What are imaginary numbers?? Why can’t the square root of a negative be real?
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A positive * a positive is positive, and a negative * a negative is positive. This means that anything times itself is always positive.
So: if anything times itself is *always* positive, what’s the square root of -4? It’s not -2, because -2 * -2 is 4.
In order to solve this, we came up with “the square root of -1 is a thing we’ll call ‘i'”. Now, we can say that the square root of -4 is “the square root of 4 * the square root of -1”, which gives us “2i”.
No real number, multiplied by itself (squared) will result in a negative number. 2×2 = 4. (-2)x(-2) also = 4. What real number can you square to get -4? There isn’t one.
However, it turns out that square roots of negative numbers are useful tools for a number of mathematical applications, so we still use them, but have to come up with some different term because they don’t belong to the set of real numbers. So, someone coined ‘imaginary’ as the opposite of ‘real’.
The reason that the square root of a negative number isn’t real, is because it violates the rules of mathematics.
The square root of 25 is 5. Or -5. Because 5 x 5 is 25, and -5 x -5 is 25 too, thus the square root of 25 is ±5.
We can’t take the square root of -25 because there’s no number that, when squared, yields a product of -25. You need to multiply a positive number by a negative number in order to get a negative product.
However, if we were to pretend that negative numbers had square roots, we’d get imaginary numbers. They behave in predictable ways. So we have defined a number, *i*, such that *i*^2 = −1
All of the real numbers can be placed on a line. 5 is bigger than 1, -6 is smaller than -3… But if I asked you to place an imaginary number on this line, you couldn’t right?
That’s why it is “imaginary”. No real number can be multiplied by itself so that it produces a negative number, because multiplying 2 positives (4 * 4 = 16) = positive and multiplying 2 negatives (-4 * -4 = 16) = positive. So if we want a number which, when multiplied by itself, equals a negative, we cannot find a real number, we needed to invent one: i.
We chose to define i such that i² = -1. We cannot place i on our line of real numbers -> it is imaginary.
Real and imaginary are just names someone gave to these numerical concepts long ago and it stuck. But they aren’t really “imaginary”. They just can’t be used to describe countable quantities of real things. You can’t have 4i grapes. They are a different class of numbers but in some branches of math they are just as “real” and useful as any other numbers.
I just want to add with all the excellent responses here is to not get too hung up on the terminology. Real is a defined sub set of numbers. Just like Imaginary is a defined subset of numbers.
Imaginary numbers are still real numbers ( lower case r) they are just not in the set if Real numbers. ( upper case r) also, Imaginary numbers are used all the time in electronics. They are real.
They could just be called Fancy numbers and Less Fancy numbers. They are all numbers.
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Imaginary numbers are an artefact of the conventions we adopt based on mathematical axioms we define.
Negative numbers are a convention we employ to describe certain relationships. But the logical bounds of this convention preclude the possibility that the square root of a negative number can be described within this same conventional framework.
Phenomena for which imaginary numbers offer a useful description are different than the conventions we create to deacribe phenomena.
The square root of a negative number can’t be a Real number because no Real number times itself equals a negative number. Any positive Real number times itself is a positive Real number and any negative Real number times itself is is also a positive Real number (and 0 * 0 is 0).
But, a long time ago, mathematicians were trying to solve certain kinds of problems and they realized that you had to be able to take the square root of a negative number, albeit temporarily, to work out these solutions. So apparently it was possible to do this and not break the rules of mathematics.
Since this was a new discovery that went against the common understanding of Mathematics at the time, there was some resistance and criticism to accepting these new kinds of numbers. Some mathematicians that were critical of the concept gave these numbers derogatory names, like “imaginary” and “useless.” Unfortunately, these ill-sounding names stuck.
So an Imaginary number is a number that, when multiplied by itself *does* equal a negative Real number, something Real numbers can’t do. We define the base imaginary number, *i*, such that i^(2) = -1 and all other imaginary numbers are multiples of i.
But don’t let the name confuse you, Imaginary numbers are just as “real” as Real numbers.
Suppose you had a real number, whose square is -1. Let us call this number i.
Now you have 3 options:
– i=0. Nope, then i*i=0 is not -1.
– i is positive, but then i*i is also positive, so not -1.
– i is negative, but then i*i is positive, also does not work.
Okay, so that does not work and we need something more than real numbers.
One idea is for any real number x, we can identify it with a matrix ((x,0), (0,x)), which just *scales* the plane with a factor x. Then multiplication, addition etc. for matrices just work. Additionally, we can look at the matrix that *rotates* everything by 90 degrees: ((0, -1), (1,0)). Now this does what we want, because rotating by 90 degrees twice gives you 180 degrees, which is the same as ((-1,0),(0,-1)).
So we now found a generalization (called extension) of the real numbers as matrices ((x,-y),(y,x)). For historical reasons, we call these “complex numbers” and write them as x+iy. That is it, nothing magical or “imaginary” about them, just matrices.
So why do we say “imaginary” and all that? Historically these came from trying to solve higher order polynomial equations like x^3 + a x +b =0. If a,b are “nice” you can write down a formula involving roots (of positive numbers) to find the solutions. However, these surprisingly sometimes also worked even if the roots supposedly did not make sense in calculations in between, e.g. calculations like 3 + sqrt{-2} -sqrt{-2}=3. People considered that rather strange or nonsensical at the time, but it worked, so they said “fine, we will use it since it works, but call it ‘imaginary’ because it seems weird”.
Complex numbers stayed controversial among mathematicians for some time, but people got used to them and nowadays they seem quite normal. However, the names stuck.
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