Imaginary Numbers in Real Life | Simple ExplanationHashtags|maths|geometry

追加: 2026-05-25

[00:00:00]What exactly is an imaginary number? An imaginary number, represented with the letter I, is not some weird fantasy value dreamed up by mathematicians to describe how many rainbow smiles it takes to fill up a unicorn. It instead helps fix a logical glitch caused by multiplying negative values. First, let's take a quick look at what happens when you multiply with negative values.

[00:00:23]If we start with a square with an area of one and combine three of them, we get three times one or positive three.

[00:00:31]Instead of a square, we can also work with a square hole. Since this hole is the absence of stuff, we can think of it as having a negative area.

[00:00:40]From the overall shape, the hole is removing one unit of area and therefore has an area of -1. If we combine three of these holes, we have 3 * -1, which gives us -3.

[00:00:54]As opposed to adding on more squares, we could also repeatedly take some away and in doing so, create a growing hole.

[00:01:02]Removing a square three times could be described as -3 * 1, which gives us -3.

[00:01:10]Or we could remove our holes.

[00:01:12]Removing a hole means we fill it in with a positive value. If we remove a hole three times, it means we're repeatedly adding area to the shape. We could describe this as -3 * -1, which gives us a change of positive three. Multiplying a negative by a negative gives us a flipped result of a positive answer.

[00:01:34]With this in mind, let's see how we could calculate the area of the squares in the first place. The area would be the length times the width, which are both one. This would give us 1 * 1 or 1 squared, which is positive one. But how would we calculate the area of the square hole? Since it's a hole, we know it has a negative area. The distance along the edge of the hole is one unit, but if we said the area is 1 * 1, we'd get the incorrect area of positive one.

[00:02:04]If we considered the edges of the hole to be negative one unit in length, we'd end up with an area of -1 * -1 or -1 squared. A negative times a negative creates a positive value. So again, we'd get the wrong answer of positive one.

[00:02:20]What we need to find is a value that when multiplied by itself would give us a negative result. The problem is that from what we've seen so far, no value times itself will give a negative result. A positive times a positive or a negative times a negative will both give a positive result.

[00:02:39]This can seem frustrating because it appears so basic, but the mechanics of multiplication don't seem to allow it.

[00:02:45]The solution mathematicians have found is to simply say there exists a different kind of value, neither negative or positive, but something logically different. If this kind of value, called an imaginary number, is multiplied by itself, it creates a negative value. The letter I can represent the length of the side of our square hole.

[00:03:06]I squared equals -1.

[00:03:09]Calculating the length of the edge of a square hole might seem like we're taking something simple and making it complicated, but imaginary numbers help us work with the negative values that show up in many different kinds of scenarios.

[00:03:21]If you look at movement, a force making something move faster has a positive effect. And a force slowing it down has a negative effect.

[00:03:30]Electrical currents can oscillate between positive and negative voltage.

[00:03:35]In these kinds of scenarios, it's important that the negative values don't get flipped to positive when multiplied together. So I is an essential tool to make calculations.

[00:03:45]Do imaginary numbers exist? Absolutely, because they logically need to exist in order for us to solve problems involving negative values.