Yes, math is culture too! Soon enough I’ll be back to discussing plays. I’ll start from the easy and progress, skipping a lot along the way. These are just a few of my favorite things. Disclaimers and credits at the end.
So simple, but still fun.
Determine if a number is divisible by 3 or 9:
Add the digits of a number together. Is this sum divisible by 3 or 9? If so, then the number itself is also.
For example, is 83941272 divisible by 3 or 9? How about 345678?
So let’s add the digits.
You could actually add these digits together again, or you could rely on your knowledge that 36 is divisible by 9. And yes, 83941272 divided by 9 gives a result of 9326808.
So how about 345678? Its digits
add up to 33. So 345678 is divisible by 3, but not by 9. In fact, 345678 divided by 3 gives a result of 115226.
So how does this work? How can you possibly know this if you are given some random number, maybe one that is thousands or millions of digits long?
Let’s take a simple case. Suppose you have the number xy. I’m not implying multiplication, but just a way to describe the number. For example, if we have the number 72, in this case x is 7 and y is 2.
In base-10 arithmetic (which we are assuming here), the number xy is equal to:
Or back to our specific example:
We already know that is divisible by 9. So add , and if they are divisible by 9, then the whole number xy must be divisible by 9.
The same applies if you have the number xyz, which is
Again, we know that the 99x and 9y must be divisible by 9, which leaves us with determining if is divisible by 9.
And yes, the same thing applies with 3 also.
which means if (x+y) is divisible by 3, then xy is too.
It’s really easy to determine if a number is divisible by 2, 3, 4, 5, 6, 8, 9, 10. The test for 7 is a bit harder. See:
For a number divisible by 7: “The last digit is multiplied by 2 and subtracted from the rest of the number. The result is either 0 or divisible by 7.”
Thus, for 224, multiply 4*2, and subtract it from 22. The result is 14. And 224 is indeed divisible by 7. Work out for yourself why this trick works.
Such a fun set of facts. Do you ever find yourself factorizing numbers in train or bus ads? Well, maybe not, but for me, between factorizing numbers and anagramming words that appear in ads, a typical commute can be quite amusing. (Honestly, although I do read books and magazines and look at my phone and stare out the window too.)
Summing Integers From 1 to N and More
Carl Friedrich Gauss’s teacher wanted to keep his class of 6-year-olds (approximately) quiet. He assigned them the task of adding the numbers from 1 to 100. CFG solved it in a minute, more or less. How? He saw this grouping:
If you know the principle of proving by induction, it’s straightforward to generalize this for any value of N as an integer. The equation for summing 1 to N is
But the fun does not stop there. What about summing the sum of the squares of the integers from to ?
How about the sum of the cubes of the integers from to ?
I’ll spare you the suspense.
Yes, it does go on and on, with higher-order equations as you progress to higher powers of N.
The Numbers e, , and -1
What is the value of e? Well, it’s approximately 2.72, but that does not tell the whole story.
e is so significant in nature, and its construction is so elegant.
For reference, a factorial, such as 6!, means 6 x 5 x 4 x 3 x 2 x 1. Similarly, N! = N x (N-1) x … 2 x 1.
Any equation that involves growth or decay, such as bacterial growth or radioactive decay of uranium, probably involves e.
What about , the number we know as approximately 3.14?
You probably know about its significance when calculating the area of a circle, or the circumference of a circle, or the volume of a sphere. But what exactly is pi? There’s no one answer. But you can express pi as a series expansion, in many different ways. My favorite way is probably the Gregory-Leibniz series:
(Both e and are transcendental numbers, which is another fun concept, but another digression.)
What is -1? If I owe you a dollar, does that mean I have -1 dollars? That’s how accountants see it, I think. You can see -1 on the number line. Wikipedia says that negative numbers are first described by the Chinese, before 200 BC. European mathematicians were way behind Chinese, Indian, and Islamic mathematicians, but had started to accept negative numbers by the 17th century. Intuitively, -1 makes sense to most of us raised in a culture that uses money.
But what about the square root of -1? You may have learned that this is the imaginary number i (and -i also works too). You might protest that such a number makes no sense. However, it does, and the concept of complex numbers (less pejorative than the imaginary epithet that Descartes threw at it, and inclusive of the concept of a number that has both a real-number and an imaginary component) is very important in electrical engineering, among other areas. There is a world where -1 has two square roots and this world is meaningful.
Skipping a whole bunch of steps, this leads us to Euler’s identity:
Such beauty, such poetry. So many intricate and beautiful concepts collapsed to a single equation.
Countable and Uncountable Infinities
How many numbers are there between 0 and 1? How many numbers are there between 1 and 100? Surely there should be a lot more numbers between 1 and 100 than between 0 and 1? But, in fact, it’s been proven there are the same number of numbers between any set interval. This is a mind-bending fact, I agree.
We understand the integers
can continue without end. Can you map these integers to the numbers in any given numeric interval, such as [0, 1]. Cantor has proven that you cannot. The integers are countable. The infinity of numbers in any given numeric interval is uncountable.
But what about the rational numbers? A rational number is any number that can be expressed as a fraction, p/q, where p and q are any integers. Surely there are many, many more rational numbers than there are integers. Between the integers 1 and 2 (or any two integers), there are an infinite number of rational numbers. But, in fact, Cantor established that the rational numbers are countable. You can set up a pattern where every rational number can be represented by and mapped to an integer. That’s not so surprising, perhaps. Consider that if you take the set of integers 5, 10, 15, … that these can be mapped in a 1-1 relationship with the integers 1, 2, 3, … But aren’t there more integers in the set than there are in the set ? It seems there are not. Cantor suffered from mental health issues; some have speculated that his exploration of different sorts of infinities exacerbated that.
The Sum of All Natural Integers is Equal to -1/12
Did we not establish above that the sum
is equal to
Surely as N increases towards infinity, the value of N(N+1)/2 is going to increase towards infinity also? How could the sum be equal to -1/12? But like the Mad Queen in Alice in Wonderland, we can believe “six impossible things before breakfast” (or arguably any N of impossible and contradictory things before breakfast).
The videos explain this much better than I can (and in fact, I’m sure most of this post can be replaced with the appropriate videos, if you hunt around). But look above at how
which hardly seems plausible either on its face. Just divide that by 12, and you have the sum of the natural numbers.
One of the videos suggested looking up the work of John Baez, who explains how this result applies to string theory. So take a look:
Douglas Adams was mistaken; 42 is not the answer to life, the universe, and everything. He somehow reversed it–24 is.
The Norwegian mathematician N. H. Abel (who sadly died at a young age, after immense contributions to group theory) said, “The divergent series are the invention of the devil”. Divergent series are definitely not easily amenable to any commonsense understanding.”
Further Explorations Beyond My Ken
Baez’s discussion of string theory mentions elliptical curves, which are toruses (doughnut shapes) constructed from parallelograms (tilted rectangles). I’m heading out of my depth, but elliptical curves are central to the proof of Fermat’s Last Theorem.
You may be familiar with Pythagorean triples like (3, 4, 5), where . Or (5, 12, 13), and so forth. It’s been proven there are an infinite number of Pythagorean triples (and you probably know that these relate to the area of a rectangle too).
Fermat noticed that while has an infinite number of solutions (he probably didn’t know the infinite part, but just the lots of solutions part), had no discoverable solutions for any integer . His perverse comment that he had discovered a proof, but couldn’t fit it in the margins of the book he was writing in at the time, has kept innumerable mathematicians awake ever since.
Until 1995, and Andrew Wiles’ proof (which turned out to have holes in it, but these were rapidly fixed by other mathematicians), there was no proof, although many, many people tried. I spoke to a former professor at the time, who worked in a related area, and he told me the proof was not fully comprehensible to him, so I am not about to try to explain it. It’s on the Web, though.
But it’s worth mentioning elliptical curves have applications related to prime numbers and cryptography too.
Disclaimers and Credits
No pretense of completeness is intended, nor are the proofs rigorous. All the math here is public domain, but I have tried to give credit where I specifically drew from a source. None of my math teachers and professors should be blamed for any errors. Numberphile’s videos about the sum of the natural numbers inspired me to post this. Wikipedia is a really excellent place to look up math. The best-looking equations are Wikipedia gifs, but I have done most of this using WordPress’s LaTeX capabilities.