My seven year old son is currently very curious about big numbers. He is interested in the number of seconds in a day, 86,400, the number of minutes in a year, 525,600, and the number of stars in the Milky Way, estimated at around 300 million. Such big numbers certainly drive his curiosity.
His older sisters on the other hand, have grown out of being impressed by magnitude. At ages 8 and 11 they already comprehend that there is not a number bigger than all the rest. They don’t get overly excited when they hear about a million, a billion, a trillion or even a quadrillion. This is 10 to the power 15 by the way; a thousand trillion or a million billion.
So if magnitude of numbers doesn’t excite them, what will? Our new mathematical frontier in the family deals with famous irrational numbers. To explore these we don't have to traverse the number line, further east than 4. But we must go deep.
Maybe you don’t know or remember the story of rational and irrational numbers. Let me briefly remind you. A rational number is a number that can be expressed as a ratio of two whole numbers, M/N, with N not 0. Look at some examples:
4/3: M = 4 and N = 3.
17: M = 17 and N = 1.
-1.8: M = -18 and N =10.
9.421: M = 9421 and N = 1000.
There is not necessarily a unique choice of M and N, but the important point about rational numbers is that there is such a choice. But, as the ancient Greeks discovered, not all numbers are rational!
The ones that are not rational are called irrational. The first irrational number ever discovered is the number such that when multiplied by itself gives 2. This number is called the (positive) square root of 2.
In a future blog post, we’ll explore the logic showing that the square root of 2 can't be rational. But for today let’s take it as a given:
You can't find whole numbers M and N such that (M/N)x(M/N) = 2.
This statement means that the square root of 2 cannot be rational. Sure, you can use a calculator to get,
But this is only an approximation to the first few digits of the square root of 2.
You see, irrational numbers have the property that when you write them as decimals you get an infinite sequence of digits that never repeats. Want 200 digits after the decimal point, no problem:
With every digit that we add on, we get a rational number that better approximates the square root of 2. But the digits will never stop nor will they form a repeating cycle. This is the nature of irrational numbers, their decimal digits continue on and on in an irregular manner.
A word of caution: Even though their digits go on forever, don’t confuse irrationals with numbers such as
1/3 = 0.33333…
7155/9990 = 7.162162162162….
For both of these numbers, the decimal digits go on forever, but they follow a predefined cyclic pattern and can be written as M/N. Both of these are rational (and not irrational).
We could explore the square root of 2 much further, especially when it comes to triangles, but let's move onto circles and the world’s most famous irrational number, denoted by the Greek letter pi:
As you may remember, pi is the ratio of a circle’s diameter to it’s circumference. Want to see pi's digits stretched to one mile? Explore this Numberphile video. A great thing to watch on pi day, March 14.
So we know about the square root of 2 and pi, great numbers to discuss and explore with our children. But what are some of the other famous irrational numbers?
The next one to consider is probably the number e:
This number, sometimes called Euler's Number, plays a key role in financial calculations, engineering, biology and just about anywhere in math. One way to arrive at the number e is to look at the sequence
What happens to the N'th term of this sequence as N grows large? Try it yourself with a calculator. Take for example N = 200 and key in
Make sure you follow the correct order of operations, and you’ll get a good approximation for e. What did you get? How does it compare to e?
So we got the square root of 2, pi and e. Are there any other famous irrational numbers between 0 and 4? The next one on the list is probably the golden ratio, denoted by the Greek letter phi:
Like e and pi, the golden ratio pops up everywhere in nature and mathematics. One way to encounter it is through the Fibonacci sequence:
Each element of this sequence is the sum of the previous two elements. It is a fun sequence for 10 year olds to compute. Try it. As a child explores Fibonacci, she immediately realizes that the sequence grows without bound. But how fast? The answer lies in the golden ratio.
Consider the sequence of ratios of neighbouring Fibonacci numbers, each number divided by it’s predecessor:
Observe that the sequence appears to converge to a limiting value. You guessed it, the golden ratio. As an exercise, try dividing the 20'th element of the Fibonacci sequence by the 19'th element to see you get something very close to the precise golden ratio. Also see this beautiful visualization of the golden ratio in nature.
OK, we got square root of 2, pi, e and the golden ratio phi. With such a neat collection, are we irrational enough to impress our kids? These four numbers indeed star in this cool talk by Christina Wallace.
But let me add a fifth famous number to our arsenal. This number is somehow my personal favourite. It is on the one hand famous enough to make it into many mathematics textbooks, but on the other hand it is far less popular than the square root of 2, pi, e and phi. It is called Euler’s constant (sometimes the Euler–Mascheroni constant) and is denoted by the Greek letter, Gamma:
A way to get this number is to first consider the sum (sometimes called the harmonic series),
Interestingly, even though we are adding smaller and smaller positive bits to this sum, it can be shown that the sum continues to grow without bound. When we are adding the million’s term to the sum, the sum only grows by 1/million. Nevertheless, mathematics shows the sum grows without bound. This is very much in contrast to this example,
The latter converges to 2, similarly to what we saw when thinking about lentils.
Well, it turns out that the harmonic series grows at a rate very similar to the natural logarithm of N. In more precise mathematical terms, the difference
approaches some constant as N grows. Guess the constant? Gamma!
As of today, humanity still doesn't know if Gamma is irrational or not. This means that there isn't a known mathematical proof showing that it is rational nor one showing that it is irrational. When it comes to the square root of 2, pi, e, and phi, the math research community has produced proofs for their irrationality (already many years ago for all these numbers). But for Gamma, not yet. No one has found such a proof yet.
Nevertheless, if you bump into a rational mathematics researcher and ask her:
"Is it rational to think that Gamma is irrational?"
She'll most probably reply,
"Yes, and it is most probably a transcendental number!
But nobody knows how to prove it yet."
But nobody knows how to prove it yet."
We'll explore transcendental numbers some other day. In the meantime, we'd love to hear what mathematical concepts interest you.
Χαράλαμπος Κ. Φιλιππίδης