# Complex Number A **complex number** is a number that can be expressed in the form `a + b * i`, where `a` and `b` are real numbers, and `i` is a solution of the equation `x^2 = −1`. Because no *real number* satisfies this equation, `i` is called an *imaginary number*. For the complex number `a + b * i`, `a` is called the *real part*, and `b` is called the *imaginary part*. ![Complex Number](https://www.mathsisfun.com/numbers/images/complex-example.svg) A Complex Number is a combination of a Real Number and an Imaginary Number: ![Complex Number](https://www.mathsisfun.com/numbers/images/complex-number.svg) Geometrically, complex numbers extend the concept of the one-dimensional number line to the *two-dimensional complex plane* by using the horizontal axis for the real part and the vertical axis for the imaginary part. The complex number `a + b * i` can be identified with the point `(a, b)` in the complex plane. A complex number whose real part is zero is said to be *purely imaginary*; the points for these numbers lie on the vertical axis of the complex plane. A complex number whose imaginary part is zero can be viewed as a *real number*; its point lies on the horizontal axis of the complex plane. | Complex Number | Real Part | Imaginary Part | | | :------------- | :-------: | :------------: | --- | | 3 + 2i | 3 | 2 | | | 5 | 5 | **0** | Purely Real | | −6i | **0** | -6 | Purely Imaginary | A complex number can be visually represented as a pair of numbers `(a, b)` forming a vector on a diagram called an *Argand diagram*, representing the *complex plane*. `Re` is the real axis, `Im` is the imaginary axis, and `i` satisfies `i^2 = −1`. ![Complex Number](https://upload.wikimedia.org/wikipedia/commons/a/af/Complex_number_illustration.svg) > Complex does not mean complicated. It means the two types of numbers, real and imaginary, together form a complex, just like a building complex (buildings joined together). ## Polar Form An alternative way of defining a point `P` in the complex plane, other than using the x- and y-coordinates, is to use the distance of the point from `O`, the point whose coordinates are `(0, 0)` (the origin), together with the angle subtended between the positive real axis and the line segment `OP` in a counterclockwise direction. This idea leads to the polar form of complex numbers. ![Polar Form](https://upload.wikimedia.org/wikipedia/commons/7/7a/Complex_number_illustration_modarg.svg) The *absolute value* (or modulus or magnitude) of a complex number `z = x + yi` is: ![Radius](https://wikimedia.org/api/rest_v1/media/math/render/svg/b59629c801aa0ddcdf17ee489e028fb9f8d4ea75) The argument of `z` (in many applications referred to as the "phase") is the angle of the radius `OP` with the positive real axis, and is written as `arg(z)`. As with the modulus, the argument can be found from the rectangular form `x+yi`: ![Phase](https://wikimedia.org/api/rest_v1/media/math/render/svg/7cbbdd9bb1dd5df86dd2b820b20f82995023e566) Together, `r` and `φ` give another way of representing complex numbers, the polar form, as the combination of modulus and argument fully specify the position of a point on the plane. Recovering the original rectangular co-ordinates from the polar form is done by the formula called trigonometric form: ![Polar Form](https://wikimedia.org/api/rest_v1/media/math/render/svg/b03de1e1b7b049880b5e4870b68a57bc180ff6ce) Using Euler's formula this can be written as: ![Euler's Form](https://wikimedia.org/api/rest_v1/media/math/render/svg/0a087c772212e7375cb321d83fc1fcc715cd0ed2) ## Basic Operations ### Adding To add two complex numbers we add each part separately: ```text (a + b * i) + (c + d * i) = (a + c) + (b + d) * i ``` **Example** ```text (3 + 5i) + (4 − 3i) = (3 + 4) + (5 − 3)i = 7 + 2i ``` On complex plane the adding operation will look like the following: ![Complex Addition](https://www.mathsisfun.com/algebra/images/complex-plane-vector-add.svg) ### Subtracting To subtract two complex numbers we subtract each part separately: ```text (a + b * i) - (c + d * i) = (a - c) + (b - d) * i ``` **Example** ```text (3 + 5i) - (4 − 3i) = (3 - 4) + (5 + 3)i = -1 + 8i ``` ### Multiplying To multiply complex numbers each part of the first complex number gets multiplied by each part of the second complex number: Just use "FOIL", which stands for "**F**irsts, **O**uters, **I**nners, **L**asts" ( see [Binomial Multiplication](ttps://www.mathsisfun.com/algebra/polynomials-multiplying.html) for more details): ![Complex Multiplication](https://www.mathsisfun.com/algebra/images/foil-complex.svg) - Firsts: `a × c` - Outers: `a × di` - Inners: `bi × c` - Lasts: `bi × di` In general it looks like this: ```text (a + bi)(c + di) = ac + adi + bci + bdi^2 ``` But there is also a quicker way! Use this rule: ```text (a + bi)(c + di) = (ac − bd) + (ad + bc)i ``` **Example** ```text (3 + 2i)(1 + 7i) = 3×1 + 3×7i + 2i×1+ 2i×7i = 3 + 21i + 2i + 14i^2 = 3 + 21i + 2i − 14 (because i^2 = −1) = −11 + 23i ``` ```text (3 + 2i)(1 + 7i) = (3×1 − 2×7) + (3×7 + 2×1)i = −11 + 23i ``` ### Conjugates We will need to know about conjugates in a minute! A conjugate is where we change the sign in the middle like this: ![Complex Conjugate](https://www.mathsisfun.com/numbers/images/complex-conjugate.svg) A conjugate is often written with a bar over it: ```text ______ 5 − 3i = 5 + 3i ``` On the complex plane the conjugate number will be mirrored against real axes. ![Complex Conjugate](https://upload.wikimedia.org/wikipedia/commons/6/69/Complex_conjugate_picture.svg) ### Dividing The conjugate is used to help complex division. The trick is to *multiply both top and bottom by the conjugate of the bottom*. **Example** ```text 2 + 3i ------ 4 − 5i ``` Multiply top and bottom by the conjugate of `4 − 5i`: ```text (2 + 3i) * (4 + 5i) 8 + 10i + 12i + 15i^2 = ------------------- = ---------------------- (4 − 5i) * (4 + 5i) 16 + 20i − 20i − 25i^2 ``` Now remember that `i^2 = −1`, so: ```text 8 + 10i + 12i − 15 −7 + 22i −7 22 = ------------------- = -------- = -- + -- * i 16 + 20i − 20i + 25 41 41 41 ``` There is a faster way though. In the previous example, what happened on the bottom was interesting: ```text (4 − 5i)(4 + 5i) = 16 + 20i − 20i − 25i ``` The middle terms `(20i − 20i)` cancel out! Also `i^2 = −1` so we end up with this: ```text (4 − 5i)(4 + 5i) = 4^2 + 5^2 ``` Which is really quite a simple result. The general rule is: ```text (a + bi)(a − bi) = a^2 + b^2 ``` ## References - [Wikipedia](https://en.wikipedia.org/wiki/Complex_number) - [Math is Fun](https://www.mathsisfun.com/numbers/complex-numbers.html)