A number line is an idealized infinite ruler, a coordinate system for the points on the line. The point associated with the number zero is called the origin, and the distance from the origin to the point associated with the number one is the unit of the ruler. Both origin and unit are arbitrary. We draw a circle at the origin and draw a square at the point corresponding to the number one; the thick segment connecting them is one unit long.
To add two numbers we use two rulers, here blue and orange, with the same units but different origins: Drag the origin of the orange ruler to the first addend on the blue scale, then drag the gray slider to the second addend on the orange scale. Read the sum as the position of the slider on the blue scale.
0.00 + 1.80 = 1.80
Similarly, to multiply two numbers we can use two rulers with the same origins but different units: Drag the square marker of the orange ruler to the first multiplicand on the blue scale (which is the new unit of the stretched orange ruler), then drag the gray slider to the second multiplicand on the orange scale. Read the product as the position of the slider on the blue scale. Be sure to notice what happens when you drag the orange square to the left of the common origin - a negative unit sometimes makes sense.
1.00 × 1.80 = 1.8000
Addition of numbers corresponds to displacing the coordinate origin. Multiplication of numbers corresponds to changing the coordinate unit. Thus, a number may play several different roles: As a coordinate, a number represents a position. As an addend, a number represents a displacement. And as a multiplier, a number represents a change of scale.
Graph paper provides a coordinate system for the points in a plane. Our unit on the plane is a square instead of a segment; one corner goes at an arbitrary origin point, then we tile the remainder of the plane into a grid of our unit squares. Two coordinate numbers are associated with each point of the plane, the row and column numbers of the point in our grid. Generally we write the coordinates as the pair $(x, y)$, the horizontal number followed by the vertical number. However, here we write $x + iy$, although we have not yet defined either the $+$ sign or the symbol $i$ that multiplies $y$. This notation will turn out to be consistent with our upcoming definitions of "addition" and "multiplication"; until then you may regard $x + iy$ as simply an eccentric way to write $(x, y)$.
By analogy with our definitions in one dimension, we will define two dimensional "addition" as changing the origin with a fixed unit square, and "multiplication" as changing the unit square with a fixed origin.
(1.00 + i0.00) + (1.80 + i0.80) = 1.80 + i0.80
To explore this idea, we need two overlapping sheets of graph paper which initially overlay each other. We choose the lower left corner of our unit square to be the origin, drawn with a circular marker here. We also need to specify which edge of our unit square corresponds to the $x$ axis (our first coordinate); here drawn with a square marker.
To add two points in the plane, we drag the origin of the orange sheet to first addend on the blue sheet, then drag the gray diamond marker to the second addend on the orange sheet. Read the sum as the position of the gray diamond on the blue sheet. We've drawn arrows corresponding to the first and second addends (blue and orange) and to their sum (gray). Note that you are to read the gray coordinates on the blue plane, not on the orange plane.
Please convince yourself that this definition of "addition" of planar coordinate pairs amounts to simply adding each of the two components separately: $$(a+ib) + (x+iy) = (a+x) + i(b+y).$$
(1.00 + i0.00) × (1.80 + i0.80) = 1.8000 + i0.8000
Multiplication is far more interesting. We can scale our unit square by dragging its lower right corner, the one we marked with a square at the point $1+i0$, to the first multiplicand on the blue sheet. Then drag the gray diamond marker to the second multiplicand on the orange sheet. Read the product as the position of the gray diamond on the blue sheet. We've drawn arrows as for addition, but this time the orange arrow always coincides with the gray arrow, so you only see the gray arrow. The orange and gray guideline rectangles from the arrow tip back to their respective coordinate axes are more useful for reading the orange and blue coordinates, respectively.
If we drag the orange square marker directly to the left or right along the $x$-axis (to points $x+i0$ where $y=0$), we change only the scale of the unit square, and this definition of multiplication exactly matches our one dimensional interpretation of multiplication with rulers when both multiplicands have $y=0$. But our orange unit square can be tilted as well as rescaled, and dragging the orange square marker off the $x$-axis does exactly that. In other words, changing unit squares in two dimensions may rotate our coordinate system in addition to merely rescaling it -- we can orient our graph paper however we like as well as making the unit square any size we like.
Consider this labeled snapshot of the interactive picture: The the distance of point V from the origin O measured in blue units represents the ratio of the distance between any two points in blue units to the distance between the same two points in orange units, because OV has length one in orange units. Hence, OW in blue units is OW in orange units times OV in blue units: The distance of the two dimensional product from O is the ordinary product of the distances of the multiplicands from O. Furthermore, the first multiplicand, OV, is at an angle UOV up from the blue $x$-axis (OU), while the second multiplicand, OW represented by orange coordinates, is at an angle VOW up from the orange $x$-axis (OV). The product OW represented by blue coordinates is at an angle UOW up from the blue $x$-axis. Since the sum of angles OUV and VOW equals angle UOW, the angle of a two dimensional product from the $x$-axis is the sum of the angles of the multiplicands.
To summarize, our two dimensional multiplication rule is that angles from the $x$-axis add and the distances from the origin multiply.
Finally, we can justify our eccentric coordinate notation $x + iy$, by interpreting all the symbols as points and operations using our two dimensional definitions of addition and multiplication: First, interpreting the two dimensional product $iy$ as $(0 + i1)(y + i0)$, the first multiplicand $i$ rotates the point at $y$ on the $x$-axis ninety degrees, so that $iy = 0 + iy$. And under two dimensional addition, interpreting $x$ as $x + i0$ means that the sum produces exactly $x + iy$. Hence, the eccentric notation turns out to exactly match the behavior of our two dimensional arithmetic operations.
Our two dimensional numbers $x + iy$, the coordinates of points in a plane in a slightly eccentric notation, are called complex numbers. Addition corresponds to changing the coordinate origin with a fixed unit square, while multiplication corresponds to changing the unit square with a fixed origin. This is a direct generalization of ordinary arithmetic, simply replacing the unit segment of a ruler with the unit square of a sheet of graph paper. You can verify these definitions of addition and multiplication satisfy the same three properties as ordinary arithmetic, namely both operations are commutative and associative, and the distributive law of multiplication over addition applies. The ordinary numbers are embedded within the complex numbers as the line where $y = 0$, the $x$-axis.
The complex number $i = 0 + i1$ is called the imaginary unit, a terrible name which came about because it has the curious property that $i^2 = -1$, which is famously impossible for any ordinary number -- hence imaginary. For us, $i$ is just the corner of the unit square diagonally opposite the corner we labeled $1$. The $x$ coordinate is called the real part of a complex number, while the $y$ coordinate is called its imaginary part. The $x$ and $y$ axes are called the real and imaginary axes, respectively. The angle from the real axis is called the argument of a complex number, while the distance from the origin is called its modulus. Hence, to add complex numbers, we separately add their real and imaginary parts, and to multiply complex numbers, we add their arguments and multiply their moduli.
Complex multiplication by $i$ represents a ninety degree rotation of the unit square (upper sketch), while multiplication by $-1$ represents a hundred eighty degree rotation of the unit square (lower sketch). Since two ninety degree rotations produce a hundred eighty degree rotation, $i^2 = -1$ is perfectly natural in two dimensions, not imaginary at all.
With this property of $i$, we can use the distributive law to find the general formula for complex multiplication: $$(a + ib)(x + iy) = ax + iay + ibx + i^2by = (ax-by) + i(ay+bx)$$ Note that we need four ordinary multiplies and two ordinary adds to carry out one complex multiply; multiplication is more complicated than addition, because rotation mixes the $x$ and $y$ coordinates.
We have defined complex addition and multiplication to correspond to physical operations of translation, rotation, and scaling. That omits one important physical operation: reflection. Reflection does not correspond to any number of add or multiply operations in two dimensions. (In one dimension, multiplying by -1 does correspond to reflection, but in two dimensions, multiplying by -1 is a hundred eighty degree rotation.) Addition and multiplication can only produce the effects of sliding the graph paper around on the plane or of scaling its unit to a different size. Reflection corresponds to flipping the graph paper over, for example by switching the order of the $x$ and $y$ coordinates, or by multiplying just one of the coordinates by -1. In complex arithmetic, the operation of reflection through the $x$-axis, that is, of changing the sign of the $y$ coordinate, is called taking the complex conjugate. In terms of our unit squares, complex conjugation flips the unit square vertically, as shown in the sketch to the left.
The product of any complex number with its conjugate gives the square of its modulus (according to the Pythagorean theorem): $(a+ib)(a-ib) = a^2 - (ib)^2 = a^2+b^2$. Note that this relation also provides the formula for the reciprocal of a complex number, $1/(a+ib) = (a-ib)/(a^2+b^2)$. In general, the quotient of two complex numbers has the quotient of their moduli and the difference of their arguments, so this reciprocal formula produces very interesting algorithms for computing differences in angle.
The key feature of complex numbers and their arithmetic is that a complex number may represent a position (coordinates in a plane), a displacement (addend), or a change of unit (multiplicand). Complex addition and multiplication are interesting because they model the physical operations of translation, rotation, and scaling in the plane. The situation is exactly analogous to the case of real numbers, which may be coordinates, displacements, or scalings on a line. Just as you think of a number line, complex arithmetic lets you think about a number plane. As the name suggests, it gets more complicated, but you won't get bored with a number plane nearly as quickly as with a number line.
w = (1.00 + i0.00)z + (0.00 + i0.00)
Here we plot ten complex numbers whose coordinates match the pattern of the constellation Orion. The unit segment, one edge of the unit square we chose, is drawn below them. You can drag the circle origin marker to translate Orion, or the square unit marker to scale and rotate him.
We can label the original coordinate complex numbers $z_k$ for $k=0$ through $9$. (Say Rigel, Orion's right foot, is $z_0$, and Betelgeuse, his left shoulder, is $z_1$, and so on in order of decreasing brightness, ending with $z_9$ for one of the nebulae in his sword.) When we select a unit, we are selecting a complex multiplicand, say $a$, and when we select an origin, we are selecting a complex addend, say $b$. In the original coordinate system, we write $w_k$ for the coordinates of the translated, scaled, and rotated points of Orion. For each point, $w_k=az_k + b.$ We could add as many points as we please to the pattern, and accomplish the common translation of all of them with the single rule, $w=az+b.$
This complex function $w = f(z) = az + b$ maps any point in the complex $z$ plane to a new point $f(z)$ in the complex $w$ plane. We call it a linear mapping. The complex constants $a$ and $b$ represent the rotation and scaling ($a$) and translation ($b$) of the mapping. Other smooth complex functions, like $w=z^2$, distort figures in the plane, but only by causing the amount of displacement, rotation, and scaling to vary smoothly from place to place -- in small regions around nearly every point every mapping is linear. Mappings which leave small shapes unchanged except for rotation and scaling are called conformal, and their study is the theory of analytic functions. Analytic function theory remains a very active and fruitful branch of mathematics.
And all of this boils down to how well we understand graph paper and rulers.