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The Beauty of Numbers
I love numbers. Smiling, I like to say, "Numbers are our friends!" But I am aware that not everyone
feels this way. Many people seem to fear numbers, or to panic at their sight.
Yet nobody fears beautiful art, or panics at its sight. And there is great beauty in numbers, and in their
patterns. The relations among numbers form pleasing and surprising patterns in much the same way
in which the colors in an Oriental rug do, or in which the relations among notes in a musical composition do.
Personally, I find the most beauty in patterned sequences and in identities, which anyone
with an aesthetic sense can appreciate. The Fibonacci numbers and Pascal's Triangle (which is composed of the
binomial coefficients, as it turns out), to give two examples, provide very simple patterns that anyone can easily see
but that nevertheless have hidden depths contained within them. And they show up in nature. (For example, a
sunflower's seeds are arranged in spirals; the number of clockwise spirals and the number of counterclockwise
spirals are two consecutive Fibonacci numbers.)
But let's start as simply as we can, with the hope that length of explanation won't destroy aesthetic appreciation.
For example, we can notice that
As it turns out, the pattern continues no matter how many odd numbers you add. So, whenever you
add up the first however many odd numbers, you get a perfect square!
Which perfect square? Well, let's see. We have
Sum to 1 equals 1 squared; 1 is the "big half" of 1
Sum to 3 equals 2 squared; 2 is the "big half" of 3
Sum to 5 equals 3 squared; 3 is the "big half" of 5
Sum to 7 equals 4 squared; 4 is the "big half" of 7
Sum to 9 equals 5 squared; 5 is the "big half" of 9
Sum to 11 equals 6 squared; 6 is the "big half" of 11
Sum to 13 equals 7 squared; 7 is the "big half" of 13
In every case, the number squared is the "big half" of the last odd number to be added. In fact, it's also the average
--the "arithmetic mean"--of the numbers added. For example, look at
1 + 3 + 5 + 7 + 9
The middle number is 5, and 1 + 3 + 5 + 7 + 9 = 52
Or look at
1 + 3 + 5 + 7 + 9 + 11
The middle numbers are 5 and 7, and 6 is right in between them--right in the middle of 1 and 11.
And 1 + 3 + 5 + 7 + 9 + 11 = 62
However, in a few moments, I'll be giving a pictorial demonstration of this identity that uses the idea of the "big half" rather
than that of the arithmetic mean.
How can we express this idea of the "big half" of an odd number? Well, add one to the last odd number added, and then cut it in half. So, for example,
(9 + 1) / 2 = 10 / 2 = 5, and 1 + 3 + 5 + 7 + 9 = [ (9 + 1) / 2 ]2. (Notice that that's the same thing you
do if you want to find the arithmetic mean of the first number being added, 1, and the last number added.) So, we have
and, in general, labeling the last odd number to be odded as "2k + 1" (both 2k -1 and 2k + 1 are odd, if k is an integer
[either a positive whole number, a negative whole number, or zero]),
(Notice that (k + 1) is the "big half" of (2k + 1).)
There's a neat pictorial demonstration of this identity. Observe:
1
3 3
1 3
5 5 5
3 3 5
1 3 5
7 7 7 7
5 5 5 7
3 3 5 7
1 3 5 7
9 9 9 9 9
7 7 7 7 9
5 5 5 7 9
3 3 5 7 9
1 3 5 7 9
You can see that we have squares of area 1, 4, 9, 16, and 25--i.e., of sides 1, 2, 3, 4, and 5--but we can generate them
by starting with the single lower-left hand corner "1," then adding the next "shell" of three "3"s, then adding the next
"shell" of five "5"s, then adding the next "shell" of seven "7"s, and finally adding the next "shell" of nine "9"s.
Now, it's clear that the squares represent the sums 1, 1 + 3, 1 + 3 + 5, 1 + 3 + 5 + 7, and 1 + 3 + 5 + 7 + 9. It's also
clear that the squares represent squares of 1 numeral, 4 numerals, 9 numerals, 16 numerals, and 25 numerals--a 1x1
square, a 2x2 square, a 3x3 square, a 4x4 square, and a 5x5 square.
To get from the 2x2 square to the 3x3 square, you add a "5" to each row, and you add a "5" to each column, and
then you add a "5" at the corner--you add 2 + 2 + 1 "5"s. To get from the 3x3 square to the 4x4 square, you add
a "7" to each row, and then you add a "7" to each column, and then you add a "7" at the corner--you add 3 + 3 + 1
"7"s. To get from a kxk square to a (k + 1)x(k + 1) square, you add a "shell" of size k + k + 1 = 2k + 1. So, 1 + 3 + 5 + . . . (2k + 1) = (k + 1)2.
When adding the "shell" of "9"s, there are five "9"s across and five "9"s down--but the "9" in the upper right-hand corner
does double duty, so that the number squared is the "big half" of 9.
Again, it turns out that the pattern continues no matter how many perfect cubes you add. So, whenever you add up the
first however many perfect cubes, you get a perfect square!
Moreover, it's easy to see that 1 = 1, 3 = 1 + 2, 6 = 1 + 2 + 3, 10 = 1 + 2 + 3 + 4, 15 = 1 + 2 + 3 + 4 + 5,
and so on; so, we get that
(If you also recognize the numbers 1, 3, 6, 10, 15, . . ., as binomial coefficients, then you'll also be able to write this as
13 + 23 + 33 + . . . + n3 = (n+1)C22
where the symbol "nC2" is read as "n choose two.")
Proofs and demonstrations can be beautiful, too. (A proof is rigorous; a demonstration isn't.) Here's a demonstration,
by way of example, that the differences between consecutive squares will be consecutive odd numbers, as we observed
above:
III
You might remember from high school algebra that the difference of two squares can be factored: x2 - y2
= (x + y)(x - y). Well, pick two consecutive squares--say, 992 and 1002. Then
1002 - 992 = (100 + 99)(100 - 99).
But 100 - 99 is just 1--since we're picking consecutive numbers to square, their difference will be 1--so we have
I.e., the difference between 992 and 1002 is 100 + 99 = 199, while the difference
between 1002 and 1012 is 101 + 100 = 201--the next odd number after 199. Always,
the difference between two consecutive perfect squares will be the sum of the two numbers being squared; and since
consecutive numbers are either even and then odd or else odd and then even, their sum will always be odd!
Another lovely little proof is a proof that
IV
0.9999999... = 1
Just notice that 0.3333333... = 1/3, as we all know, and then multiply both sides of the equation by 3. The left becomes
0.9999999... , while the right becomes 3/3, or just 1. So, 0.9999999... = 1.
This might all seem too simple to count as really beautiful, but remember, we're starting very simply. Twinkle,
Twinkle, Little Star might not have the kind of beauty that The Four Seasons has, but one doesn't begin
listening to music by listening to Vivaldi. However, lest you retort that anyone can enjoy The Four Seasons
without himself being a musician, I'll note that both the Fibonacci numbers and Pascal's Triangle (the binomial
coefficients) can be appreciated even by a non-mathematician. Have a look!