Why is the square root of -4 an imaginary number instead of -2?
In: Mathematics
Which number when multiplied to itself will give you -4?
We can simplify this even further. Remember that -4 is really (-1)*(4). So our aforementioned question now becomes a two part question.
(i) which number when multiplied to itself gives you 4? The answer is easy. It is 2.
(ii) which number when multiplied to itself gives -1? We can’t solve it. This is because when a negative number is multiplied to another negative number, you get a positive number as a result. Since we don’t have any answer but we need an answer, we give the answer a name. We call it the imaginary number. In short the ‘i’.
So now putting the answers from (i) and (ii) above, we get 2i as the square root of -4.
The way my maths teacher described it to me is that if you take a set of axes and draw a 2×2 square in the top right quadrant its area is 4. And if you mirror it diagonally into the lower left quadrant it’s -2 x -2. But it’s area is the same as the first square you drew. So the areas are both 4. Meaning the square of a negative must be positive as well. Therefore in order to have the square of a number be a negative we have to create a new world where the root could be negative in order to get on and deal with the rest of the problem. Hence why we refer to it as an imaginary number.
(-2)^2 = (-2)*(-2) = 4
So -2 cant be the square root of -4. Because -2 times -2 is 4.
Essentially, with imaginary numbers, mathematicians “created” a solution to the sqrt of negative numbers. By defining a number i, which has the property that i*i = -1, you can now solve any square roots, in terms of that number i.
That alone doesnt sound very exciting, but it turns out that i has the nice property of going in circles if multiplied with itself (e.g. i = i, i*i = 1, i*i*i = 1*i = 1), and that we can express trigonometric functions with it, which makes them very useful in describing things which rotate and/or repeat itself, such as waves, alternating currents, etc.
Because a negative times a negative is a positive, and a positive times a positive is a positive, therefore, without the square root of negative one (the imaginary number [i] in this case), it would be mathematically impossible to reach -4 by squaring any *real* number.
The square of “-2” isn’t “-4”, but instead it is “4”. Thus, “-4” doesn’t have a real square root
Have you tried multiplying -2 by -2? Hopefully you did, and you would see that this equals 4 and not -4. That’s really the only answer you need. If you want to know why it doesn’t equal -4 well now you’ve gone down the rabbit whole of asking why math is math, and the short answer is because it is a system that when used in the real world, *actually works to solve problems*
These two questions are the same:
– What is the square root of -4?
– What number, when multiplied by itself, is -4?
If you multiply -2 by -2, you get 4 (that is, positive 4).
That’s why it’s not -2.
Why does it have to be a different kind of number? Well:
– Zero times zero is zero. It can’t be zero.
– Positive times positive is positive. It can’t be positive.
– Negative times negative is negative. It can’t be negative.
So at some point, someone said “What if invent had something that, when multiplied by itself, gives us a negative result?” That’s basically where imaginary numbers come from.
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