What makes an imaginary number

But then people researched them more and discovered they were actually useful and important because they filled a gap in mathematics 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. And it may be a little surprising that this number is — i!

This means that if we ever want to divide a number by i , we can just multiply it by — i instead. For other complex numbers, the arithmetic may get a little harder, but the reciprocal idea still works. The product of the complex number and its conjugate is a real number!

This property of conjugates helps us compute the reciprocal of any complex number. The introduction of this one new non-real number — i , the imaginary unit — launched an entirely new mathematical world to explore.

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. This cycle will continue through the exponents, also known as the imaginary numbers chart.

Knowledge of the exponential qualities of imaginary numbers is useful in the multiplication and division of imaginary numbers. After grouping the coefficients and the imaginary terms, the rules of exponents can be applied to i while the real numbers are multiplied as normal. The same is done with division. By applying the usual multiplication and division rules, imaginary numbers can be simplified as you would with variables and coefficients. Imaginary numbers have also made an appearance in pop culture.

Complex number calculator.