
$(S_{d},\circ_{d})$ is a CCS.

  * •

Note that if $x\in S$, then $x_{d}=[(y\circ x,y)]\in S_{d}$.

  * •

Identity: Define $e_{d}$ to be $[(x,x)]$ for some $x$ in $S$. For all
$\mathfrak{a}$, we have
$e_{d}\circ_{d}\mathfrak{a}=\mathfrak{a}\circ_{d}e_{d}=\mathfrak{a}$.

  * •

Inverses: If $\mathfrak{a}=[(x,y)]$, define $\mathfrak{a}^{-1}$ to be
$[(y,x)]$. We have
$\mathfrak{a}\circ_{d}\mathfrak{a}^{-1}=e_{d}=\mathfrak{a}^{-1}\circ_{d}\mathfrak{a}$.

Comparing the theory of fractions and the theory of accounts suggests that we
can define a unified theory for adding inverses and, if none is present,
identity elements.

CCS | $(\mathbb{N}^{+},+)$ | $(\mathbb{N}^{+},\cdot)$
---|---|---
inverses | $\times$ | $\times$
Identity element | $\times$ | $\surd$
CCS | $(\mathbb{Z},+_{i})$ | $(\mathbb{Q}^{+},\cdot_{r})$
with identity and inverses | $\surd$ | $\surd$

Just like we built $(\mathbb{Z},+)$ with identity $0_{i}$ and inverses
$-\alpha$ from $(\mathbb{N}^{+},+)$, we can build $(\mathbb{Q},+_{r_{i}})$
with identity and inverses from $(\mathbb{Q}^{+},+_{r})$. But we cannot build
$\mathbb{R}$ this way. The real numbers need to be constructed (using
approximations and limits like Dedekind cuts, Cauchy sequences, etc.). This
brings us to what is the foundations of mathematics? The foundation of
mathematics is reasoning about whether the infinitesimal is sound. Euclid’s
Elements developed mathematics in geometric terms and anything not expressible
in such terms was excluded. Geometry could accommodate the whole numbers and
their ratios as well as irrational magnitudes. Think for example of the spiral
of Theodorus of Cyrene which established that the square roots of non square
integers from 3 to 17 are irrationals.

$\sqrt{2}$$\sqrt{3}$$\sqrt{4}$$\sqrt{5}$$\sqrt{6}$$\sqrt{7}$$\sqrt{8}$$\sqrt{9}$$\sqrt{10}$$\sqrt{11}$$\sqrt{12}$$\sqrt{13}$$\sqrt{14}$$\sqrt{15}$$\sqrt{16}$$\sqrt{17}$

###  1.2 Proofs by Pebbles/Diagrams

Knorr [5] suggests that the original proofs were proofs as diagrams using
_pebble diagrams_. It is known that the ancient Greeks did arithmetic by
counting with pebbles, and pebble diagrams give these calculations by
representing the pebbles by using small circles.

###### Example 1

Here are some statements and their proofs:

  * •

The square of an odd number is 1 + a multiple of 4.
The square of an even number is a multiple of 4.

  * •

_If as many odd numbers as we please be added together, and their multitude be
even, then the sum is even._

$A$$B$$C$$D$$E$

The Greeks also mastered the use of geometric proofs:

###### Example 2

The geometric proof of the Pythagorean Theorem: $c^{2}=a^{2}+b^{2}$. The left
square shows $(a+b)^{2}=2ab+c^{2}$ while the right one shows
$(a+b)^{2}=2ab+a^{2}+b^{2}$.
Hence, $2ab+c^{2}=2ab+a^{2}+b^{2}$ and $c^{2}=a^{2}+b^{2}$.

$b$$c$$a$$c$$a$$b$$b$$a$$c$$b$$c$$a$$b$$a$$a$$a$$b$$a$$b$$b$

###  1.3 Proofs by Contradiction

According to Knorr [5], the change from proofs using diagrams/pebbles to
proofs as sequences of statements occurred with the discovery of
incommensurability:

###### Theorem 1

There is no unit which measures exactly the side and diagonal of a square.

Key results needed for the incommensurability proof relate to Pythagorean
triples and the theory of Odd/Even Numbers:

###### Definition 2

Pythagorean triples are triples of positive whole numbers representing the
lengths of two legs and the hypotenuse of a right triangle. I.e., a
Pythagorean triple is a triple of positive integers $(a,b,c)$ if and only if
$a^{2}+b^{2}=c^{2}$.

E.g. (3, 4, 5), (6, 8, 10), (5, 12, 13), (9, 12, 15), (8, 15, 17).

The following are the results needed to prove incommensurability theorem 1.
Assume $(a,b,c)$ is a Pythagorean triple.

  * 1.

If $c$ is even, then both $a$ and $b$ are even.

  * 2.

If $c$ is even, then
$(\frac{\displaystyle{a}}{\displaystyle{2}},\frac{\displaystyle{b}}{\displaystyle{2}},\frac{\displaystyle{c}}{\displaystyle{2}})$
is also a Pythagorean triple.

  * 3.

If $c$ is a multiple of four, then so are $a$ and $b$.

  * 4.

If $c$ is odd, then one of $a$, $b$ is odd and the other is even.

  * 5.

If any two of $a,b,c$ is even, then the third is also even.

  * 6.

If one of $a,b,c$ is odd, then two are odd and one is even.

$1\cdots 6$ above can be shown using diagrams/pebbles. However, theorem 1
itself needs a proof by contradiction:

Proof. Suppose there is such a unit in terms of which, the side of the square
is $a$ and the diagonal is $c$.
Then, we have a right triangle $c$$a$$a$ and so $(a,a,c)$ is a Pythagorean
triple. Now c must either be even or odd.

  * •

Suppose $c$ even. Then, by 1., $a$ is even. So by 2., we can double the unit
and halve all the dimensions. Clearly, we cannot do this indefinitely, since
otherwise the unit will grow larger than $a$.

  * •

So we must have a Pythagorean triple of the form $(a,a,c)$ in which $c$ is
odd. But then, by 4., $a$ is both even and odd, a contradiction. $\Box$

The proof of incommensurability is believed to be the first proof by
contradiction in the history of mathematical proofs. The proof cannot be
“seen” by looking at a diagram: it is necessary to follow a sequence of
sentences with reasons.

Theorem 1 implies that $\sqrt{2}$ is not a rational number.
Proof: Assume $\sqrt{2}=\frac{\displaystyle{p}}{\displaystyle{q}}$, then
$2q^{2}=p^{2}$. Hence $(q,q,p)$ forms a Pythagorean triple. $p$$q$$q$ Hence
there is a unit which measures exactly the side and diagonal of a square. This
contradicts the incommensurability theorem. $\Box$

###  1.4 Numbers and Magnitudes

With the incommensurability results, the notion of “number” as a discrete
collection of units (e.g., naturals or rationals) was no longer enough. There
arose a need for numbers that are continuous. The Greeks did not know how to
handle these continuous quantities. The main problem was that they treated
mathematical objects as given and did not conceive of constructing them. And
so, they juggled with two notions:

  * •

Their notion of “numbers” (as a multitude of units, Definition 2 of Book VII).

  * •

The so-called “magnitudes” (which include things like lines and areas and
volumes, etc.).

The Greeks developed arithmetic for their numbers, but treated their
magnitudes geometrically. However, although they had not thought of
constructing new mathematical objects, they did introduce a procedure for
approximating ratios. Such approximations were helpful for the much later
constructions of magnitudes (e.g., the real numbers).

Before explaining how the Greeks developed approximations, we explain the
anthyphairesis concept. Anthyphairesis is composed of two Greek terms:
${\upsilon\phi\alpha\iota\rho\epsilon\omega}$ (meaning subtract) and
${\alpha\nu\tau\iota}$ (meaning alternating/reciprocal) and hence
${\alpha\nu\theta\upsilon\phi\alpha\iota\rho\epsilon\sigma\iota\varsigma}$
stands for alternated/reciprocal subtraction. So, given whole numbers $r_{0}$
and $r_{1}$, repeatedly subtract $r_{1}$ from $r_{0}$, $r_{0}-r_{1}$,
$r_{0}-r_{1}-r_{1}$, $\cdots$ until $r_{2}<r1$ remains, then repeat the
process for $r_{1}$ and $r_{2}$, and so on.

$r_{1}$$r_{1}$$r_{1}$$r_{1}$$r_{1}$…$:$$r_{1}$$r_{2}$$r_{0}$

Euclid used anthyphairesis to check whether two numbers are prime to one
another. He proved that anthyphairesis applied to two relatively prime numbers
leads to the unit.

Proposition 1. of Book VII of the _Elements_ Two unequal numbers being set
out, and the less being continuously subtracted in turn from the greater, if
the number left never measures the one before it until a unit is left, the
original numbers will be prime to one another.

###### Example 3

Here is why 17 and 3 are prime to one another.

$\begin{array}[]{|lllllll|}\hline\cr 17&=&5&\times&3&+&2\\\ \hline\cr
3&=&1&\times&2&+&①\\\ \hline\cr 2&=&2&\times&1&+&0\\\ \hline\cr\end{array}$

The ratio and continued fraction are respectively:
$[5,1,2]$ and
$\frac{\displaystyle{17}}{\displaystyle{3}}=\mbox{\framebox{$5$}}+\frac{\displaystyle{1}}{\displaystyle{\mbox{\framebox{$1$}}+\frac{\displaystyle{1}}{\displaystyle{\mbox{\framebox{$2$}}}}}}.$

$3$$3$$3$$3$$3$$3$$1$$1$$1$$2$$2$$17$

Euclid proved that anthyphairesis applied to non relatively prime numbers
gives their greatest common divisor (GCD).

Proposition 2. of Book VII of the _Elements_ Given two numbers not prime to
one another, to find their greatest common measure.

###### Example 4

As we see below, 136 and 6 are not prime to one another and their greatest
common divider is 2.

$\begin{array}[]{|lllllll|}\hline\cr 136&=&22&\times&6&+&4.\\\ \hline\cr
6&=&1&\times&4&+&②.\\\ \hline\cr 4&=&2&\times&2&+&0.\\\ \hline\cr\end{array}$

The ratio and continued fraction are respectively:
$[22,1,2]$ and
$\frac{\displaystyle{136}}{\displaystyle{6}}=\mbox{\framebox{$22$}}+\frac{\displaystyle{1}}{\displaystyle{\mbox{\framebox{$1$}}+\frac{\displaystyle{1}}{\displaystyle{\mbox{\framebox{$2$}}}}}}.$

$6$$6$……………………..22 repeats of $6\times 6$
square……………………..……………………..$6$$1$$1$$1$$4$$4$$136$

###### Example 5

  * •

12 and 5 are prime to one another.

$\begin{array}[]{|lllllll|}\hline\cr 12&=&2&\times&5&+&2\\\ \hline\cr
5&=&2&\times&2&+&①\\\ \hline\cr 2&=&2&\times&1&+&0\\\ \hline\cr\end{array}$

The ratio and continued fraction are respectively:
$[2,2,2]$ and
$\frac{\displaystyle{12}}{\displaystyle{5}}=\mbox{\framebox{$2$}}+\frac{\displaystyle{1}}{\displaystyle{\mbox{\framebox{$2$}}+\frac{\displaystyle{1}}{\displaystyle{\mbox{\framebox{$2$}}}}}}.$

$5$$5$$5$$1$$1$$1$$2$$2$$2$$12$

  * •

22 and 6 are not prime to one another and their greatest common divider is 2.

$\begin{array}[]{|lllllll|}\hline\cr 22&=&3&\times&6&+&4.\\\ \hline\cr
6&=&1&\times&4&+&②.\\\ \hline\cr 4&=&2&\times&2&+&0.\\\ \hline\cr\end{array}$

The ratio and continued fraction are respectively:
$[3,1,2]$ and
$\frac{\displaystyle{22}}{\displaystyle{6}}=\mbox{\framebox{$3$}}+\frac{\displaystyle{1}}{\displaystyle{\mbox{\framebox{$1$}}+\frac{\displaystyle{1}}{\displaystyle{\mbox{\framebox{$2$}}}}}}.$

$6$$6$$6$$6$$2$$2$$2$$4$$4$$22$

The Greeks also applied anthyphairesis to magnitudes. They showed that two
magnitudes are commensurable if and only if anthyphairesis terminates and that
if the anthyphairesis procedure of finding the ratio or GCD of two numbers is
applied to incommensurable magnitudes, it will not terminate.

Proposition 2 of Book X of the _Elements_. If, when the less of two unequal
magnitudes is continuously subtracted in turn from the greater, that which is
left never measures the one before it, the magnitudes will be incommensurable.

###### Example 6

We show that $\sqrt{2}$ is incommensurable.

$C$$B$$A$$1$$1$$\sqrt{2}$$C$$B$$A$$D$$F$$\sqrt{2}-1$$\sqrt{2}-1$$1$$2-\sqrt{2}$

$C$$B$$A$$D$$G$$F$$E$$3-2\sqrt{2}$$\sqrt{2}-1$$\sqrt{2}-1$$C$$B$$A$$D$$G$$H$$F$$E$$I$

Geometrically, we assume the isosceles rectangular triangle $BCA$ below and
take $BD$ of length $\sqrt{2}-1$. From $D$ we draw the perpendicular to $AB$
meeting $BC$ on $F$. We get an isosceles recltangular triangle $BDF$. We
repeat the process obtaining isosceles recltangular triangles $BEG$, $BHI$,
and so on. In this repetition, the less of two unequal magnitudes is
continuously subtracted in turn from the greater, yet what is left never
measures the one before it. This can be repeated infinitely and $\sqrt{2}$ is
incommensurable. Using anthyphairesis:

$\ddots$$\vdots$$1$$1$$3-2\sqrt{2}$$\sqrt{2}-1$$\sqrt{2}-1$$\sqrt{2}-1$$\sqrt{2}$

The ratio of $\sqrt{2}$ to 1 is $[1,2,2,\cdots]$ and the continued fraction is
$\sqrt{2}=1+\frac{\displaystyle{1}}{\displaystyle{2+\frac{\displaystyle{1}}{\displaystyle{2+\ddots}}}}.$

$\sqrt{2}$ is called a quadratic irrational because it is the solution to the
quadratic equation $x^{2}-2=0$. Note that these continued fractions provide an
approximation to $\sqrt{2}$ as follows:

  * •

$\sqrt{2}\approx 1$,

  * •

$\sqrt{2}\approx 1+\frac{\displaystyle{1}}{\displaystyle{2}}=1.5$,

  * •

$\sqrt{2}\approx
1+\frac{\displaystyle{1}}{\displaystyle{2+\frac{\displaystyle{1}}{\displaystyle{2}}}}=1.4$,

  * •

$\sqrt{2}\approx
1+\frac{\displaystyle{1}}{\displaystyle{2+\frac{\displaystyle{1}}{\displaystyle{2+\frac{\displaystyle{1}}{\displaystyle{2}}}}}}=1.417$,

  * •

$\sqrt{2}\approx
1+\frac{\displaystyle{1}}{\displaystyle{2+\frac{\displaystyle{1}}{\displaystyle{2+\frac{\displaystyle{1}}{\displaystyle{2+\frac{\displaystyle{1}}{\displaystyle{2}}}}}}}}=1.4139$
etc.

Infinite repetitions/approximations were a useful part of Greek’s Mathematics
but, anthyphairesis had its limitations. E.g., the obvious theorem below
cannot be proved with it:

If the ratio of $A$ to $C$ is the same as the ratio of $B$ to $C$, then $A=B$.

To overcome the problems, Eudoxus, defined proportion (having the same ratio)
for magnitudes instead of ratios. He invented the method of exhaustion which
was used by Archimedes and Euclid (see Sections 1.7 and 1.8). Theodorus of
Cyrene used Eudoxus approximation in his spiral of irrational numbers pictured
earlier.

###  1.5 The Greeks’ problems with infinitesimals/limits

The Greeks were puzzled by limits and infinitesimals. They needed
approximations but faced obstacles they could not explain. For example, in the
diagram below, the length of the stepped line is clearly $2s$ no matter how
many steps there are. But as the number of steps increases, the stepped line
seems to approach the diagonal whose length is $\sqrt{2}s\not=2s$.

$s$

They demonstrated many paradoxes like the following:

Zeno’s Dichotomy Paradox  There is no motion, because what moves must arrive
at the middle of its course before it reaches the end.

For example, to leave the room, you first have to get halfway to the door,
then halfway from that point to the door, etc. No matter how close you are to
the door, you have to go half the remaining distance. Hence, there is no
finite motion because always going half way while in motion is infinite.

Suppose the distance is 1 meter and the object moves at 1 meter per second. It
must reach halfway ($\frac{1}{2}$ meter from the starting point) in
$a_{1}=\frac{1}{2}$ second. Let $t_{1}=a_{1}$. From this halfway point, the
object moves halfway to the end, which is $a_{2}=\frac{1}{4}$ meters. The
total time so far is $t_{2}=a_{1}+a_{2}=\frac{1}{2}+\frac{1}{4}$. We clearly
have the following infinite sequences:

$a_{1},a_{2},a_{3},\ldots=\frac{1}{2},\frac{1}{4},\frac{1}{8},\ldots$
$t_{1},t_{2},\ldots=\frac{1}{2},\frac{3}{4},\frac{7}{8},\ldots\mbox{ where
each }t_{n}=a_{1}+a_{2}+\cdots+a_{n}.$

Zeno concluded that the total time which is the sum of an infinite sequence
must be infinite and we can never reach our destination. This is incorrect
since we can reach our destination in a finite time. So, where did Zeno get it
wrong?

In modern notation, we see that:

  * •

$t_{n}=\frac{2^{n}-1}{2^{n}}=1-\frac{1}{2^{n}}<1\mbox{ and
}\lim_{n\to\infty}t_{n}=1.$

  * •

$2\Sigma_{n=1}^{\infty}a_{n}=2a_{1}+2\Sigma_{n=2}^{\infty}\frac{1}{2^{n}}=1+\Sigma_{n=1}^{\infty}\frac{1}{2^{n}}=1+\Sigma_{n=1}^{\infty}a_{n}.$

  * •

$\mbox{Hence, }\Sigma_{n=1}^{\infty}a_{n}=1\mbox{ and
}lim_{n\to\infty}t_{n}=\Sigma_{n=1}^{\infty}a_{n}=1.$

Despite the complications of limits, the Greeks continued to use them to
measure magnitudes. Both Archimedes and Euclid (see Sections 1.7 and 1.8) used
Eudoxus theory of proportions which is a geometric method based on exhaustive
approximations designed to overcome the difficulties obtained from the
discovery of the irrationals.

###  1.6 The area a regular polygon

For both Archimedes’ theorem and Euclid’s theorem, we need a general formula
for the area of a regular polygon (i.e., a polygon where all angles (resp. all
sides) are equal). Let us start with the area of a square of side $s$.

$s$h

Instead of simply taking $s^{2}$, take the bottom of the 4 triangles obtained
by the diagonals. Note that the altitude
$h=\frac{\displaystyle{1}}{\displaystyle{2}}s$. The area $A$ of the square =
4$\times$ area of triangle =
$4\times\frac{1}{2}hs=\frac{1}{2}h(4s)=\frac{1}{2}hp$. where $p$ is the
perimeter of the square.
Note that $A=\frac{1}{2}hp=\frac{1}{2}\frac{s}{2}(4s)=s^{2}$.

Now let us consider a regular octagon. If we divide it into triangles the same
way, we get eight triangles, each of whose areas is $\frac{1}{2}hs$. If we
take all eight triangles and note that here $p=8s$, we get for the area
$A=\frac{1}{2}h(8s)=\frac{1}{2}hp$.

We saw this for the square and the regular octagon, but it holds for every
regular polygon:

The area of any regular polygon is one-half the altitude to a side times the
perimeter, or $\frac{1}{2}hp$.

Now we come to the area of a circle. Note that the above polygon was inscribed
in the circle with circumference $C$. If we keep increasing the number of
sides, the perimeter will approach the circumference $C$ and the altitude will
approach the radius $r$. By the above, this suggests that the formula for the
area of a circle should be

$A=\frac{1}{2}rC.$

And since $\pi$ is defined to be the ratio of the circumference of a circle to
twice its radius, we have

$\pi=\frac{C}{2r},$

Hence

$A=\frac{1}{2}r(2\pi r)=\pi r^{2}$

This must have seemed obvious to the ancient Greeks from an early period in
the history of their geometry. But how could they prove it? At one time some
of them argued that a circle is a regular polygon with infinitely many sides,
but they eventually decided that this kind of reasoning is not immune to
attacks by sophists. For just because regular polygons with an increasing
number of sides seems to be approaching a circle, does not automatically
justify in deducing this formula for the area of a circle. They found evidence
like this to be misleading. Recall the stepped line which wrongly gave the
impression that $\sqrt{2}s=2s$.

###  1.7 Euclid on Areas of Circles and Squares

It took a long time for the proof that $A=\frac{1}{2}rC$ to be given. Although
this was obvious to the Greeks, a proof was hard to find. Before that proof
was given (by Archimedes), Euclid proved that the areas of circles have the
same proportion as the squares on their diameters (Proposition 2 of Book XII
of _Elements_). The proof uses Proposition 1 of Book XII.

Proposition 1 of Book XII of the _Elements_. Similar polygons inscribed in
circles are to one another as the squares on the diameters of the circles.

Similar figures are those which have the same shape. In similar polygons the
corresponding angles are equal and the corresponding sides all have the same
proportion.

The areas $A$ of similar polygons are proportional to: • The squares of their
altitudes $h$. • The squares of their perimeters $p$. • The squares of any of
their linear parts.
$\frac{\displaystyle{p_{1}}}{\displaystyle{p_{2}}}=\frac{\displaystyle{h_{1}}}{\displaystyle{h_{2}}}$
and
$\frac{\displaystyle{A_{1}}}{\displaystyle{A_{2}}}=\frac{\displaystyle{h_{1}}}{\displaystyle{h_{2}}}\frac{\displaystyle{h_{1}}}{\displaystyle{h_{2}}}=\frac{\displaystyle{h_{1}^{2}}}{\displaystyle{h_{2}^{2}}}=\frac{\displaystyle{p_{1}^{2}}}{\displaystyle{p_{2}^{2}}}$.

The proof of Proposition 1 of Book XII uses the above and the fact that $AGB$
is similar to $A^{\prime}G^{\prime}B^{\prime}$ below and hence
$(\frac{\displaystyle{AB}}{\displaystyle{A^{\prime}B^{\prime}}})^{2}=(\frac{\displaystyle{AG}}{\displaystyle{A^{\prime}G^{\prime}}})^{2}=\frac{\displaystyle{A_{1}}}{\displaystyle{A_{2}}}$.

O$B$$A$$D$$C$$E$$F$$G$$O^{\prime}$$B^{\prime}$$A^{\prime}$$D^{\prime}$$C^{\prime}$$E^{\prime}$$F^{\prime}$$G^{\prime}$

Now we look at Euclid’s proposition 2 and its proof:

Proposition 2 of Book XII of the _Elements_. Circles are to one another as the
squares on the diameters.

Euclid starts his proof as follows:

Let $ABCD$, $EFGH$ be circles, and $BD$, $FH$ their diameters; I say that, as
the circle $ABCD$ is to the circle $EFGH$, so is the square on $BD$ to the
square on $FH$. For, if the square on $BD$ is not to the square on $FH$ as the
circle $ABCD$ is to the circle $EFGH$, then, as the square on $BD$ is to the
square on $FH$, so will the circle $ABCD$ be either to some less area than the
circle $EFGH$ or to a greater.

Euclid’s strategy is to prove his result by contradiction. In fact, it will be
a double proof by contradiction. He will first assume that it will be in the
ratio to a smaller area $S$, derive a contradiction from that, then assume
that it will be in the ratio to a larger area $S$, and then derive a
contradiction from that area as well. As a result, the only possibility left
will be the result stated in the proposition.

We will not repeat the proof here (see [2, 4]). We must mention however that
Euclid’s method is based on Eudoxus exhaustion which infinitely inscribes and
circumscribes polygons inside the circles. First, Euclid assumes it to be in
that ratio to a less area $S$ and shows that the square $EFGH$ inscribed in
the circle $EFGH$ is greater than half of the circle $EFGH$. He shows this by
noting that the circumscribed square, which includes area outside the circle,
has twice the area of the inscribed square.

Then, he bisects the circumference $EF$, $FG$, $GH$, $HE$ at the points $K$,
$L$, $M$, $N$ and joins $EK$, $KF$, $FL$, $LG$, $GM$, $MH$, $HN$, $NE$ and
proves that the new circumference (in effect inscribing a new regular polygon
with twice the number of sides as the previous one), is more than half the
area inside the circle but outside the previous polygon. By bisecting the
remaining circumferences and joining straight lines, and by doing this
continually, one is left with some segments of the circle which will be less
than the excess by which the circle EFGH exceeds the area S.

In modern notation, let the circles have areas $a$ and $b$ respectively, and
let the ratio of the squares of their diameters be $k$. Let the areas of the
polygons inscribed in the circle with area $a$ (resp. $b$) have areas
$a_{1},a_{2},\ldots$ (resp. $b_{1},b_{2},\ldots$). We have
$0<a_{1}<a_{2}<\ldots<a_{n}<\ldots<a$ and
$0<b_{1}<b_{2}<\ldots<b_{n}<\ldots<b$.

  * •

For each $n$, we have

    * –

$k=\frac{\displaystyle{a_{n}}}{\displaystyle{b_{n}}},\mbox{ so that
}\frac{\displaystyle{a_{n}}}{\displaystyle{k}}=b_{n}.$

    * –

$(a-a_{n+1})<\frac{\displaystyle{1}}{\displaystyle{2}}(a-a_{n})$ and
$(b-b_{n+1})<\frac{\displaystyle{1}}{\displaystyle{2}}(b-b_{n}).$

  * •

We want to prove $k=\frac{\displaystyle{a}}{\displaystyle{b}}.$

  * •

If $k\not=\frac{\displaystyle{a}}{\displaystyle{b}}$, then
$k=\frac{\displaystyle{a}}{\displaystyle{S}}$, where $S<b$ or $S>b$.

    * –

Suppose $S<b$. Choose $N$ so that $b-b_{N}<b-S.$ The number $N$ represents the
number of times the number of sides of the inscribed polygon was doubled. Then
$S<b_{N}.$ But
$S=\frac{\displaystyle{a}}{\displaystyle{k}}>\frac{\displaystyle{a_{N}}}{\displaystyle{k}}=b_{N},$
a contradiction.

    * –

Suppose $S>b$. This is similar to the above case with $a$ and $b$ reversed.

It follows that
$k=\frac{\displaystyle{a}}{\displaystyle{b}}.\hfill\mbox{$\Box$}$

###  1.8 Archimedes’ Measurement of a Circle

Archimedes used Eudoxus’ exhaustion to prove the following proposition (and
hence its corollary that the area of a circle of circumference $C$ and radius
$r$ is $A=\frac{\displaystyle{1}}{\displaystyle{2}}rC$).

Proposition 1 of Archimedes’s Book “Measurement of a Circle”. _The area of any
circle is equal to a right-angled triangle in which one of the sides about the
right triangle is equal to the radius, and the other to the circumference of
the circle._

As we see from the begin of its proof, an infinite number of polygons will be
inscribed/circumscribed in the circle.

Let $ABCD$ be the given circle, $K$ the triangle described. Then, if the
circle is not equal to $K$, it must be either greater or less. I. If possible,
let the circle be greater than $K$. Inscribe a square $ABCD$, bisect the arcs
$AB$, $BC$, $CD$, $DA$, and then bisect (if necessary) the halves, and so on,
until the sides of the inscribed polygon whose angular points are the points
of division subtend segments whose sum is less than the excess of the circle
over $K$.

Let us write the proof in modern notation.

Let $K=\frac{\displaystyle{1}}{\displaystyle{2}}rC$ (the area of the
triangle). If $A\not=K$, then:

  * I.

Suppose $A>K$.

    * –

Inscribe a square with side $s_{1}$, altitude to the side $h_{1}$, and
perimeter $p_{1}$. The area of the square is
$a_{1}=\frac{\displaystyle{1}}{\displaystyle{2}}h_{1}p_{1}.$

    * –

Now, double the number of sides of the inscribed polygon, and keep doubling
it. For polygon $n$ with side $s_{n}$, altitude to the side $h_{n}$, and
perimeter $p_{n}$, the area is
$a_{n}=\frac{\displaystyle{1}}{\displaystyle{2}}h_{n}p_{n}.$

    * –

From the geometry of the situation, we have that
$h_{1}<h_{2}<\ldots<h_{n}<\ldots r,$
$p_{1}<p_{2}<\ldots<p_{n}<\ldots<C,$ and
$a_{1}<a_{2}<\ldots<a_{n}<\ldots<A.$

    * –

Now choose $N$ so that
$A-a_{N}<A-\frac{\displaystyle{1}}{\displaystyle{2}}rC.$ It follows that
$\frac{\displaystyle{1}}{\displaystyle{2}}rC<a_{N}.$

    * –

But since $h_{N}<r$, $p_{N}<C$, and
$a_{N}=\frac{\displaystyle{1}}{\displaystyle{2}}h_{N}p_{N}$, we have
$a_{N}<\frac{\displaystyle{1}}{\displaystyle{2}}rC,$ a contradiction.

  * II.

Suppose, on the contrary, that $A<K$.

    * –

Circumscribe a square with perimeter $P_{1}$; then the area is
$A_{1}=\frac{\displaystyle{1}}{\displaystyle{2}}rP_{1}.$

    * –

Double the number of sides of the circumscribed figure, and keep doing it. If,
for the $n$th polygon, the perimeter is $P_{n}$, then the area is
$A_{n}=\frac{\displaystyle{1}}{\displaystyle{2}}rP_{n}.$

    * –

From the geometry, we have
$C<\ldots<P_{n}<\ldots<P_{2}<P_{1}$ and
$A<\ldots A_{n}<\ldots<A_{2}<A_{1}.$

    * –

Choose $N$ where $A_{N}-A<\frac{\displaystyle{1}}{\displaystyle{2}}rC-A.$
Then $A_{N}<\frac{\displaystyle{1}}{\displaystyle{2}}rC.$

    * –

But $C<P_{N}$ and $A_{N}=\frac{\displaystyle{1}}{\displaystyle{2}}rP_{N}$, so
$\frac{\displaystyle{1}}{\displaystyle{2}}rC<A_{N},$ another contradiction.

It follows that $A=K=\frac{\displaystyle{1}}{\displaystyle{2}}rC$. $\Box$

###  1.9 Eudoxus, the infinitesimal and the limit

We saw the use of Eudoxus’ exhaustion method in the proofs of Euclid and
Archimedes. This method infinitely constructs new objects that would
eventually only differ in infinitesimal amounts. It can be used to develop a
definition of the limit of a sequence and a function. Historically, the
development of calculus and analysis in European mathematics occured before a
definition of the real numbers. At the time of Descartes, Leibniz and Newton,
it had not even been settled whether or not there were infinitely small
quantities. For centuries before and after, infinitesimals oscillated between
being accepted and being rejected. They were _introduced_ in 450 BC, _banned_
by Eucledian mathematicians because of the problems they faced with them, used
by Kepler to calculate the area of an ellipse as the infinite sum of vertical
lines contained in the ellipse, _banned again_ in the 1630s by religious
clerics in Rome. They still _flourished_ in the 17th century444In the ideas
that a curved line is made of infinitely small straight line segments, and
quantities that differ by an infinitely small quantity are equal. and were
_crucial_ for the development of calculus by Newton and Leibniz. They were
thought to exist by Cauchy who used them in his approach to calculus, then
they were _abandoned again_ in the 19th century due to their unclear logical
status to be _revived again_ in the 20th century especially in Robinson’s non-
standard analysis. Nowadays, they take center stage in the foundations of
mathematics which many people define as a sound theory of infinitesimals.

The next graph demonstrates how a curved line is made of infinitely small
straight line segments.

The next example explains a cleric position on infinitesimals.

###### Example 7

To find the derivative $f^{\prime}(2)$ at $x=2$ of $y=f(x)=x^{2}$, we assume
$x\not=2$. Then we calculate:

$\frac{\displaystyle{\Delta y}}{\displaystyle{\Delta
x}}=\frac{\displaystyle{f(x)-f(2)}}{\displaystyle{x-2}}=\frac{\displaystyle{x^{2}-2^{2}}}{\displaystyle{x-2}}=\frac{\displaystyle{(x+2)(x-2)}}{\displaystyle{x-2}}=x+2.$

Since we are only able to conclude that the quotient is equal to x + 2 on the
assumption that $x\not=2$, we appear to have taken an illegal step. We justify
this by saying that we are taking its limit as $x\rightarrow 2$ and write:
$\frac{\displaystyle{dy}}{\displaystyle{dx}}=\lim_{x\to
2}\frac{\displaystyle{\Delta y}}{\displaystyle{\Delta x}}.$ Newton calls
$\frac{\displaystyle{dy}}{\displaystyle{dx}}=\lim_{x\to
2}\frac{\displaystyle{\Delta y}}{\displaystyle{\Delta x}},$ _ultimate value_
or _value at instant of disappearance_. Sarcastically, this is called _the
ghosts of a departed quantity_ in a critique [1] by Bishop Berkeley addressed
to a certain “Infidel Mathematician”. [1] examined whether the object and
principles of the modern Analysis are more distinctly conceived, or more
evidently deduced, than religious mysteries and points of faith.

###  1.10 Infinitesimals and the birth of analysis

At school, after studying arithmetic and elementary algebra, you are
introduced to geometry (_the study of shapes_) and trigonometry (_the study of
side lengths and angles of triangles_) and then you move to a _pre-calculus
course_ which combines advanced algebra and geometry with trigonometry. After
all this, you are introduced to _calculus_. Calculus (originally called
_infinitesimal calculus_) is the mathematical study of continuous change. The
infinitesimal part is important. It is believed that if Descartes had
expressed rather than supressed the infinitesimals and infinites in his
method, he would have invented the calculus before Newton and Leibniz.

Calculus formalizes the study of continuous change, while analysis provides it
with a rigorous foundation in logic. As we saw, the Greeks dealt with discrete
numbers arithmetically and with continuous magnitudes geometrically. But
continuous systems can be subdivided indefinitely, and their description
requires the real numbers. This infinite subdivision was influenced by
Eudoxus’ and Archimedes’ approximations. The real numbers were not present in
the historic approach to define limits and develop the calculus. The ancient
Greeks separated whole and rational numbers, which are discrete, from
continuous magnitudes. They had different kinds of magnitudes for lengths,
areas, volumes, angles, etc., and never multiplied two lengths to get another
length. The beginning of algebra and the reduction of geometrical problems
into algebraic and arithmetical ones in the 9th century [6, 7] paved the way
for Descartes innovative ruler-and-compass construction for multiplying two
lengths to get a length. This allowed Algebra to be a science concerned with
numbers rather than geometric magnitudes.

Here is how the ruler-and-compass construction works:

###### Example 8

The length of $AB$ is $a$. On a line $AC$ through $A$ and at an angle to $AB$,
let the length of $AC$ be a unit, and construct $E$ on the same line so that
the length of $AE$ is $b$. Join $C$ and $B$ with line segment $BC$, and
construct a line through $E$ parallel to $BC$; let this line intersect the
extension of $AB$ at $D$. Then triangles $ABC$ and $ADE$ are similar. Hence,
$AE$ is to $AC$ as $AD$ is to $AB$. I.e., $AD=ab$.

The move to generalise the geometric concepts and methods of the calculus to
more algebraic forms continued into the 18th century. But the field was still
rife with disagreements on the need and use of infinitesimals and
mathematicians began to worry about the lack of rigorous foundations of the
calculus (recall that the foundations of mathematics is a sound reasoning
about the infinitesimal). This would change due Cauchy’s ideas of _function
and limit_ which led to a more _rigorous_ formulation of the calculus,
limit/continuity/_real numbers_. And, due to the emerging _exact definition of
real numbers_ the rules for reasoning with real numbers became even more
precise. However, all this historical background of the development of
analysis is rarely reflected in the modern teaching of the subject. Instead,
students are introduced to methods that they find challenging, like the
$\epsilon-\delta$/ $\epsilon-N$ proofs of limits without background material
on why limits, infinites, approximations and infinitesimals were developed.
From our experience, an evolutionary and somewhat historic approach is
helpful. This is why we embarked on a book [4] that introduces mathematical
analysis by employing the evolution of this area of mathematics to first
develop fundamental concepts of mathematical analysis and to only introduce
formal definitions after the concepts are understood.

The landscape of mathematics would change forever during the 19th century and
the commitment to rigorous foundations would lead to the discovery of
computability and its limits. Rigorous foundations also shed light on the
holes that started to appear in Euclid’s historic work which led some to
question the deductive structure of the Elements. Such logical inaccuracies
have been addressed in the work of Hilbert [3] who wrote 20 postulates
adequate to prove all the theorems in the Elements. Here we go through some of
these holes.

  * •

Look at Proposition 1 of Book I of Euclid’s Elements:

_To construct an equilateral triangle on a given finite straight line._

Let $AB$ be the finite straight line. The proof draws two circles with radius
$AB$, and center $A$ (resp. $B$). The circles intersect at $C$ and the
triangle $ABC$ is equilateral.

ABDEC

There is a problem in this proof. At first glance, there does not appear to be
any doubt that the construction given there constructs the desired equilateral
triangle and that the proof proves that it is an equilateral triangle.
However, there is a gap in the proof. There is, in fact, no proof that the
point $C$ exists. We can construct a _model_ of geometry in which all of the
postulates and axioms are satisfied but Proposition 1 is not.

  * •

Euclid’s Postulate 5 (the parallel postulates) is less obvious than the other
postulates.

  * •

Euclid used a number of statements as facts in his Elements even though they
had neither been proved nor been introduced as postulates. For example:

A straight line that intersects one side of a triangle but does not pass
through any vertex of the triangle must intersect one and only one of the
other sides.

Based on this statement, Pasch proved that Euclid’s formulation was not
complete in the sense that there are statements that should hold but which
cannot be proven from Euclid’s formulation.

    1. 1.

A straight line passing through the center of a circle must intersect the
circle.

    2. 2.

Given 3 different points on the same line, one of them is between the other
two.

Having introduced the discrete (natural, rational and integer) numbers, and
having emphasised the historical treatment of continuous magnitudes and the
need for real numbers in the development of analysis, we now discuss the real
numbers.

###  1.11 What are the real numbers?

Recall Proposition 2. of Book VII of the _Elements_ and the approximations for
$\sqrt{2}$ in Section 1.4. You can think of $\sqrt{2}$ as all the rational
numbers strictly less than it. I.e., as:
$\\{1,1+\frac{\displaystyle{1}}{\displaystyle{2}},1+\frac{\displaystyle{1}}{\displaystyle{2+\frac{\displaystyle{1}}{\displaystyle{2}}}},1+\frac{\displaystyle{1}}{\displaystyle{2+\frac{\displaystyle{1}}{\displaystyle{2+\frac{\displaystyle{1}}{\displaystyle{2}}}}}},\cdots\\}$.
All irrational numbers have infinitely distinct approximations like
$\sqrt{2}$. Hence, the real numbers can be defined as non empty subsets of the
rationals which satisfy some properties (see below). Real numbers will be
defined as elements of a complete ordered field. Hence the following
definitions.

###### Definition 3

A _field_ is a set $(S,+,\cdot)$ such that $S$ is closed under $+$ and $\cdot$
and satisfies distributivity $a(b+c)=ab+ac$, commutativity and associativity
of $+$ and $\cdot$, and existence of identity elements $0$ and $1$ ($a+0=a$
and $a\cdot 1=a$) and inverses $-a$ and $a^{-1}$ (for each $a$ except for $0$
under $\cdot$).

###### Example 9

None of $\mathbb{N}^{+}$ or $\mathbb{Z}$ is a field but $\mathbb{Q}$ is a
field.

###### Definition 4

A field is _ordered_ (by $<$) if for all $a$, $b$, $c$:

  * •

exactly one of $a<b$, $a=b$, and $b<a$ holds.

  * •

if $a<b$ and $b<c$, then $a<c$.

  * •

if $0<a$ and $0<b$, then $0<a+b$ and $0<ab$.

  * •

$a<b$ if and only if $0<b+(-a)$.

The next axiom is important for the real numbers.

Axiom of Completeness [AC]
Every nonempty set of quantities that has an upper bound has a least upper
bound.

Now we give the definition of the Real Numbers $\mathbb{R}$.

###### Definition 5

Our quantities form an ordered field that satisfies the Axiom of Completeness
AC. We will refer to them as real numbers and denote their collection by
$\mathbb{R}$.

Recall that the real numbers are continuous whereas the
natural/integer/rational numbers are discrete. The following help us to see
some differences between these numbers.

Archimedes Law [AL] For any two quantities $a$ and $b$ where $b>a>0$, there is
a positive integer $n$ such that $b<an$.

###### Definition 6

An ordered field which also satisfies AL is called an _Archimedean ordered
field_.

###### Example 10

$\mathbb{Q}$ is an Archimedean ordered field.

  * •

Completeness implies the Archimedean Property Assume $a$ and $b$ are real
numbers such that $a>0$. There is a positive integer $n$ such that $an>b$.

  * •

We can approximate real numbers by rational numbers.

Density of rationals If $a$ and $b$ are any two real numbers with $a<b$, then
there is a rational number $r$ such that $a<r<b$.

## References

  * [1] George Berkeley.  The Analyst: or A Discourse Addressed to an Infidel Mathematician. First printed in 1734.  In A. A. Luce and T. E. Jessop, editors, The Works of George Berkeley Bishop of Cloyne, volume 4, pages 53–102. Nelson, London, 1951.
  * [2] Heath.  The 13 Books of Euclid’s Elements.  Dover, 1956.
  * [3] David Hilbert.  The Foundations of Geometry.  The Open Court Publishing Co, 1902.
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