Is the number 5 imaginary? Look at the 5 fingers on your left hand. You don't need to imagine them, you can see them. So unless all that we experience is a big dream, 5 probably isn't imaginary, it exists.
Now, how about -1, is it imaginary? We don't see -1 on our hands, in the forest, or in the street. So maybe -1 is imaginary. Is it?
Adding -1 to a number is another way for describing subtraction of 1. Look for example at the equation,
7 + Z = 6.
What is the solution?
What number do we need to add to 7 to get 6? The answer is clearly Z = -1. So is -1 imaginary? If you understand the equation 7 + Z = 6, then -1 probably isn't imaginary for you. It exists as the solution of that equation.
Have you tried exploring negative numbers with youngsters? Negatives add excitement to the arena of early mathematics, especially at first encounters. Next time you get to spend time with kids ask them what they think about negative numbers. Responses are always very interesting.
In fact, can you remember the first time you thought about negative numbers? This memory is probably not as vivid as the first time you rode a bicycle, touched snow or swam in the sea, but maybe it is. Can you recall?
OK, so we assume you accept -1. Imaginary or not, it is part of life. But, there is another number in mathematics that is often formally called the imaginary number. It is denoted by i, yes, the lower case letter "i" (electrical engineers use "j" for it instead).
As opposed to -1, something that you surely occasionally use, i is in many ways less common in the day to day life of a non-specialist. It may be something that teenagers are exposed to in their mathematics education, but unlike -1, it doesn't make it often to the dinner table conversation at home. Why not actually?
The number i is often used in physics, signal processing, electrical engineering, statistics and many other fields. Experts working in all these fields don't think it is imaginary. It is part of their day to day life. But what is "i"?
Actually you don't need to be a genius to understand i, just as you don't need to be a genius to understand -1. Just consider the equation
Z x Z = -1.
Do you know of a number Z such that when multiplied by itself it gives -1? An alternative way to ask this is,
What is the square root of -1?
If you were presented with a similar equation, such as for example Z x Z = 9, you would probably ponder for an instant and then conclude that Z = 3.
We may then even ask, "are there any other Z's that satisfy this equation?" As a hint, let us remind you of the following rule:
A negative times a negative equals a positive.
With this at hand, you would say, "oh yeah, also Z = -3". By the way, there is a great Mathologer video about this rule. Watch this video if you follow the rule of not taking rules for granted but understanding where they come from.
Now back to Z x Z = -1. You can't find a "real solution" to this equation. If you look for a positive Z, then Z x Z is positive and can't be our -1. If you look for a negative Z, then Z x Z is again positive. Z = 0 won't work either because 0 x 0 = 0. Here comes the number i:
The number i is a solution of Z x Z = -1.
At this point you can say, "hey you can't just imagine stuff up!".
But think about it. The number -1 is as "imagined up" as the number i. When humanity "made up" -1, it allowed us to understand equations such as 7 + Z = 6. Now with i, we can also understand Z x Z = -1.
The difference between -1 and i, is simply that -1 is used by almost everybody. On the other hand, i is mainly used by "STEM specialists". But as such specialists would attest, having the number i is critical to the mathematics that drives physics, engineering and science. Teenagers may study i at school, but "i" can actually be introduced to younger children, just as well. After all, kids have a great imagination.
To see that we can do something simple and fun with -1 and i, let's agree that when we multiply them together we have
(-1) x i = -i.
Now ask, "what is (-i) x i ?" Well i times itself is -1 and that times -1 (coming from the -i) is 1. Hence,
(-i) x i = 1.
Now use these rules to fill out the question marks below. What is the emerging pattern?
Isn't it cool that it goes round and round?
You see, with "real numbers" such as -1, 3, 0, or pi, we can't produce such cyclic behaviour with a cycle of length 4. In fact, the only real number that can cycle using repeated multiplications by itself is -1. Try it with any other real number and you'll either never get back to where you started, or get stuck where you started (with 0 or 1).
It turns out that a whole lot more can be done with the "imaginary number" i. One famous mind-blowing equation, connects it to -1 through pi and e. Do you remember these irrationals from an earlier blog post? This equation is called Euler's identity. Here it is:
Alongside with E = mc^2, it is arguably one of the most famous and beautiful identities discovered to date. We will explore it in future blog posts, but if you can't wait, then don't hesitate to view this captivating Mathologer video.
Concluding this blog post, we want to say that us both, Inna and Yoni, often feel humbled at the doorstep of mathematics. We can touch -1 and we can feel and work with i. With some effort we can even comprehend much more involved concepts. But as we do that, we know that there is world of mathematical truths, imaginary or real that is waiting to be discovered. We'll never get to do it all, but we can still imagine. And we love it.
On that note let us share a very mathematical poem by a great Russian poet, Osip Mandelstam, in the English translation:
And I go out from space
Into the abandoned garden of values,
And pick off fake constancy
And self-consciousness of reasons.
And your, infinity, textbook
I read alone in my solitude —
The leafless, wild medical manual,
Assignment book of enormous roots.
(November 1933, Moscow)