Why is i the square root of -1?

Why is “i” the square root of -1? How can there even be a square root of a minus number? Any minus number multiplied by itself gives a positive result, every schoolchild knows that! So how could anything squared ever be negative? To find out, we're going to go back to basics and ask what are numbers?

Maths in general, and numbers in particular, are very abstract concepts. But we all learned numbers as young children. How do you teach children abstract concepts? You have to apply them to something they can see and touch. So we take everyday objects, such as wooden blocks, and ask them to count them.

As they get older, we teach them addition and subtraction.

"If you have six blocks and you give two of them to me, how many blocks will you have left?" etc...

As they get older still, we teach them multiplication and division.

"If I have two blocks and you have four, what is the ratio of the number of blocks you have compared to the number of blocks I have?" etc...

That's all fine and covers everything they need to know about maths, at least while they are young. But then we hit a problem.

"If I only have two blocks and you want me to give you three blocks... uhh..... oh!"

So this way of teaching them numbers only covers the positive integers.

By the time they reach this age, you might have started giving them pocket money, and all of the sudden, our budding little mathematicians discover that positive numbers are not the only numbers out there.

"Dad?.... Could you eh... possibly lend me some money???"

The moment money is involved, we encounter the concept of debt. Therefore we need a new set of numbers, negative numbers, to represent the idea of money owed.

Counting and money are not the only ways we can apply the abstract concept of maths and numbers to the real world. For example, we could also use geometry.?

In the same way, as there are four mathematical operators: Addition, Subtraction, Multiplication & Division, there are also four geometric transformations: Translation, Rotation, Scaling & Reflection.

The four mathematical operators can be applied in the world of geometry to perform the four transformations.

Addition and subtraction translate an object. For example, addition can move the object to the left and subtraction can move it back to the right again.

Multiplication and division can be used to scale an object. Multiplication makes it larger and division makes it smaller.

Multiplication can also reflect an object if we multiply that object by a special number. That special number is -1.

What about squaring a number? Whenever we square a number, we could think of it as starting out with the unit square (that’s a square whose length and height are both 1) and applying the same multiplication twice to that unit.

So, if I want to represent 22, I scale the unit square by 2 and then by 2 again.

What happens if I wanted to represent -22? Again, we start with our unit square, only this time I am scaling it by -2. The minus sign reflects the object it and the 2 scales it. Then I have to do the same again. The minus sign reflects the object it and the 2 scales it.

The two reflections mean that I end up with exactly the same shape as I did when I squared positive 2. This is why any negative number squared always gives a positive result.

If this is true, how can anything squared ever equal -1? In other words, if we ask the same question in a geometric way: what transformation can we possibly perform twice that will result in a single reflection?

Scaling doesn't help us and neither does reflection as the squaring operation always ends up reflecting the object back into the domain of positive numbers. Translation can only be done with addition or subtraction, so that doesn't help either.

What's left?

Only rotation.

I can rotate the object once by 90° then again by another 90°. The same transformation, repeated twice just as the squaring operation requires.

But in the diagram above, my object disappeared on the first 90° rotation. Where did it go?

We're used to living in a 4-dimensional world. The position of any object in our world can be described by 3 spatial coordinates: X, Y, and Z, and a 4th coordinate which is time.

However, in the abstract world of maths, we can have as many dimensions as we like. The only problem is, we sometimes have trouble representing them when we try to draw them on a 2-dimensional piece of paper.

So as we only have 3 spatial dimensions to play with, I am going to have to borrow one of those dimensions to represent this strange new dimension.?

Now we can see where my object went in that first 90° rotation. It rotated into a new dimension which we couldn't see on my two-dimensional diagram. However, once we draw the same diagram in 3 dimensions, we can see it. However, this third dimension is not the regular third dimension that we're all familiar with from our everyday 3D world. I've only drawn it as such here so that I can represent it on a diagram. This new dimension is something else entirely.

For centuries, mathematicians suspected that this new dimension might exist. However, they were reluctant to believe it.? René Descartes, the 17th-century philosopher, and mathematician, who gave us the Cartesian coordinate system, wrote about it in 1637 and was rather derogatory about the idea. He disliked it so much, that he called it the imaginary dimension, for he too was hard-pressed to believe in its existence.

It wasn't until the nineteenth century and the work of Leonhard Euler on his famous identity, that the imaginary dimension became more widely accepted. You can find out more about Euler in a video I've produced all about Euler's Identity.

Any number in the imaginary dimension is known as an imaginary number and is given the symbol “i”. So just as we have 1, 2, 3, etc. on the x, y, and z axes, we have “i”, “2i”, “3i” etc. on the imaginary axis.

In geometric terms, just as multiplying by -1 can be thought of as a reflection, so multiplying by “i” can be thought of as a rotation around the imaginary axis.

So if I want to use a squaring operation to transform my unit square so that its length is -1, I have to multiply it by “i” to rotate it by 90°, and then by “i” again to rotate it by another 90°.?

Therefore “i2” is a rotation of 180° around the imaginary axis which equals -1 and that is why “i” is the square root of -1.

Now you might think that imaginary numbers are just a load of abstract mathematical twaddle. If so, you're in good company as most of the eminent mathematicians who were born before the 19th century would agree with you. However, as I said at the beginning of the article, numbers are an abstract concept until you apply them to the real world where it turns out that imaginary numbers can be extremely useful. They enable us to solve all kinds of problems that were either difficult or impossible to solve before their discovery.?

For example, the 90° rotation of “i” makes them ideal for use with sines and cosines in very important algorithms like the Fourier Transform. If you'd like to find out how, please visit my online course on How the Fourier Transform works.