Eli5: What are real and non real roots?

Question

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

In: 18

Answer 1

YouTube channel Welch Labs has [an incredible series of short, ELI5-level videos](https://www.youtube.com/playlist?list=PLiaHhY2iBX9g6KIvZ_703G3KJXapKkNaF) on exactly this topic. I recommend you watch it.

Here’s some quick vocabulary:

– Expression: A calculation you can do if you know the values of all your variables.
– Polynomial: A certain kind of expression that’s a sum of terms, where each term is a constant times a variable raised to a whole-number power. Example: 2x^3 + 3x^2 – 14x – 15.
– Root: A number that makes an expression equal zero. Example: x = 2.5 makes the above expression 0, it turns into 31.25 + 18.75 – 35 – 15 which is 0.
– Real number: Numbers that are on the number line.

Most of the numbers you encounter in your life are real numbers. Examples of real numbers:

– The counting numbers (1, 2, 3, …) are real numbers
– Zero is a real number
– Fractions (1/3, 7/9, 1043/221) are real numbers
– Irrational numbers like π, e or √2 that can’t be expressed as fractions
– Negative numbers (-1, -2, -11/15)

There’s another number line perpendicular to the real number line. The numbers on that second number line are called “imaginary numbers.” The imaginary numbers are based on multiplying real numbers by √-1 or i for short.

So if you have an equation like x^2 = 4, you have two real solutions (x = 2 and x = -2). These correspond to the two real roots of the expression x^2 – 4. (To find this expression, you subtract one side of the equation to get something = 0, then the roots — the things that make the expression = 0 — are solutions to the equation.)

If you have an equation like x^2 = -4, there are no real solutions (you can’t square a real number and get a negative number). However there there are two imaginary solutions (x = 2i and x = -2i). These values (2i and -2i) are non-real roots of the expression x^2 + 4, just like 2 and -2 are real roots of x^2 – 4.

By adding real numbers and imaginary numbers you can form an entire plane called the complex plane. So the non-real roots might be complex, for example 2+3i and 2-3i are the roots of x^2 – 4x + 13.