Eli5: What are real and non real roots?

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I keep googling it and I still don’t understand what they are and their differences 😭😭

In: 18

A root of an equation is a number where that equation is equal to zero. Some expressions, like x^2 – 4, have real roots. In this case the expression is zero when x = 2 or x = -2.

Other expressions, like x^2 + 4 have no real numbers that can make them equal zero. There are, however, complex numbers that can. Recall a complex number is a number that includes the imaginary unit i, where i = sqrt(-1). We can set x = 2i, and that means x^2 = -4. So the expression x^2 + 4 becomes -4 + 4 = 0 and we say that the expression has a complex root at x = 2i.

A root of a number X is finding the number that you have to multiply with itself to find X. The most common is the square root, where you have to multiply a number with itself once to find the target X. The square root of 4 is 2 (and -2) because 2×2=4. (You can of course have ‘higher’ roots.) The difference arises when you try to get a root of a target number X that is negative. Let’s say you want the square root of -4. Is it 2? No because 2×2 is not -4. Is it -2? No because -2x-2=4.
Damn.
The solution to this are non-real numbers, namely imaginary numbers, which is a very difficult topic I know nothing about and is out of the scope of ELI5.
This means that the square root of -4 is non real, i.e. a non-real root.

‘Real’ is a math definition. It basically means the regular normal numbers you know

‘Imaginary’ is a math definition. It sort of expands the Real numbers.

‘complex’ is a math definition. It’s the combination of Real and Imaginary. It’s sort of 2-dimensional numbers. One axis being the real and the other axis being the Imaginary. Look up some pictures of a complex plane.

To understand that you first need to understand what a real number is.

A natural number is all the numbers you can count. (1,2,3…).

A whole number does include the negative numbers and zero.

A rational number is any number that can be written as the ratio of two whole numbers. For example 1/3 or 22/7.

A real number is any number on the number line. Famous examples are pi and e, but also numbers like root(2) are real numbers but not rational numbers.

When taking the root you are asking “What number can I multiply with itself to get the number under the root?”.

If you are multiplying a positive number with itself you will get a positive result. If you multiply a negative number with itself it will also be a positive number (negative*negative=positive).

The answer to root(-1) seems impossible to find. And when you are looking at the real numbers it is. So mathematicians invented complex numbers. They said that root(-1)=i. You are no longer dealing with the number line but with a number plane.

When you take the root of any negative number you can get an answer on the number plane. There is a complex number (also known as an imaginary number, that might be where i is from).

Complex numbers are very useful in things like electronics, but in some fields you don’t want to look for complex answers, so you just say that this root doesn’t have an answer in the real numbers and you leave it at that.

If you ever see a big electric motor and you look at it’s plate you will see cos(phi) on there. That phi is the angle how far the apparent power is off the number line in complex math.

Real roots have no imaginary component, non real roots can have imaginary components

An example. The 4th roots of 1. You can multiply 1•1•1•1 to get 1. You can also multiply -1•-1•-1•-1 to get 1. Those are real roots.

Then there’s also i•i•i•i = 1 and -i•-i•-i•-i = 1 which are non real roots

An easy way to solve for all of them is polar form. Say you want to find the cube roots of 1. Well the polar form is 1∠0 which is equal to 1∠360 which is equal to 1∠720 (you can keep going, but you’ll just get repeated answers in further steps since we’re using the cube root which will have 3 roots)

Take those, and since it’s the cube root which is the third power, divide all of those angle by 3. You’ll end up with 1∠0 ; 1∠120 ; and 1∠240. If the angle is 0 or 180 it is a real root, if it is 90 or 270 it is a purely imaginary root, and anything else is a complex root. The way you get the final answer is converting the polar coordinates back to rectangular. So 1∠0 becomes 1•(cos(0) + sin(0) • i) = 1 {real root}; 1∠120 becomes 1•(cos(120) + sin(120) • i) = -0.5 + 0.866i; And 1∠240 becomes 1•(cos(240) + sin(240) • i) = -0.5 – 0.866i {both complex roots}

The reason it would repeat if you kept going is because if you go to 1∠1080 and divide 1080 by 3, you get 360 and 1∠0 would be equal to 1∠360 and just give you 1 again as a final answer