What is a Complex Number?

The history of complex numbers begins, as with many stories in mathematics, with the ancient Greeks. They were the first to try and find solutions to polynomials, and they discovered that none of the numbers they knew of could solve ones like $x^2+1=0$x2+1=0. These sorts of polynomials arose from seemingly solvable, physical problems - a famous one was proposed by Diophantus of Alexandria (AD 210 – 294 approx):

“A right-angled triangle has a perimeter of $12$12 units and an area of $7$7 squared units. What are the lengths of its sides?”

Here is such a triangle:

Letting $AB=x$AB=x and $AC=h$AC=h, the area of the triangle is then given by

$\text{Area }=\frac{1}{2}xh$Area =12​xh

and the perimeter is 

$\text{Perimeter }=x+h+\sqrt{x^2+h^2}$Perimeter =x+h+√x2+h2

Try this yourself before revealing the solution:

Does this polynomial have real solutions? What does that mean for Diophantus' triangle?

Another mathematician named Jerome Cardan (1501 - 1576) also thought about problems like these.  He tried to solve the problem of finding two numbers, $a$a and $b$b, whose sum is $10$10 and whose product is $40$40:

$a+b=10$a+b=10, $ab=40$ab=40

Eliminating $b$b gives $a\left(10-a\right)=40$a(10−a)=40, which expands to $a^2-10a+40=0$a2−10a+40=0. Solving this quadratic would give the solutions:

$a=\frac{1}{2}\left(10\pm\sqrt{60}\right)=5\pm\sqrt{-15}$a=12​(10±√60)=5±√−15

But since there are no real numbers whose square is $-15$−15, the term $\sqrt{-15}$√−15 has no meaning in the real number system - we say there are no real solutions to the problem Cardan posed. That said, if we treat the "numbers" $a=5+\sqrt{-15}$a=5+√−15 and $b=5-\sqrt{-15}$b=5−√−15 like ordinary numbers, they are solutions! They sum to $10$10:

$a+b=\left(5+\sqrt{-15}\right)+\left(5-\sqrt{-15}\right)=\left(5+5\right)+\left(\sqrt{-15}-\sqrt{-15}\right)=10+0=10$a+b=(5+√−15)+(5−√−15)=(5+5)+(√−15−√−15)=10+0=10

... and they multiply to $40$40:

$ab=\left(5+\sqrt{-15}\right)\left(5-\sqrt{-15}\right)=5^2+5\times\sqrt{-15}-5\times\sqrt{-15}-\left(\sqrt{-15}\right)^2=25-\left(-15\right)=40$ab=(5+√−15)(5−√−15)=52+5×√−15−5×√−15−(√−15)2=25−(−15)=40

In each instance the $\sqrt{-15}$√−15 term disappeared, cancelling nicely as we performed the algebra. So everything in mathematics still works, and most mathematicians of the time thought this was a silly, slightly eerie "trick". It wasn’t until the nineteenth century that the power of these sorts of solutions began to be fully understood. 

The advent of the complex numbers

While complex numbers were created almost by accident when solving a series of abstract questions, they have ended up being critical to a broad range of applications in contemporary mathematics.  Wi-fi systems, telephone networks, electrical circuits, electromagnetism, any areas that use physics and differential equations together and more all rely heavily on complex numbers.  

The other key area is in any field that relies on the theory of self-similarity, commonly referred to as fractals. It is a very modern field, requiring enormous computing power to even represent, and the results are stunningly beautiful. Computer-generated imagery in movies and games have complex numbers as a foundational cornerstone. 

This fractal is the famous Mandelbrot Set.

Definitions

Complex numbers are built on the concept that there is an object, called $i$i, that is the square root of $-1$−1.

$i=\sqrt{-1}\equiv i^2=-1$i=√−1≡i2=−1

This is not a real number, since the square of any real number is always positive. But using this single object we can define an entire new dimension for the number line.

In the previous section, we came across the value $5+\sqrt{-15}$5+√−15. Rewriting this as $5+\sqrt{-1\times15}=5+\sqrt{-1}\times\sqrt{15}$5+√−1×15=5+√−1×√15 then allows us to express it more concisely:

$5+\sqrt{-15}=5+i\sqrt{15}$5+√−15=5+i√15.

Instead of inventing a new symbol for "the square root of $-15$−15", we just re-use the symbol $i$i from before. Generally, all complex numbers $z$z can be written in the form $z=x+iy$z=x+iy, where $x$x and $y$y are real (and therefore familiar) numbers. And just like $$ is used to denote the set of all real numbers, we use the symbol $$ to denote all the complex numbers.

So a number like $5+3i$5+3i is a complex number.  It has both real and imaginary components. 

 

The real part of $z$z is $Re\left(z\right)=x$Re(z)=x, and the imaginary part of $z$z is $Im\left(z\right)=y$Im(z)=y.

So for $5+3i$5+3i, $Re\left(5+3i\right)=5$Re(5+3i)=5 and $Im\left(5+3i\right)=3$Im(5+3i)=3.

Every real number $x$x can be written as $x+i*0$x+i*0, which means every real number is also a complex number - in other words, the set of real numbers is a subset of the set of complex numbers.  

Writing numbers in complex form

To be able to write some numbers in complex form, a little algebraic manipulation may be necessary, mostly involving the fact that $\sqrt{-1}=i$√−1=i or that $-1=i^2$−1=i2.

Example 1

Rewrite  $\sqrt{-50}$√−50 in the form $z=x+iy$z=x+iy

$\sqrt{-50}$√−50	$=$=	$\sqrt{-1\times50}$√−1×50	using properties of surds: $\sqrt{ab}=\sqrt{a\times b}=\sqrt{a}\times\sqrt{b}$√ab=√a×b=√a×√b
	$=$=	$\sqrt{-1}\times\sqrt{50}$√−1×√50
	$=$=	$i\sqrt{50}$i√50	and now simplify the $\sqrt{50}$√50: $\sqrt{50}=\sqrt{25\times2}=\sqrt{25}\times\sqrt{2}=5\sqrt{2}$√50=√25×2=√25×√2=5√2
	$=$=	$i5\sqrt{2}$i5√2

So in the form $x+iy$x+iy, the real component is $x=0$x=0 and the imaginary component is $y=5\sqrt{2}$y=5√2, so $x+iy=0+i5\sqrt{2}$x+iy=0+i5√2.

Practice questions

QUESTION 1

QUESTION 2

QUESTION 3