The Very Next Step - Squares and the Power of Two
It is Hip to be Square
We carry on now from what we learned in The Maths They Never Taught Us - Part Three , and I ask You, remember the Multiplication Table we did earlier, where we went from one to one hundred ? Now the numbers which are bold with an asterisk *, forming the diagonal on this table, you may know as squares. You may also remember the mention and use of them before, with a promise we shall see more of them later. Well, it’s later now, so here we are. I didn't get where I am today by keeping or breaking my promises. Let’s recreate this table, showing all numbers, just to refresh your memory.
Table Of Squares
1
| 2
| 3
| 4
| 5
| 6
| 7
| 8
| 9
| 10
| ||
1
| 1*
| 2
| 3
| 4
| 5
| 6
| 7
| 8
| 9
| 10
| |
2
| 2
| 4*
| 6
| 8
| 10
| 12
| 14
| 16
| 18
| 20
| |
3
| 3
| 6
| 9*
| 12
| 15
| 18
| 21
| 24
| 27
| 30
| |
4
| 4
| 8
| 12
| 16*
| 20
| 24
| 28
| 32
| 36
| 40
| |
5
| 5
| 10
| 15
| 20
| 25*
| 30
| 35
| 40
| 45
| 50
| |
6
| 6
| 12
| 18
| 24
| 30
| 36*
| 42
| 48
| 54
| 60
| |
7
| 7
| 14
| 21
| 28
| 35
| 42
| 49*
| 56
| 63
| 70
| |
8
| 8
| 16
| 24
| 32
| 40
| 48
| 56
| 64*
| 72
| 80
| |
9
| 9
| 18
| 27
| 36
| 45
| 54
| 63
| 72
| 81*
| 90
| |
10
| 10
| 20
| 30
| 40
| 50
| 60
| 70
| 80
| 90
| 100*
|
Comments on Squares and Powers
You basically get a square number, as I said before, when a number is multiplied by itself. For example, 3 × 3, which equals nine, is also known in the maths business as three squared, or three to the power of two, and designated 3². Now you may also be aware, that the small numeral 2, raised up and to the right of the 3, is known as an exponent, power, or index, and the reason the exponent here is 2, is because when we multiply, we have two lots of three to be multiplied together, thus : 3 × 3. Now, were we to have 3 × 3 × 3, we would render it as 3³, and so on.
It is suffice to say, that just as 3 + 3 = 3 × 2, then 3 × 3 = 3², so that what multiplication is to addition, raising a number to a power is to multiplication, forasmuch as multiplication is repeated addition, then raising a number to an exponential power is repeated multiplication. Square numbers such as nine, sixteen, and twenty five, are called so, because they can easily be formed into a square, which will be of sides equal to the number that is squared, while this square number will equal the square’s area.
For example :
We see here that a square with sides each of three inches will have a total area of nine square inches, and it is this number nine that we refer to as our square number. Let us see what else we can find out about these special numbers.
Table Showing Numbers 1 - 10 With their Squares beside them
Discussion about the Table
So, as you can see, as mentioned above, that each square is indeed the sum of successive odd numbers. But just as a matter of interest, how do you know which odd to sum to, in order to get the square you want ? All you have to do, is double the number you wish to square, subtract one, and this is the last odd you add to all others up to it to get your square. i.e., all odds from one to this last odd number you have worked out.
That is, if, for example, you want to know to which odd to sum to, to work out 9², you double 9 to get to 18, then subtract one. Therefore, to work out 9², you would sum all the odd numbers from one to 17, inclusive, and this would equal 9², or 81.
There is in fact a formula I have developed for summing either consecutive or alternate numbers, from a known initial number to a final one :
( x² + xy - a² + ay ) ÷ 2y
where a is the first number of your sequence, x is the last, and y is the common difference between all the numbers. So with respect to the example above of finding 9² by summing all odds up to 17 :
a = 1 ( the first odd ) ; x = 17 ( the last odd ) ; y = 2
( x² + xy - a² + ay ) ÷ 2y
( 17² + 17 × 2 - 1² + 1 × 2 ) ÷ 2 × 2
= ( 289 + 34 - 1 + 2 ) ÷ 4 = 324 ÷ 4 = 81
We shall in fact be seeing this and similar summation formula in future. As you continue to study Maths, especially if, like me, you do so for the enjoyment of it (and yes, although I never always used to think so, Mathematics can be enjoyable), you will find many interesting patterns such as these, and as a matter of fact, there is yet another right before our very eyes.
Consider the amounts which are the difference between a number and its own square. If I make a list of these numbers, I find the following :
0, 2, 6, 12, 20, 30, 42, 56, 72, 90.
Now these numbers don’t appear all that significant, until we halve them, and then we get :
0, 1, 3, 6, 10, 15, 21, 28, 36.
What, still don’t recognise them ? I know. This is why I decided to come up with this book, to share such information, and only then shall you learn, Grasshopper.
This second set of numbers are in fact sums of all numbers in succession from one to whichever number you wish to go up to. For example, 15 is the sum of all numbers from one to five inclusive, or that is to say,
15 = 1 + 2 + 3 + 4 + 5.
In the table, the sums are highlighted in green, and are also known as Triangular Numbers, and why this is so we shall see a bit later. Squares are marked in red, while the two numbers in this table in particular that are both Triangular, and square, are marked in both ways.
Table of Squares and Triangular Numbers
Now as it works out, with respect to the first list of numbers we had before halving them, zero is the difference between one and one squared. After this, the list goes all the way up to 90, this being the difference between 10 and 10², while 45, which is half of ninety, is the sum of all the numbers from one to nine inclusive. Thus, if we express this algebraically : (x² - x) ÷ 2 = Σ [ 1 to (x - 1) ]
The Greek letter Σ sigma here indicates sum of, and the more general formula for the sum of all numbers from one to a given number symbolised by x, is the following : ( x² + x ) ÷ 2.
This means, for example, that if you want to sum all numbers from one to ten inclusive, you add 10² to 10, and then divide by two, thus :
10² ( = 100 ) + 10 ( total = 110) ÷ 2 = 55,
which is indeed the answer. But with the formula before this, we see that ( x² - x ) ÷ 2 equates to the sum of all numbers from one to one less than x, so in order to express this mathematically, we write :
[ ( x - 1 )² + ( x - 1 ) ] ÷ 2
= [ x² - 2x + 1 + x - 1 ] ÷ 2
= ( x² - x ) ÷ 2,
which is certainly what we said before. Thus, mathematically, the formula works out.
It is also an interesting fact that if the sums of two successive numbers, ( those sums obtained by calculating ( x² + x ) ÷ 2 ), are added together, the resulting number equals a square.
So :
SUMS ADDED SQUARE SUMS ADDED SQUARE Σ1 + Σ2 = 1 + 3 4 = 2² Σ2 + Σ3 = 3 + 6 9 = 3² Σ3 + Σ4 = 6 + 10 16 = 4² Σ4 + Σ5 = 10 + 15 25 = 5² Σ5 + Σ6 = 15 + 21 36 = 6² Σ6 + Σ7 = 21 + 28 49 = 7²
And so you get the idea. There are patterns just falling out of the sky, and all we as mathematicians need to do is pick them up. And yes, there is a mathematical way even of expressing the fact we just saw, how successive sums equal a square
( x² + x ) ÷ 2 + [ ( x - 1)² + ( x - 1 ) ] ÷ 2
= ( x² + x ) ÷ 2 + ( x² - x ) ÷ 2
= [ ( x² + x ) + ( x² - x ) ] ÷ 2
= ( 2x² ) ÷ 2 = x²
This is because the square is of the greater of the two numbers summed. For instance, to find the two sums of numbers which equal 4², we had to add the sums of all numbers up to 4, and those up to 3, so for the purpose of our formula, in this case, x would equal 4, and ( x - 1 ) would be equal to 3.
What we see when we do add successive sums to equal a square also works out mathematically, which is what you should expect. Get into the habit of testing your observations and hypotheses in this way, to make sure that what you observe one way, you can also perceive in another.
Now, let’s say we do the same as before to the idea which we saw a little while before, that successive squares have a difference of an odd number between them. Let’s call a given number x, and its square x². This means then that the next square in line can be given the designation ( x + 1 )², since it is the next square in line, being the square of the number one more than x.
Now, it is also important not to confuse ( x + 1 )² with x² + 1, since most of the time this is not the same thing. Now let’s check my current statement, that the difference between successive squares is odd :
( x + 1 )² - x²
= x² + 2x + 1 - x²
= 2x + 1.
This tells us that the difference between the square of any given number, and the square of the next one is one more than twice the lesser number. If you think about it, 2x + 1 is always an odd number, because the 2x part is even, and then adding one to any even number will certainly make it odd.
Let’s check :
x = 4, ( x + 1 ) = 5 ; x² = 16,
so if what I have just worked out is true, ( x + 1 )², or that is, 5², ought to equal :
16 + (2 TIMES 4) + 1 = 16 + 9 = 25,
which is right. And this works for the difference between any successive squares. In fact, you could work out the difference between squares that are not even of successive numbers, but this will give you a number to add other than 2x + 1, which, if you practise at this kind of thing a bit, you will be able to find.
For instance :
x = 6, x² = 36 (x + 3) = 9.
( x + 3 )² - x² = x² + 6x + 9 - x = 6x + 9,
so that the difference between the squares of any two numbers three apart in value is always 6x + 9. And thus we have : the difference between 6² and 9²
= ( 6 TIMES 6 ) + 9 = 45
36 + 45 = 81 = 9².
So, as I have said before, mathematics is simply full of interesting discoveries like this, that show how numbers form intricate and varied patterns with themselves in the most seemingly unlikely of places. What’s more, we shall definitely be seeing a lot more of these patterns as we progress in our study of this brilliant and useful subject.
As another example, if I wanted to add together just two successive squares, the sums of these would differ from each other by a consistent pattern.
0 + 1 = 1 ; 1 + 4 = 5 ; 4 + 9 = 13 ; 9 + 16 = 25
You might notice that the difference between each sum here increases by four each time. This works also if you add three, four, or even five squares in a row ; there is a pattern of some sort, and in addition, the numbers alternate odd and even, just like squares themselves do, and the diagram below shows clearly how each square is the sum of successive odd numbers.
Gnomons
Comments on Gnomons
The L - shaped lines between the dots are referred to as Gnomons. We see the dots are divided into sections, where each L – shaped group of dots between two Gnomons is an odd number, This means that the sum of all the dots bordered in this case above and to the left is a square number, so that between each odd number is added in turn to each square number to get the next one. In addition, each odd number is itself the sum of two successive numbers : 3 = 1 + 2 ; 5 = 2 + 3 ; 7 = 3 + 4 ; 9 = 4 + 5, and so on. But also, if we add certain alternating odd numbers to get triangular ones, we can have a pair of these triangular numbers to add together to get the square they are the sum of. For example, take the Gnomon equal to 1, add to it the one equal to five, then the one equal to 9, this gives us a sum of fifteen, which is a Triangular Number. The sum of all dots included up to where the odd number 9 is, will be the square number 25, the sum of the two triangular numbers 10 and fifteen.
There is also a way that one can arranges the cubes of numbers into a pattern known as a Cubic Gnomon. Since we shall move on to look at Cubes in another Hub, we will show it there.
Patterns among Squares of Numbers
If you square an odd number, then subtract one from it, your answer will always be a multiple of eight, and if you square an even number, which itself is not divisible by three, then also subtract one, your answer this time will be a multiple of three.
1² - 1 = 8×0 ; 3² - 1 = 8×1 ; 5² - 1 = 8×3 ; 7² - 1 = 8×6
2² - 1 = 3×1 ; 4² - 1 = 3×5 ; 8² - 1 = 3×21 ; 10² - 1 = 3×33
Now the numbers we get to multiply eight by in the case of odd numbers happen to be sums of numbers, ( triangular ), which we have just seen as half the difference between a number and its own square, and we shall come across them again later.
Last Word
The numbers [in braces] represent the gaps between our main values. These also happen to equal the sums of the differences between successive squares, so that the first number, eight, is equal to 3 + 5 ; 3 being the difference between 1² and 2², and five that between 2² and 3². In like manner, the 2nd number, 16, is equal to 7 + 9, with seven the difference between 3² and 4², while nine is that difference between 4² and 5².
The chart for even numbers squared then reduced by one also holds a pattern, but I prefer not to show it here. You may wish to make up your own tables like the one above, and find out yourselves.
This is it so far for Squares. We now move on to Higher Powers such as Cubes, and this discussion is found in the next Hub, And then there were Three - a Study on Cubes , and if You are curious, take a look at the other Hubs, The Maths They Never Taught Us - Part One, The Maths They Never Taught Us - Part Two , The Maths They Never Taught Us - Part Three, Moving on to Higher Powers - a First look at Exponents, The Power of Many More - more on the Use of Exponents, Mathematics - the Science of Patterns , More on the Patterns of Maths, Mathematics of Cricket , The Shape of Things to Come , Trigonometry to begin with, Pythagorean Theorem and Triplets, Things to do with Shapes, Pyramids - How to find their Height and Volume, How to find the Area of Regular Polygons, The Wonder and Amusement of Triangles - Part One, The Wonder and Amusement of Triangles - Part Two, the Law of Missing Lengths, The Wonder and Amusement of Triangles – Part Three : the Sine Rule, and The Wonder and Amusement of Triangles - Part Four : the Cosine Rule.
Also, feel free to check out my non Maths Hubs :
Bartholomew Webb , They Came and The Great New Zealand Flag
Just take a good look then see if you can come up with anything along the same lines. By all means Your comments are very welcome, as well as feedback and suggestions which would be given due credit, or indeed have a go and publishing Your ideas Yourselves, but firstly, by all means, add Your comments - it's a free Country.
Disclaimer
Any reference to any Copyright or Registered Trademark is credited as such. Some discoveries are my own, but may also have been found independently by others. Some information has been referenced in a number of publications, most in the public domain, as well as on Wikipedia ( copyright 2013 Wikimedia Foundation ). Also use of the words I didn't get where I am today, are of course those used by the late John Barron ( 1920 - 2004 ), as Charles Jefferson ( CJ ) in the Rise and Fall of Reginald Perrin and the Legacy of Reginald Perrin.
