## The Useless Number

In this exciting video, the “Useless Number” refers to complex numbers, which were first thought to be useless when first discovered. However, nowadays everyone knows that complex numbers are very useful in engineering, physics and science!

Also, check out these related posts on complex numbers:

An interesting book about complex numbers is An Imaginary Tale: The Story of [the Square Root of Minus One] (Princeton Science Library). Styled like a storybook, this book tells the history of the imaginary numbers, which was discovered as early as during ancient Egypt. However, people didn’t realize the immense usefulness of complex numbers until much later. Click the image below to read more!

# What is i to the power of i?

When you first learnt that $\boxed{i=\sqrt{-1}}$, you have entered the mysterious world of complex numbers.

A mystifying question would be to ask, what is i to the power of i? Is it a complex number?

The surprising answer is that $i^i$ is a real number! Let us explain it here:

The key step is to use Euler’s formula: $\boxed{e^{i\pi}=-1}$. This has been voted as the most beautiful equation in mathematics by many people.

Then, $i=\sqrt{-1}=(-1)^{1/2}={(e^{i\pi})}^{1/2}=e^{i\pi /2}$

Hence, $i^i=e^{i^2\pi /2}=e^{-\pi /2}\approx 0.208$

It is really amazing that an imaginary number to the power of an imaginary number gives a real number, isn’t it? Leave your comments below!

## H2 Maths Tuition: Complex Numbers Notes

H2 Maths: Complex Numbers 1 Page Notes

 Modulus Argument Cartesian Form Draw diagram first, then find the appropriate quadrant and use (can use GC to double check) Polar Form Exponential Form

When question involvespowers, multiplication or division, it may be helpful toconvert to exponential form.

Please write Ƶ and 2 differently.

De Moivre’s Theorem

Equivalent to

Memory tip: Notice that arg behaves similarly to log.

Locusof z is aset of pointssatisfying certain given conditions.

in English means:The distance between (the point representing)and (the point representing)

Means the distance offromis a constant,.

So this is acircular loci.

means that the distance offromis equal to its distance from

In other words, the locus is theperpendicular bisectorof the line segment joiningand.

represents ahalf-linestarting frommaking an anglewith the positive Re-axis.

(Exclude the point (a,b) )

Common Errors

– Some candidates thought thatis the same asand thatis the same as.

– The “formula”for argumentsdoes not workfor points in the 2ndand 3rdquadrant.

– Very many candidates seem unaware that their calculators will work in radians mode and there were many unnecessary “manual” conversions from degrees to radians.