Complex numbers and the Argand plane

Why introduce complex numbers?

The source motivates complex numbers using physics, engineering and circuit calculations: complex notation can simplify calculations involving a source, current and impedance into addition and multiplication.

Its introductory circuit sketch includes an alternating source, resistor R, inductor L and capacitor C; it is a motivation, not an additional circuit-solving task.

The opening equation $x^2+1=0$ has no real root because a real square is nonnegative. Defining $i^2=-1$ extends the number system and gives two roots, $\pm i$.

Follow the power cycle

The source derives

$$\begin{aligned}i^3&=i^2i=-i,\\i^4&=(i^2)^2=(-1)^2=1,\\i^5&=i^4i=(i^2)^2i=i,\\i^6&=(i^2)^3=(-1)^3=-1.\end{aligned}$$

For positive integers k,

$$\begin{aligned}i^{2k}&=(i^2)^k=(-1)^k,\\i^{2k+1}&=i^{2k}i=(i^2)^ki=(-1)^ki.\end{aligned}$$

A general complex number is $a+ib$, where a,b are real; a is its real part and b its imaginary part.

“Complex” means made of connected parts, as in the source’s shopping-complex analogy.

For positive real a, the principal square root is $\sqrt{-a}=\sqrt{-1}\sqrt a=i\sqrt a$.

This special rule does not justify multiplying principal square roots of arbitrary complex numbers without checking branches.

The notes credit the notation to Leonhard Euler and refer to his biography in Appendix A, source page 36. That historical pointer is retained with the original notes.

Place the number sets inside one another

Name: Natural numbers

• Symbol: $\mathbb N$
• Source description: Counting numbers $\{1,2,3,\ldots\}$; this source convention starts at 1.

Name: Integers

• Symbol: $\mathbb Z$
• Source description: $\{\ldots,-3,-2,-1,0,1,2,3,\ldots\}$

Name: Rational numbers

• Symbol: $\mathbb Q$
• Source description: $p/q$, where p,q are integers and $q\ne0$

Name: Irrational numbers

• Symbol: No separate symbol in the table
• Source description: Real numbers with neither a finite nor a recurring decimal representation; examples $e,\pi,\sqrt2$

These types are real numbers. The complex numbers are

$$\mathbb C=\{a+ib:a,b\in\mathbb R\}.$$

The source Venn diagram nests $\mathbb N$ inside $\mathbb Z$, then $\mathbb Q$, then $\mathbb R$, then $\mathbb C$.

Irrational numbers occupy the real region outside the rationals.

The source extension callout treats complex numbers as extending the real number line; their two real coordinates are represented in a plane.

The source QR links to further understanding.

Read Cartesian components and the Argand plane

Read the real coefficient and the coefficient of the imaginary unit separately. Use those two real coordinates to place the point on the Argand plane.

Cartesian, algebraic and rectangular form all refer here to $z=x+iy$, with real x,y.

The parts are $\operatorname{Re}(z)=x$ and $\operatorname{Im}(z)=y$.

The imaginary part does not contain i; this is the explicit Teacher reminder and the key to completing the example table.

If the imaginary part is zero, z is real.

If the real part is zero, z has form iy.

The source calls this purely imaginary; some conventions exclude zero from that label.

Zero has both parts zero and is real regardless of that convention.

The ordered pair $(x,y)$ uniquely specifies z.

It can be represented by a point $P(x,y)$ or the position vector $\overrightarrow{OP}=(x,y)$.

In an Argand diagram the horizontal x-axis is the real axis, and the vertical y-axis is the imaginary axis.

The source diagram callout compares this with Cartesian coordinates and the real number line.

Its historical note names Jean-Robert Argand (1768–1822), described there as a Parisian bookkeeper and amateur mathematician.

Use a point, a vector and its direction

The source shows three Argand views of the same $z=x+iy$: a point $P(x,y)$ with dashed coordinate projections; the position vector $\overrightarrow{OP}=(x,y)$; and a vector of length r making angle $\theta$ with the positive real axis. These are different descriptions of one complex number. The length and angle will be developed further as modulus and argument.

Compare components for equality

For real a,b,c,d,

$$a+ib=c+id\quad\Longleftrightarrow\quad a=c\text{ and }b=d.$$

For example, $4+3i=p+iq$ is equivalent to $p=4$ and $q=3$. The source equivalence callout uses this definition to solve unknown coefficients.

As the Teacher note suggests, connect it with equality of two-dimensional column vectors: both corresponding entries must agree.

Retain the four basic-operation rows

Let $z_1=x_1+iy_1$ and $z_2=x_2+iy_2$, with all four components real.

Addition

$$\begin{aligned}z_1+z_2&=x_1+iy_1+x_2+iy_2\\&=(x_1+x_2)+i(y_1+y_2).\end{aligned}$$

Subtraction

$$\begin{aligned}z_1-z_2&=x_1+iy_1-(x_2+iy_2)\\&=(x_1-x_2)+i(y_1-y_2).\end{aligned}$$

Multiplication by a real constant k

$$\begin{aligned}kz_1&=k(x_1+iy_1)\\&=kx_1+iky_1.\end{aligned}$$

Multiplication of two complex numbers

$$\begin{aligned}z_1z_2&=(x_1+iy_1)(x_2+iy_2)\\&=x_1x_2+ix_1y_2+ix_2y_1+i^2y_1y_2\\&=(x_1x_2-y_1y_2)+i(x_1y_2+x_2y_1),\end{aligned}$$

because $i^2=-1$. This complex product is different from the dot product of the coordinate vectors, as Example 2 demonstrates.

The source warns against comparing complex numbers with the usual real-order symbols $<,\le,\ge,>$. Its example $1+4i<3+4i$ is not a valid usual numerical comparison. You can compare real quantities such as their real parts or moduli when the question requests that comparison; the usual ordered-field structure of the reals does not extend to all complex numbers.

Interpret the operations geometrically

Negation is a 180° rotation about O. Addition and subtraction follow vector parallelogram laws and can be viewed as translations of a point by another complex number. Multiplication by i is a 90° anticlockwise rotation; by −i it is a 90° clockwise rotation. Draw Argand diagrams to scale, using equal unit lengths on both axes, to preserve these angles.

Use the source calculator procedure

The printed graphing-calculator steps are: press MODE and select a+bi to obtain algebraic-form answers $x+iy$; use 2nd, then the decimal-point key, to enter i. The source output screen retains these Example 2 results: $-8-6i$, $-5+15i$, $-4-3i$, $5+6i$ and $-3$. Calculator key sequences are the source model’s procedure; the algebra above explains the results independently.

Distinguish a principal square root from both roots

First decide what is being requested: one chosen square-root value or all solutions of a square equation. For algebraic roots, compare real and imaginary components and check both signs.

On the nonnegative real domain, $f(x)=\sqrt x$ is strictly increasing and gives a unique nonnegative value.

Solving $w^2=z$ asks for algebraic square roots: a nonzero complex $z$ has two, $w$ and $-w$; zero has only one distinct root.

Example 4 finds $4+i$ and $-4-i$ for $15+8i$.

Set $w=a+bi$ with real $a,b$, expand, then compare components.

A calculator square-root command shows one selected root; negate it for the other.

Understand conjugation as a reflection

Recall the surd pair $a+\sqrt b$ and $a-\sqrt b$.

The notes motivate the complex analogue by writing $2+3i=2+3\sqrt{-1}$, whose conjugate is $2-3\sqrt{-1}=2-3i$. For real $x,y$, define

$$z=x+iy\quad\Longrightarrow\quad z^*=x-iy.$$

The real part stays the same; the imaginary coefficient changes sign.

The source pair $1+3i$ and $1-3i$ lies at $(1,3)$ and $(1,-3)$: reflecting the point and its position vector in the real axis gives the conjugate.

A real number is unchanged because its imaginary coefficient is zero.

For the calculator illustrated in the source, use MODE → a+bi, then MATH → CMPLX → 1:conj(. The screen gives $\operatorname{conj}(1+3i)=1-3i$ and $\operatorname{conj}(-1-4i)=-1+4i$.

Derive the first eight conjugate properties

Let $z=x+iy$ with real $x,y$. These are the source rows (a)–(h), including their derivations.

(a) Definition. $z^*=x-iy$ by definition.

(b) Conjugate twice.

$$(z^*)^*=(x-iy)^*=x+iy=z.$$

(c) Product with the conjugate.

$$\begin{aligned}zz^*&=(x+iy)(x-iy)\\&=x^2-ixy+ixy-(iy)^2\\&=x^2-i^2y^2\\&=x^2+y^2.\end{aligned}$$

(d) A real scalar passes through conjugation. For $k\in\mathbb R$,

$$\begin{aligned}(kz)^*&=[k(x+iy)]^*\\&=(kx+iky)^*\\&=kx-iky\\&=k(x-iy)\\&=kz^*.\end{aligned}$$

The assumption that $k$ is real matters; it is not a rule for an arbitrary complex scalar without conjugating that scalar too.

(e) Add conjugates to recover the real part.

$$z+z^*=(x+iy)+(x-iy)=2x=2\operatorname{Re}(z).$$

(f) Subtract conjugates to recover the imaginary coefficient.

$$z-z^*=(x+iy)-(x-iy)=2iy=i[2\operatorname{Im}(z)].$$

The factor $i$ is essential: $\operatorname{Im}(z)$ itself is a real number.

(g) Conjugation preserves addition. With real components $x_1,y_1,x_2,y_2$,

$$\begin{aligned}(z_1+z_2)^*&=[(x_1+iy_1)+(x_2+iy_2)]^*\\&=[(x_1+x_2)+i(y_1+y_2)]^*\\&=(x_1+x_2)-i(y_1+y_2)\\&=(x_1-iy_1)+(x_2-iy_2)\\&=z_1^*+z_2^*.\end{aligned}$$

(h) Extend the addition rule to a finite sum. Apply (g) repeatedly, separating the last term at each stage:

$$\begin{aligned}(z_1+\cdots+z_n)^*&=(z_1+\cdots+z_{n-1})^*+z_n^*\\&=(z_1+\cdots+z_{n-2})^*+z_{n-1}^*+z_n^*\\&\;\vdots\\&=z_1^*+z_2^*+\cdots+z_n^*.\end{aligned}$$

Complete the conjugate product rules

For $z_1=x_1+iy_1$, $z_2=x_2+iy_2$ with real components, the remaining source properties are (i)–(k).

(i) Conjugation preserves multiplication. Expand, conjugate the collected form, and regroup into the two conjugates:

$$\begin{aligned}(z_1z_2)^*&=[(x_1+iy_1)(x_2+iy_2)]^*\\&=[x_1x_2+ix_1y_2+ix_2y_1+i^2y_1y_2]^*\\&=[(x_1x_2-y_1y_2)+i(x_1y_2+x_2y_1)]^*\\&=(x_1x_2-y_1y_2)-i(x_1y_2+x_2y_1)\\&=x_1x_2-ix_1y_2-y_1y_2-ix_2y_1\\&=x_1(x_2-iy_2)+(-i)^2y_1y_2-ix_2y_1\\&=x_1(x_2-iy_2)+(-i)y_1(-iy_2+x_2)\\&=(x_1-iy_1)(x_2-iy_2)\\&=z_1^*z_2^*. \end{aligned}$$

Here $(-i)^2=-1$ is the key regrouping identity.

(j) Extend to a finite product. Apply (i) repeatedly:

$$\begin{aligned}(z_1z_2\cdots z_n)^*&=(z_1\cdots z_{n-1})^*z_n^*\\&=(z_1\cdots z_{n-2})^*z_{n-1}^*z_n^*\\&\;\vdots\\&=z_1^*z_2^*\cdots z_n^*. \end{aligned}$$

(k) Conjugate a positive integer power. Regard $z^n$ as a product of $n$ identical factors:

$$\begin{aligned}(z^n)^*&=(\underbrace{z\cdot z\cdots z}_{n\text{ factors}})^*\\&=\underbrace{z^*\cdot z^*\cdots z^*}_{n\text{ factors}}\\&=(z^*)^n. \end{aligned}$$

Make a complex denominator real

Check that the denominator is nonzero before manipulating the fraction. Multiply the whole fraction by the conjugate over itself, then collect the real and imaginary parts.

The surd analogy multiplies $1/(\sqrt a+\sqrt b)$ by $(\sqrt a-\sqrt b)/(\sqrt a-\sqrt b)$ to obtain $(\sqrt a-\sqrt b)/(a-b)$, for real $a,b\ge0$ and $a\ne b$.

Taking $a=5,b=1$ gives the source split $\sqrt5/4-1/4$.

The source says “integer” denominator; generally it is real, and it is an integer only when $a-b$ is an integer.

The case $a=b>0$ needs a direct simplification instead.

For division by $w=x+iy\ne0$, multiply numerator and denominator by $w^*=x-iy$.

Since $ww^*=x^2+y^2>0$, the denominator becomes real and nonzero.

This is what the notes call realising the denominator.

Example 7 then splits its fraction to identify the real part and the imaginary coefficient separately.

Solve simultaneous equations by substitution

Use the same substitution or elimination principles as for real equations.

Example 8 first writes $z=(1-i)w$, substitutes into the second equation and realises the remaining denominator; its instruction forbids a graphing calculator.

When a conjugate remains, as in Example 9, set $z=a+bi$ and $z^*=a-bi$ with real $a,b$, then compare components.

A ratio such as $w/z$ requires $z\ne0$.

Count polynomial roots with multiplicity

Check the coefficient type before using conjugate pairs. Count repeated roots with their multiplicities, then use known roots to build factors and reduce the degree.

A polynomial of degree $n\ge1$, with complex coefficients and nonzero leading coefficient $a_n$, has exactly $n$ complex roots counting multiplicity.

Some roots can repeat; the statement does not promise $n$ distinct roots.

For a polynomial with real coefficients, a nonreal root $p$ is accompanied by its conjugate $p^*$.

A real root equals its own conjugate and does not require a second distinct root.

To see why the real-coefficient condition matters, start with

$$a_np^n+a_{n-1}p^{n-1}+a_{n-2}p^{n-2}+\cdots+a_1p+a_0=0.$$

Conjugate both sides and use the sum, product and power rules:

$$\begin{aligned}(a_np^n+a_{n-1}p^{n-1}+a_{n-2}p^{n-2}+\cdots+a_1p+a_0)^*&=0^*,\\(a_np^n)^*+(a_{n-1}p^{n-1})^*+(a_{n-2}p^{n-2})^*+\cdots+(a_1p)^*+a_0^*&=0,\\a_n^*(p^n)^*+a_{n-1}^*(p^{n-1})^*+a_{n-2}^*(p^{n-2})^*+\cdots+a_1^*p^*+a_0^*&=0. \end{aligned}$$

Each coefficient is real, so $a_k^*=a_k$. Therefore

$$a_n(p^*)^n+a_{n-1}(p^*)^{n-1}+a_{n-2}(p^*)^{n-2}+\cdots+a_1p^*+a_0=0.$$

Thus $p^*$ is a root of the same polynomial.

Check the coefficient condition before assuming a conjugate root

If any coefficient is nonreal, the proof cannot replace every $a_k^*$ by $a_k$; the conjugate may fail to be a root. The source counterexample is $3z^2+iz-2=0$. The quadratic formula gives

$$\begin{aligned}z&=\frac{-i\pm\sqrt{i^2-4(3)(-2)}}{2(3)}\\&=\frac{-i\pm\sqrt{23}}6\\&=\frac{\sqrt{23}}6-\frac i6\quad\text{or}\quad-\frac{\sqrt{23}}6-\frac i6. \end{aligned}$$

Both imaginary coefficients are $-1/6$, so these two roots are not a conjugate pair.

Combine conjugate factors into a real quadratic

For real $x,y$, conjugate roots $x+iy$ and $x-iy$ give

$$\begin{aligned}[z-(x+iy)][z-(x-iy)]&=z^2-z(x+iy)-z(x-iy)+(x+iy)(x-iy)\\&=z^2-2xz+(x^2+y^2). \end{aligned}$$

This quadratic has real coefficients.

Pairing all nonreal roots this way shows that every nonconstant real polynomial factors over the reals into real linear and quadratic factors, together with its nonzero leading constant.

A real cubic has either three real roots or one real root and a nonreal conjugate pair, counting multiplicity.

Example 10 uses the pair $2\pm i$ to form $z^2-4z+5$, then long division gives the remaining factor $z+1$.

Measure modulus and principal argument

Plot the point first. Find its distance from the origin, identify its axis or quadrant, and only then select the angle in the stated principal interval. Zero has no direction to select.

For $z=x+iy$, plot $P(x,y)$ on the Argand plane.

The position vector $\overrightarrow{OP}$ has length $r$ and direction $\theta$ from the positive real axis.

The source triangle labels the horizontal component $x$, vertical component $y$, hypotenuse $r$ and angle $\theta$; the point/vector/direction figure above retains these roles.

The modulus is the distance from $O(0,0)$ to $P(x,y)$:

$$|z|=r=\sqrt{x^2+y^2}.$$

An argument describes the direction of this vector.

Anticlockwise angles are positive and clockwise angles negative.

Arguments differ by multiples of $2\pi$; the notes choose the principal argument in $(-\pi,\pi]$.

In particular the negative real axis uses $\pi$, not $-\pi$.

The source procedure is: draw an Argand diagram and identify the quadrant; find the basic angle $\alpha=\tan^{-1}\lvert y/x\rvert$ for $x\ne0$; then choose the correct principal angle from the cases below.

Treat points on the axes separately.

Principal argument $\theta$

Case: 1

• Position of a nonzero point: Positive real axis
• Principal argument: $0$

Case: 2

• Position of a nonzero point: Negative real axis
• Principal argument: $\pi$

Case: 3

• Position of a nonzero point: Positive imaginary axis
• Principal argument: $\pi/2$

Case: 4

• Position of a nonzero point: Negative imaginary axis
• Principal argument: $-\pi/2$

Case: 5

• Position of a nonzero point: First quadrant
• Principal argument: $\alpha=\tan^{-1}\lvert y/x\rvert$

Case: 6

• Position of a nonzero point: Second quadrant
• Principal argument: $\pi-\alpha=\pi-\tan^{-1}\lvert y/x\rvert$

Case: 7

• Position of a nonzero point: Third quadrant
• Principal argument: $-\pi+\alpha=-\pi+\tan^{-1}\lvert y/x\rvert$

Case: 8

• Position of a nonzero point: Fourth quadrant
• Principal argument: $-\alpha=-\tan^{-1}\lvert y/x\rvert$

The eight diagrams show the same cases, with signed angle arrows from the positive real axis and basic-angle arcs in the quadrants. Their points illustrate the position cases rather than prescribing particular numerical coordinates.

The argument of zero is undefined. The slope expression would require the undefined ratio $0/0$, and geometrically the zero vector has no direction from which to measure an angle.

Its modulus is still $|0|=0$.

Apply the modulus and argument rules with their conditions

Appendix C is on PDF/printed 38–39 in both copies.

The earlier calculator note points to page 42; that printed pointer is stale.

The appendix is labelled non-examinable in the earlier note; retain that source label while using these rules to understand the worked examples.

For z=x+iy with real x,y, and for the products/quotients below where defined, the nine source rows are:

Row: (a)

Rule, with the necessary conditions:

$$zz^*=\lvert z\rvert^2=x^2+y^2$$

Row: (b)

Rule, with the necessary conditions:

$$\lvert z^*\rvert=\lvert z\rvert$$

Row: (c)

Rule, with the necessary conditions:

For nonzero z:

$$\arg(z^*)\equiv-\arg z\pmod{2\pi}$$

Choose the principal representative.

Row: (d)

Rule, with the necessary conditions:

$$\lvert z_1z_2\rvert=\lvert z_1\rvert\lvert z_2\rvert$$

Row: (e)

Rule, with the necessary conditions:

$$\lvert z^n\rvert=\lvert z\rvert^n$$

For positive integer n; negative integer n needs z≠0.

Row: (f)

Rule, with the necessary conditions:

$$\lvert z_1/z_2\rvert=\lvert z_1\rvert/\lvert z_2\rvert$$

With z₂≠0.

Row: (g)

Rule, with the necessary conditions:

$$\arg(z_1z_2)\equiv\arg z_1+\arg z_2\pmod{2\pi}$$

With both numbers nonzero.

Row: (h)

Rule, with the necessary conditions:

$$\arg(z^n)\equiv n\arg z\pmod{2\pi}$$

With nonzero z and integer n.

Row: (i)

Rule, with the necessary conditions:

$$\arg(z_1/z_2)\equiv\arg z_1-\arg z_2\pmod{2\pi}$$

With both numbers nonzero.

The source prints the argument rows as equalities without branch conditions.

For principal arguments in $(-\pi,\pi]$, add or subtract multiples of 2π until the result lies in that interval.

For example, both −1+i and −1−i have argument magnitude 3π/4.

Squaring −1+i gives −2i: the raw sum 3π/2 must become −π/2.

The quotient $(-1+i)/(-1-i)=-i$ gives the same adjustment after subtracting arguments.

The power $(1+i)^6=-8i$ likewise has principal argument −π/2, not 3π/2.

For z=−1, conjugation leaves z unchanged, so both principal arguments are π; the unadjusted −π is excluded.

Zero has modulus 0 but no argument.

The nine worked examples use 1+i and √3+i.

Their results are 2,√2,−π/4,2√2,4,√2/2,5π/12,π andπ/12 in source order.

In these particular examples all raw angles are already principal.

Open the linked appendix exercise for every printed calculation and the relevant condition at each step.

Read the Euler and equation-history appendices

This unit is historical reading and enrichment. Read the credits as the source’s simplified chronology; the equation catalog is not a list of additional assessed techniques.

Appendices A and B are on PDF/printed 37 in both copies; the earlier Appendix A pointer to page 36 is stale.

Appendix A describes Leonhard Euler (1707–1783), born 15 April 1707 in Basel, as an academy scholar whose work influenced geometry, trigonometry, calculus and physics.

It notes his hundreds of publications and continued productivity after losing his sight, before his death on 18 September 1783.

The original adaptation credits Biography.com; ETH Library’s account confirms the dates and continuing work after blindness.

The source QR links to further understanding.

Appendix B recommends Ian Stewart’s Seventeen Equations that Changed the World.

Its 2013 date fits the paperback edition; the publisher’s copyright page identifies the first British publication as 2012.

The source highlights the square-root-of-minus-one and polyhedron rows associated with Euler.

These are reading prompts about mathematical history, rather than additional assessed complex-number tasks.

Explore the seventeen equation topics

The embedded source chart is not searchable text in the original PDF. The table below retains all seventeen topics and its listed credits/years.

Those labels are a simplified source chronology, not claims of first discovery.

The equations are illustrative forms, with physical assumptions and units needed in their respective subjects.

Two mathematical corrections are explicit: the calculus limit has one equals sign before the limit expression, and the normal density has a negative quadratic exponent and positive scale σ.

The original graphic’s extra equals sign, positive exponent and ambiguous scale glyph remain in the retained source image/transcription.

Topic 1: Pythagoras’s Theorem

Equation, with reviewed notation:

$$a^2+b^2=c^2$$

Credit/year listed by the source: Pythagoras,530 BC

Topic 2: Logarithms

Equation, with reviewed notation:

$$\log xy=\log x+\log y$$

Credit/year listed by the source: John Napier,1610

Topic 3: Calculus

Equation, with reviewed notation:

$$\frac{df}{dt}=\lim_{h\to0}\frac{f(t+h)-f(t)}h$$

Credit/year listed by the source: Newton,1668

Topic 4: Law of Gravity

Equation, with reviewed notation:

$$F=G\frac{m_1m_2}{r^2}$$

Credit/year listed by the source: Newton,1687

Topic 5: The Square Root of Minus One

Equation, with reviewed notation:

$$i^2=-1$$

Credit/year listed by the source: Euler,1750

Topic 6: Euler’s Formula for Polyhedra

Equation, with reviewed notation:

$$V-E+F=2$$

Credit/year listed by the source: Euler,1751

Topic 7: Normal Distribution

Equation, with reviewed notation:

$$\phi(x)=\frac1{\sigma\sqrt{2\pi}}e^{-\frac{(x-\mu)^2}{2\sigma^2}}$$

Credit/year listed by the source: C.F.Gauss,1810

Topic 8: Wave Equation

Equation, with reviewed notation:

$$\frac{\partial^2u}{\partial t^2}=c^2\frac{\partial^2u}{\partial x^2}$$

Credit/year listed by the source: J.d’Alembert,1746

Topic 9: Fourier Transform

Equation, with reviewed notation:

$$\widehat f(\omega)=\int_{-\infty}^{\infty}f(x)e^{-2\pi i x\omega}\,dx$$

Credit/year listed by the source: J.Fourier,1822

Topic 10: Navier-Stokes Equation

Equation, with reviewed notation:

$$\rho\left(\frac{\partial\mathbf v}{\partial t}+\mathbf v\cdot\nabla\mathbf v\right)=-\nabla p+\nabla\cdot\mathbf T+\mathbf f$$

Credit/year listed by the source: C.Navier,G.Stokes,1845

Topic 11: Maxwell’s Equations

Equation, with reviewed notation:

$$\nabla\cdot\mathbf E=0;\quad\nabla\cdot\mathbf H=0;\quad\nabla\times\mathbf E=-\frac1c\frac{\partial\mathbf H}{\partial t};\quad\nabla\times\mathbf H=\frac1c\frac{\partial\mathbf E}{\partial t}$$

Credit/year listed by the source: J.C.Maxwell,1865

Topic 12: Second Law of Thermodynamics

Equation, with reviewed notation:

$$dS\ge0$$

Credit/year listed by the source: L.Boltzmann,1874

Topic 13: Relativity

Equation, with reviewed notation:

$$E=mc^2$$

Credit/year listed by the source: Einstein,1905

Topic 14: Schrodinger’s Equation

Equation, with reviewed notation:

$$i\hbar\frac\partial{\partial t}\Psi=H\Psi$$

Credit/year listed by the source: E.Schrodinger,1927

Topic 15: Information Theory

Equation, with reviewed notation:

$$H=-\sum p(x)\log p(x)$$

Credit/year listed by the source: C.Shannon,1949

Topic 16: Chaos Theory

Equation, with reviewed notation:

$$x_{t+1}=kx_t(1-x_t)$$

Credit/year listed by the source: Robert May,1975

Topic 17: Black-Scholes Equation

Equation, with reviewed notation:

$$\frac12\sigma^2S^2\frac{\partial^2V}{\partial S^2}+rS\frac{\partial V}{\partial S}+\frac{\partial V}{\partial t}-rV=0$$

Credit/year listed by the source: F.Black,M.Scholes,1990