Posted by Mark on Apr 12, 2015 in Algebra, Explore Maths, Interesting Facts, Secondary, Upper Primary | 0 comments

Odds are high that you’ve not given much thought to numbers, I’d wager.

You encounter them everyday and you use them, knowing the purposes they serve in your situation.

Beyond that, perhaps you have to solve problems with them. Like if you’re a parent helping your kid with Maths sums.

Still, they may just be numbers to you. A bunch of digits, or simply one, which can convey size and quantity; represent monetary value; label things (e.g. house numbers, car licence plates and product serial numbers), etc.

Now, aren’t they convenient and a boon to all of us?

I’d like to take you beyond that and look at the beauty in the numbers…

**A word from Mark:** *“I love puns, so I didn’t let up on the opportunity for word play with the words ‘odd’ and ‘even’. Plus others too, as you’d soon discover. If you find any part of this article confusing, re-read it (in a punny mood, perhaps) and it should all clear up.”*

When shown the number sequence 1, 3, 5, 7, 9, …, we’d recognize these numbers are odd numbers. That’s a basic fact we learnt in school.

But to really understand odd numbers, we need to look at even numbers.

An even number is simply an integer* multiple of 2; like 2, 4, 10, 28, etc. When we list consecutive numbers, even numbers will have other numbers between them: these are called the odds.

So, odd numbers are neighbours to even numbers and vice versa. And every integer is either odd or even — this is referred to as parity.

From this, we know that an odd number is simply one more than the even number preceding it (when looking at numbers in ascending order). Or one less than the even number after it.

Expressed as a formula, an odd number is:

2`n`

+ 1

where `n`

is any positive integer* or zero.

In case you are wondering, that 2`n`

part signifies a multiple of two, i.e. an even number.

So, when `n`

= 0, we get 1 as the answer — this is the first odd number.

When `n`

= 1, we have 3 or the second odd. And when `n`

= 4, we get 9, which is the fifth odd number. And so on.

**Note: To simplify the discussion, I’ve stayed with natural numbers ^{#} only. Indeed, n can also be a negative integer; so -3 and -11 are also odd while -6 and -24 are even.*

^{#}Confusion alert: There’s no universal agreement on whether 0 is considered a natural number, or what we call “whole number”. Here, I’ve chosen to regard zero as one.

In a sense, the `n`

in the formula tells us what the (n+1)-th odd number is, in the number series that goes 1, 3, 5, 7, 9, …

If we want to know the n-th odd number, we can plug “`n`

– 1″ into that 2`n`

+ 1 formula; viz: 2(`n`

– 1) + 1 = 2`n`

– 2 + 1, after expanding.

Rearranging, we’d get 2`n`

+ 1 – 2, which is both familiar and logical. Familiar because 2`n`

+ 1 is the definitive odd number; and logical, as consecutive odds always differ by 2.

With this, we get this neat trick below…

To get the n-th odd number, use:

2`n`

– 1

From simple observation, we have the following rules when operating with numbers:

- odd + odd = even
- odd – odd = even
- odd X odd = odd

Again for simplicity, I’ve omitted the more complicated division operation.

The first 2 rules aren’t difficult to understand if we perceive odd numbers as offset by one position from their even siblings. Pictorially, we have the following:

So, if the first addend (for Rule 1) or the minuend (for Rule 2) is an odd number, we are starting with an offset position.

And if we depict adding on or taking away an odd number as a “shifting” operation by one position — left or right is immaterial here — we are back to a non-offset position.

In other words, we’ve just toggled the parity, from odd back to even. Hence the first two rules are what they are, naturally.

Multiplication is a little harder to see, but if we think about it as repetitive addition it becomes obvious. Multiplying by 2 means adding a copy of a number to itself — once. By 3 means adding on two copies. And so forth.

So when we multiply an odd number by another odd, we add on an even number of copies of either number. Returning to our picture model, adding an even number means nothing gets shifted; so the result is an odd number, intuitively.

Odd numbers aren’t queer at all. Really.

Consider these mathematical facts…

In the strict sense, odd numbers make up half** of all whole numbers out there; the rest belong to the evens. If they didn’t, then these numbers would have been few and far between. Like prime numbers, for example.

**Ok, to be really correct, it’s ever so close to 50% as the “odd number” (read “unusual”) zero is considered even. However, if we stretch out to infinity, then odd/even (dis)parity pretty much evens out. See?

Every prime number is an odd number except for the first, which is the number 2.

By its very definition, a prime number does not have other factors except for 1 and itself; thus, it’s not hard to understand that multiples of other numbers can’t be prime.

On that same token, even numbers — remember, these are actually multiples of 2 — aren’t prime too except for one odd man out. Which explains clearly the fact about prime numbers.

At my WittyCulus blog, I had previously written about primes. Like this article about the largest prime; and a related one over here. I hope you’ll check them out.

Now, let’s take it one step further and delve into squares. Not just any square, but squares of consecutive numbers.

Did you know that the difference between the squares of two consecutive numbers is always odd?

More interestingly, did you know that these differences would form the odd number series when all the whole numbers are involved? My boy “discovered” this and I thought it’ll be a fun topic to write about.

Let’s dive in…

Consecutive numbers differ by 1. For two consecutive numbers, we can denote the smaller number by `a`

; its neighbour would then be `a`

+ 1.

And we’d write their squares as `a`

^{2} and (`a`

+ 1)^{2} respectively.

Using a little algebra — parents, I hope you aren’t rusty here — we can expand the larger square of (`a`

+ 1)^{2} into `a`

^{2} + 2`a`

+ 1.

Then, the difference between the two consecutive squares (I’ll call this DoCS for short) becomes:

`a`

^{2} + 2`a`

+ 1 – `a`

^{2} = 2`a`

+ 1

Recognize the result as the formula for an odd number? We’ve just verified the first fact that DoCS is always odd!

For the second fact, study the diagram below first…

How do we explain this?

Consider four consecutive numbers and call them `b`

, `b`

+1, `b`

+2 and `b`

+ 3.

Their squares are `b`

^{2}, (`b`

+ 1)^{2}, (`b`

+ 2)^{2} and (`b`

+ 3)^{2}.

Now, computing each DoCS, we get:

- 2
`b`

+ 1, like we have done earlier. Let’s write this as 2`b`

+ 1 + 0 - (
`b`

+ 2)^{2}– (`b`

+ 1)^{2}=`b`

^{2}+ 4`b`

+ 4 – (`b`

^{2}+ 2`b`

+ 1) = 2`b`

+ 3 = 2`b`

+ 1 + 2 - (
`b`

+ 3)^{2}– (`b`

+ 2)^{2}=`b`

^{2}+ 6`b`

+ 9 – (`b`

^{2}+ 4`b`

+ 4) = 2`b`

+ 5 = 2`b`

+ 1 + 4

We can see that the differences increase by two (as those double-underscored numbers show), constantly. This holds true if we compute the next 10 DoCS; or next 100 or thousand. Ad infinitum, in fact.

Which essentially means the DoCS are part of a sequence — an arithmetic progression to be exact.

In this case, it’s the odd number series! Magical!!!

Where does this knowledge bring us to? For one, if we know any square, it isn’t hard to mentally compute another which is very close by.

As an example, we can tell that 11^{2} is 100 + 21 = 121, because 10^{2} is an easy 100 and the DoCS here is simply 2 X 10 + 1. Got it?

Here’s our next short-cut…

To get the square of the next consecutive number, use:

`n`

^{2} + 2`n`

+ 1

where `n`

is a number you know the square of.

If we are looking at a number which is 2 more, how can we get its square? There are two ways we can do this:

- Compute the square of the next consecutive number as usual. Keep this square (for adding to later); compute the DoCS based on this same consecutive number — this is shown doubly-underlined below.

Let’s see how this works using the same starting point as before, but now we want the square of 12. We got 11^{2}as 121; the DoCS in this case would be 2 X 11 + 1 = 23. Thus, 12^{2}= 121 + 23 = 144. - We use a new trick here: forget the original DoCS but compute 2
`n`

+ 2 instead; then, we double the answer before adding to the known square.

Here’s how to do 12 squared: we know the square of 10; now 2 X 10 + 2 = 22 and double that is 44. Hence 12^{2}= 100 + 44 = 144. Neat!

We can actually generalize this new trick, as such…

To get the square of a number `m`

counts larger, use:

`n`

^{2} + `m`

X (2`n`

+ `m`

)

where `n`

is a number you know the square of; and (`n`

+ `m`

)^{2} is what you want to solve for.

Now, don’t let this slightly more complex shortcut faze you. A few examples will quickly make it look easy, I promise.

- We know 4
^{2}= 16 and want to solve for 7^{2}- m = 7 – 4 = 3
- 2 X 4 + 3 = 11
- 3 X 11 = 33
- 16 + 33 = 49 = 7
^{2}

- How do we get 137
^{2}?- We know 13
^{2}= 169, so 130^{2}= (13 X 10)^{2}= 169 X 100 = 16900 - m = 137 – 130 = 7
- 2 X 130 + 7 = 267
- 7 X 267 = 1869
- 137
^{2}= 16900 + 1869 = 18769

- We know 13
- What is the square of 35?
- 30
^{2}= 900 - 2 X 30 + 5 = 65
- 5 X 65 = 325
- Answer = 900 + 325 = 1225

- 30

If you are game, go figure out how to square a number which is one less. It’s not that hard if you’ve come this far.

And with that mini-challenge, I end this super long article about odd numbers. I’m hopeful you’ve seen some of the beauty of numbers and have learned a couple of useful mental tricks.

Most importantly, I really hope you had fun with this study.

Do let me know in the comment box below about your views of what I’d written and the kinds of topics we could touch on next time. My thanks to you for visiting No Problem Sums, as always.

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