Read complex numbers in the plane

Chapter preview — later methods: For division, multiply numerator and denominator by the denominator’s conjugate. Use both coordinates to select an argument, and state a chosen argument interval.

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.