I keep googling it and I still don’t understand what they are and their differences 😭😭
In: 18
To put it simply, we can create equations where there are no answers that work using our normal numbers.
To start, what do I mean by “normal numbers”? Well the math term for it is ‘real numbers’. These are numbers that you could use in every day life, and they themselves have a few different types (which I won’t go into here in detail), but these could include natural numbers like 1, 2 and 3, integers like -10, 0 and 100, rational numbers like 1/2 or 1/3 and irrational numbers like pi.
So the real numbers encompass a lot of different types of numbers, so surely they must be usable for all our equations, right? Unfortunately no, there are some equations where no real numbers will work. A basic example is the equation x^2 = -1
This equation is saying “what number can I multiply by itself to get -1?”. If you think back to the rules of positive and negative numbers, you’ll see where the issue is. If we multiply a positive number by a positive number, we get a positive number. If we multiply a negative number by a negative number, we still get a positive number. With what we know so far, the only way we can multiply two numbers together to get a negative answer is if one of the numbers is negative. But if one of the two numbers is positive and one is negative, then we can’t use that for our equation as the numbers need to be the same.
How do we solve this equation then? Based on the rules we know, it doesn’t seem like there is any real numbers that could work? And you would be right in thinking that. So mathematicians did what they often like to do, and they make something up to help fix this problem. Their solution was ‘imaginary numbers’, and these are numbers that are separate from the real numbers, so they behave differently. At their most basic level, we have an imaginary number that acts as an alternative to 1, except this number gives a negative answer when multiplied by itself. Specifically, this is a number called i, and is defined as i^2 = -1. This is the number that answers our equation from earlier, and this answer is what we call a non-real root of the equation x^2 = -1.
Although these imaginary numbers were created as sort of a band-aid to an issue we found in math, it turns out that they are incredibly useful in higher level math and engineering applications. There are many more complicated equations out there that may have a combination of real and non-real roots, and using these imaginary numbers helps solve a lot of other issues in mathematics and allows us to approach complex issues from other angles.
As a side note, one of the most profound instances of imaginary numbers is in something called Euler’s Identity, which is a formula that states if you raise the math constant e to the power of i times pi, and then add 1, you get 0. Or in math language: e^(i*pi) + 1 = 0 This simple formula is a bizarre example of how seemingly unrelated things in math can be linked together.
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