Yarnell's World of Math

---

---

---

unique visitors since March 2009

---

FP1: Complex Numbers

---

The rich tapestry of numbers that forms our history and future are all closely linked to reality. To make a fraction, just cut an apple into quarters; to see pi, just look at a circle. All numbers are entwined in our physical world in this way. But there is one number that has a more complicated relationship with reality. This is an imaginary number. Despite its name, it is very real indeed. [from The Book Of Numbers by Peter J. Bentley]

Definition of Complex Numbers

What number, when multiplied by itself gives 2? The answer is √2. The "√" symbol was invented to solve such equations as x² = 2.

What number, when multiplied by itself gives -1?

You should already know that there is no such number. But hold on... Pythagoras and his mates refused to accept surds like root 2, but we accept them nowadays. Why shouldn't this question have an answer? Mathematicians invented an answer to the question and called it "i" - the imaginary number.

So i² = -1

Italian mathematician and quack Gerolamo Cardan came up with a problem: Divide 10 into 2 equal parts, the product of which is 40. Have a bash at solving this.

The puzzle has no answer, of course. Half of 10 is 5 and 5² is 25.

Actually, you're (almost) wrong. The puzzle has no real solution, but that doesn't mean that there is no solution at all.

Cardan answered his own puzzle: "divide 10 into 2 equal parts makes 5. But 5×5 = 25. Now, 25-40 = -15, so we need to add and subtract √-15 to 5 to get the solution."

Check this by working out what (5 + √-15) × (5 - √-15) is. Although these two numbers clearly do add to 10 and also multiply to 40, it is not clear whether they can be considered equal parts. We'll learn more about this idea (of equality) later.

We could write the solution to the puzzle as 5 + 15i and 5 - 15i. This are complex numbers.

A complex number has both a real part and an imaginary part. In the complex number 5 + 15i, 5 is the real part and 15 is the imaginary part.

Arithmetic of Complex Numbers

Addition and Subtraction

These are easy! Just add/subtract the real bits and add/subtract the imaginary bits...

eg. (2 + 3i) + (5 - 2i) = (7 + i)

Multiplication

This works just like multiplying brackets, so (2 + 3i) × (5 - 2i) = 10 + 11i -6i².

But i² = -1, so this is 16 + 11i. Easy!

Division

This is a wee bit trickier. Think of it like Rationalising the denominator.

So (2+3i)/(5-2i) = (2+3i)(5+2i) / (5-2i)(5+2i) = (4+11i) / 29 = 4/29 + 11/29 i.