$i$ is called an imaginary number. It's unreal, because there is no real number whose square is $-1$. It doesn't match reality. In reality, a square of any non-zero number is always positive. Or is it?

Have you ever wondered why the result of multiplying a negative number with a negative number must be positive?

Multiplying two negative numbers together doesn't look natural to me. In real life we multiply when we want to count things, but I can't think of any situations where we want to multiply a negative number of things a negative number of times. Thus, in order to make sense of negative number multiplications, we have to forget reality and turn into the realm of abstraction.

We want to know why $(-x) * (-y) > 0$, for all positive $x$ and $y$. This question eventually boils down to: how can we prove that $(-1) * (-1) = +1$, which bothered me for quite sometime. Tell me, why isn't it $-1$ or even $-2$?

Even the great Euler resorted to an thoroughly unconvincing argument to answer this question. He reasoned that $(-1) * (-1)$ must be either $-1$ or $+1$, but it can't be $-1$, because $-1 = (-1) * (+1)$.

It took a long time before mathematicians realized that this rule, which is called the rule of signs, can't be "proved". In fact it's a definition created by mathematicians to preserve the fundamental laws of arithmetic. Mathematicians want to ensure that adding negative numbers to natural numbers doesn't mess up with how humans have been doing calculations. The existing laws must keep working. Let's see why this preservation desire makes $(-1) * (-1)$ equal to $+1$.

Let's take a look at this series of manipulations:

0 = -1 * 0 (see below)
   = -1 * (1 - 1) (definition of -1, which is the additive inverse of 1)
   = -1 * 1 + (-1) * (-1) (distributive law of integers)
   = -1 + (-1) * (-1) (see below)

If we accept that $-1 * 0 = 0$, and $-1 * 1 = -1$ (more on these two laws in a moment), for the distributive law to stay correct $(-1) * (-1)$ must be the additive inverse of $-1$ which is $+1$; otherwise we would come up with a contradiction that is $0 = -2$.

But why $-1 * 0 = 0$? The only natural properties of $0$ are $0 + x = x + 0 = x$ and $x + (-x) = 0$ (which is the definition of negative number). This is how we count. If you have $3$ dollars, adding $0$ dollars you still have 3 dollars. If I owe you $3$ dollars, and I pay you $3$, I would end up with $0$ debt. $0$ looks trivial, but its discovery is actually a significant event in the history of mathematics. For a long time, most people including mathematicians didn't welcome $0$.

Now if we apply the distributive law again, we can see that

(-1) * 0 + (-1) * 0 = (-1) * (0 + 0)
                             = -1 * 0
                             = 0 + (-1) * 0

We can eliminate $(-1) * 0$ from both sides, and conclude that $(-1) * 0$ must be $0$. The same technique can be used to prove that $x * 0 = 0 * x = 0$ with all $x$. Isn't that cool? Have you ever thought that this is provable?

Now the last mystery is why $-1 * 1 = -1$? One can say that it must be $-1$ because $-1 * -1 = +1$, but that's tautology. This is actually something we cannot prove, but we have to accept it as another law to keep everything working correctly. We accept as an axiom that in the ring of integers, $x * 1 = 1 * x = x$ for all $x$.

Thus, if we want to maintain the rules of arithmetic we must assign $-1 * -1$ to be $+1$. Otherwise everything would collapse, and any calculations mixing negative and positive numbers wouldn't make any sense. In other words if we accept the rules of arithmetic as axioms, we can deduce that $-1 * -1$ must be $+1$, but if we don't it could be an arbitrary value.


How does betterexplained.com explain this? Much better than I do, of course. It doesn't give a proof or anything, but it gives an intuitive explanation why the result must be +1.

It shows that there are two ways of making sense of multiplication: repeated addition or scaling.

The former is what we were taught in school, but it doesn't work well when we encounter negative numbers (let alone complex ones!). How do you repeatedly add a number to itself a negative number of time? No sense.

The later, on the other hand, is a great way to think about and visualize multiplication or any other arithmetic operations. For each multiplication, we always start at 1, and scale to the next position on the number line according to the multiplicand. Now, multiplying with a negative number is a scale-then-flip operation. For example to calculate 4 * -3, we start at 1, scale to position +4, scale to position +12, then flip back to -12.

If we start at 1, multiply by -1, we scale by 1 so we stay at the same place, then we flip to -1. At -1, if we multiply by another -1, we scale by 1, stay at the same place which is -1, then flip back to 1. Thus -1 * -1 can be seen as equal to +1.


I hope you now understand why $(-1)^2= +1$. But what else have we learned? This little thought experiment tells us that we've taken so many rules for granted, but we actually have no ideas why they are true. We are like the monkeys in this little story that I was told a long time ago.

Pure math, if there's a way to distinguish it from applied math, is a game of mind. Mathematicians create some rules, and keep playing with them until they found something interesting; otherwise they go back and change the rules.

An interesting result doesn't necessarily have any useful real world applications, but somehow a lot of them do. This is why the success of math to effectively describe or found applications in the physical world is considered unreasonable. For example, complex number was invented to solve equations like $x^2 = -1$, but soon people discovered that it can be used to model many physical interactions. Or take elliptic curves. People started investigating them just because they wanted to solve some integral problems, which are entirely pure math and useless. A hundred years later it's discovered that the same objects can be used to factor integers and do crypto. In fact every time we connect to Gmail, we're using the math of elliptic curves.

In the early days of mathematics, a set of rules are considered useful if it allows mathematicians to solve equations. Note that mathematicians not only want to find useful rules, they also want the exact ones. They don't want 3 rules, if 2 already do the trick. Let's see which rules we need to solve $x + 3 = 6$:
x + 3 = 6 (given)
-3 + x + 3 = -3 + 6 (adding -3)
x + (-3) + 3 = -3 + 6 (commutative law)
x + (-3 + 3) = -3 + 6 (associative law)
x + 0 = -3 + 6 (definition of -3)
x = -3 + 6 (property of 0)
x = 3

Thus, if we want to generalize the set of integers and the addition operation (which is the definition of the additive group $\mathbb{Z}$), we must at least keep the associative law and the existence of an element $e$ such that $x + e = x$ for all $x$. It turns out that this is enough to define a group, which is a much more abstract and general concept than $\mathbb{Z}$. It turns out that there are a lot of groups (or rings or fields) out there in physics, in computer science, in engineering, etc. If you prove a result in group theory, you can use it in any groups in any other settings. Prove once, use everywhere!

This is the deep insight of abstract algebra that wasn't understood for thousands of years. This is the power of abstraction that once unleashed shall enlighten humanity for eternity.

Suddenly $i$ is no more weird or strange or whatever, isn't it? $i^2 = -1$ is just another rule, and as long as it doesn't violate existing rules, but even allows us to solve more problems, it is welcome to join the party of cool axioms. Actually, $i$ is not more imaginary than any other numbers. All numbers, such as 0, 1, 6, $\pi$, $e$, and $i$, exist in our minds only. There is no physical entity that is the number 3. Doesn't matter! As long as they exist in our minds, they're real.

After all, what is reality if not an invention of our minds?


Thuong said…
It is interesting to discuss about these topics. I would say the closure of (finite) additions in a group is also very important. Cancel the "finite" conditions, we may get trouble. For example: e = 1 + 1/1! + ... + 1/n! + ... , and e does not lie in Q, though any term in this series lies in Q. In this comment, we will discuss something about this "trouble", and some other systems of "numbers". The example above shows us that there are "holes" in Q. That means, there exists sequences, whose terms lie in Q, but its limit does not lie in Q anymore.

Kronecke said: "God made the integers, all else is the work of man". From integers, we can build Q, because Q is the fraction field of Z. Now, how to construct R? Just filling out all holes. For example, \pi = 3,141592... we can see that, in fact, \pi is a limit of a sequence {3; 3,1; 3,14; 3,141; ...}. And it is tempting to think that a real number is the limit of a sequence in Q, such sequence is called Cauchy's sequence. In this case, a sequence {a_n} in Q is called Cauchy sequence if for sufficient large integer N, a_n is very "close" to a_m, for all m, n larger than N (in a more formal language: for all \epsilon > 0, there exists an integer N_{\epsilon}, such that for all m, n > N_{\epsilon}, |a_m - a_n| < \epsilon).

Let S be the set of all Cauchy sequence in Q, we define an equivalent relation r in this set. Say, {a_n} r {b_n} if lim {a_n-b_n} = 0 (Think: e = 2.71828... = {2; 2,7; 2,71, ...} = 1 + 1/1! + ... + 1/n! + ... = {1; 2; 2,5; ...}, the two sequences are different, but they just represent the same number e). And what we do now is taking representative of each equivalent class S/r. It is our R, the field of real number. By this construction (due to Cauchy), R is "complete". That is, any Cauchy sequence in R converges in R.

By the completeness of R, we can do the same process to complete any "metric" space, which is a set, where elements are called points, with distance function defined between two points, that satisfies some axioms:

1/ The distance is non-negative real number.
2/ If x != y are two points, then their distance is positive real number. And the distance from x to x is zero.
3/ The distance from x to y is equal to the distance from y to x.
4/ The distance satisfies the triangle inequality.

We can see, the usual metric on Q is the absolute value. That is d(x,y) (denoted the distance between two points x, y in Q) = |x-y|. But it is not the only metric on Q. We have other metric on Q, that brings much arithmetic information than the usual metric. It is p-adic metric.

We will start with Z, let p be a prime in Z. For any a in Z, we can write a = q.p^r, where (gcd(p,q)=1). And we denote |a| = 1/p^r (that means, the higher power of p that a has in its factorization, the smaller |a| is). For example, let p = 2, then |18| = 1/2, |12| = 1/4. And the distance in this case is d(a,b) = |a - b|. It is a metric on Z. And |a-b| \le 1/p iff a = b (mod p). We say, the (p-adic) distance between the two integers is smaller or equal to 1/p iff p | (a-b). It is cool! We have just interpreted arithmetic information to analysis language.

Now, it is easy to extend this metric to Q, by |a/b| = (c/d)p^r, where gcd(c,p) = gcd(d,p) = 1, and r is an integer (note that r can be negative). We now complete Q by this p-adic metric to obtain the field of p-adic number Q_p. It brings much more arithmetic information than R, and it is used to study "WHAT MADE BY GODS" (the integers :D).

It is the analytic construction of Q_p. There is also an algebraic construction, and of course, they are the same. But looking an object by many ways can help us understand more throughly about its properties, and its relations to other objects.

For other example, we now come back to the other aspects of the first part of the post, what can we understand about the multiplication in Z? I think it is an interesting question to think of and discuss.
Unknown said…
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Thai Duong said…
Thanks for the comment, Thuong. It's the first time I see such a definition of R and it's really cool :).

Can you please shed some light on what you think about multiplication in Z?
Thuong said…

It is great to see your interest. I would love to share my thought on the topic. But please give me more time, I cannot write a full comment at this time.