## 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! Mandelbrot Set

## 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 of from is a constant, .

So this is acircular loci.

Centre: , radius =   means that the distance of from is equal to its distance from In other words, the locus is theperpendicular bisectorof the line segment joining and .  represents ahalf-linestarting from making an angle with the positive Re-axis.

(Exclude the point (a,b) ) Common Errors

– Some candidates thought that is the same as and that is 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.