How can imaginary numbers be used

But think of it this way. You could owe me five apples, or five dollars. Once people started doing accounting and bookkeeping, we needed that concept. Another way to look at negative numbers — and this will come in handy later — is to think of walking around in a city neighborhood, Moore says. If you make a wrong turn and in the opposite direction from our destination — say, five blocks south, when you should have gone north — you could think of it as walking five negative blocks to the north.

Imaginary numbers and complex numbers — that is, numbers that include an imaginary component — are another example of this sort of creative thinking. As Moore explains it: "If I ask you, what is the square root of nine, that's easy, right? The answer is three — though it also could be negative three," since multiplying two negatives results in a positive.

But what is the square root of negative one? Is there a number, when multiplied by itself, that gives you in negative one? But Renaissance mathematicians came up with a clever way around that problem. Let's give it a name. Once they came up with the concept of an imaginary number, mathematicians discovered that they could do some really cool stuff with it. Remember that multiplying a positive by a negative number equals a negative, but multiplying two negatives by one another equals a positive.

But what happens when you start multiplying i times seven, and then times i again? Because i times i is negative one, the answer is negative seven. But if you multiply seven times i times i times i times i , suddenly you get positive seven. Now think about that. You took an imaginary number, plugged it into an equation multiple times, and ended up with an actual number that you commonly use in the real world.

It wasn't until few hundred years later, in the early s, that mathematicians discovered another way of understanding imaginary numbers, by thinking of them as points on a plane, explains Mark Levi.

It is a strange world, where squares can be negative, but one whose structure is very similar to the real numbers we are so familiar with.

And this extension to the real numbers was just the beginning. The quaternions are structured like the complex numbers, but with additional square roots of —1, which Hamilton called j and k.

For instance, will the system be closed under multiplication? Will we be able to divide? Hamilton himself struggled to understand this product, and when the moment of inspiration finally came, he carved his insight into the stone of the bridge he was crossing:.

People from all over world still visit Broome Bridge in Dublin to share in this moment of mathematical discovery. The other products can be derived in a similar way, and so we get a multiplication table of imaginary units that looks like this:. Notice this means that, unlike with the real and complex numbers, multiplication of quaternions is not commutative.

Multiplying two quaternions in different orders may produce different results! To get the kind of structure we want in the quaternions, we have to abandon the commutativity of multiplication.

This is a real loss: Commutativity is a kind of algebraic symmetry, and symmetry is always a useful property in mathematical structures. But with these relationships in place, we gain a system where we can add, subtract, multiply and divide much as we did with complex numbers.

To add and subtract quaternions, we collect like terms as before. To multiply we still use the distributive property: It just requires a little more distributing. And to divide quaternions, we still use the idea of the conjugate to find the reciprocal, because just as with complex numbers, the product of any quaternion with its conjugate is a real number. Thus, the quaternions are an extension of the complex numbers where we can add, subtract, multiply and divide.

And like the complex numbers, the quaternions are surprisingly useful: They can be used to model the rotation of three-dimensional space, which makes them invaluable in rendering digital landscapes and spherical video, and in positioning and orienting objects like spaceships and cellphones in our three-dimensional world.

And just as with the quaternions, we need some special rules to govern how to multiply all the imaginary units. As in the representation for the quaternions, multiplying along the direction of the arrow gives a positive product, and against the arrow gives a negative one. Like the quaternions, octonion multiplication is not commutative. But extending our idea of number out to the octonions costs us the associativity of multiplication as well.

And that is also how the name " Real Numbers " came about real is not imaginary. Those cool displays you see when music is playing? Yep, Complex Numbers are used to calculate them! Using something called "Fourier Transforms".

In fact many clever things can be done with sound using Complex Numbers, like filtering out sounds, hearing whispers in a crowd and so on. AC Alternating Current Electricity changes between positive and negative in a sine wave.