And then there were Three - a Study on Cubes
One Lump or Two ?
So, having looked at squares as numbers multiplied by themselves, or raised to the power of two, as in The Very Next Step - Squares and the Power of Two , it makes sense to go on to the next power, which is three, the raising of numbers to which gives us what we know as cubes. Why do we call them cubes ? For pretty much the same sort of reason we give squares their name.
Consider for example the box in the illustration above, three feet by three feet by three feet. This is 3³, or 27 cubic feet. Here it can be seen that we have added another dimension to what we previously had with squares and area. This time we are dealing with volume, which is x times x².
The following is a list of numbers and their cubes :
Chart of Numbers and their Cubes
We can see for a start how large cubes do get very quickly, since this time around we are multiplying the same number by itself, and then by itself again.
An interesting thing to note, is that a number can be both a square and a cube at the same time. For example, we have 64, which is not only 8², but also happens to be 4³. As a matter of fact, 64 also equals 26, but we will get to higher powers of numbers in a later Hub, when we look at the Laws of Exponents.
So, in the way that an ordinary number, such as five, which is in fact 51, gives us the idea of length, squares, like 5² = 25 give us area, and cubes, where 5³ = 125, furnish us with volume.
Incidentally, if you’re curious to know what 50 is, this equals one. As a matter of fact, any number other than zero, which has been raised to the power of zero, will also equal one, and once again, we shall leave this to our discussion on the Laws of Exponents. And by discussion I mean that this is where I speak, and you listen.
One thing to note with powers, is that, just because, for example, 2 × 3 = 3 × 2, this does NOT mean that 2³ = 3², and in fact it doesn’t, since the first equals eight, while the second gives us nine. There are in fact some examples where ab= ba, since 24= 4² = 16, but otherwise you need to be aware that most of the time the power is NOT interchangeable with its base.
The simple reason is because we are multiplying different numbers a given number of times. That is, for example, if we have 2³, we are going 2 × 2 × 2, which is divisible by 2, but not 3, whereas 3², or 3 × 3, is divisible by 3, but is not divisible by 2.
Now the reason 24 just happens to equal 4², is because 4 also divides by two, but most of the time we are dealing with bases not evenly dividable by their exponents, and vice versa. It is only the base which determines what the answer is a product of, while the exponent is simply how many times the base is to be multiplied by itself, and is not itself normally a factor of the final answer.
There is another important fact to note when dealing with both areas and volumes, and that is what happens when you fiddle with the sides of either a square or a cube.
Say you have a simple length of five inches. If I double this, my length is simply doubled, and it becomes an even ten inches. But imagine the five inches this time represent the sides of a square, so that its total area is equal to 5² inches², which is twenty five square inches. What happens when I double both sides, so my square is now 10 by 10 ?
Well now my area is equal to 10² inches², or 100 square inches. The area has not doubled, but rather quadrupled. It is now four times what it was before. How does this work ? Think about it - you’re dealing with area, where things are now squared, so the amount by which you multiply your sides is also squared, and since you timesed it by two, your area gets to be multiplied by 2², which of course is 4.
Realise that this only occurs if you double both length and width. If you double only one, then you will only double the area. Also, if I triple my length and width from five inches to fifteen, I then end up with 3² = 9 times the area.
Now, if we are dealing with volume, we can see the pattern, so that, if we have a cube of say five inches by five by five, its volume equals 125 cubic inches³. We see that if we double all three dimensions to equal ten inches, our volume increases to 10³ = 1000 cubic inches, which this time is eight times the volume, simply because 2³ = 8, because we are cubing the amount by which we are multiplying the side, also, just as we squared before.
So, we cube the factor by which we multiply all three edges. And again, if we do not multiply all these edges of our cube, well, it is best then to take all three new dimensions, and multiply accordingly to find our new volume. The idea of cubing the amount by which we multiply all three edges is to be used as a shortcut to find the new volume, as long as all three edges have been multiplied by the same amount.
Now, just as squares are sums of successive odd
numbers, there’s a similar pattern to with our cubes :
And as with the squares, where there is a way to work out up to which odd number you need to add to find it, we can work out from which odd number, and to which other odd number we need to sum, in order to work out our cube.
We see for a start that there are as many odds to add as the number you are cubing. That is, 5³ is the sum of five odds, and so forth.
But the way to find both odd numbers that are the first and last of the summation sequence to be added with all the odds in between, is to find your value of x to be cubed, multiply it by ( x - 1 ) then add one, and this is your initial number, then to find the final one, you multiply x by ( x + 1 ), then subtract one, so that your first and last numbers in the sequence are given by x² - x + 1, and x² + x - 1, and that these become the bounds of your sum.
For example, if you wish to work out from which initial odd number to which last odd number to sum in order to determine 9³, we go : 9 × 8 + 1 ( = 73 ), and then 9 × 10 - 1 ( = 89 ), so that the nine odd numbers you sum to work out 9³ are : 73, 75, 77, 79, 81, 83, 85, 87 and 89, all of which, like 9³, equals 729.
The formulas x² - x + 1 and x² + x - 1 were worked out by looking at the list of sums of odds for each cube, and finding the pattern.
Another way to find the cube of a given number is to get the sums of the odd numbers used to find the square of that particular number, and then multiply it by that number, since after all, 3³ is just the same as saying 3² × 3.
For Example :
Patterns amongst Cubes
The pattern we get from these is that when we have at least three of them, we can see that each of the numbers added in parentheses ( ), which will give us the sum equal to the cube, are the same distance apart. For example, where we add (3 + 9 + 15), each of these has a difference of 6 between itself and the one after it. In this case, for the sum that is equal to 3³, this alternative gap of 6 equals 3 × 2, where x = 3, and 2 is the normal gap we get between the odd numbers we sum in order to get the square of whatever x is.
Thus we should be able to conclude that in each case if we want to find the numbers which sum to equal x³, the first number to which we add the next ones will be x itself, and then the gap between this and the next number we add will be 2x, which as noted, is the gap between all successive numbers after that in the sum to x³.
As another example : 2² = (1 + 3) ; × 2 = (2 + 6), and 2 here in parentheses equals x, since 2 is the number we are trying to find the cube of. The gap from the 2 to the 6 that are in the parentheses is 4, which is 2 times 2, or if you like, 2x.
If we analyse these numbers in parentheses, these multiples of the odd numbers summed to equal a square, which now add to be a cube, we see a pattern. They become sums of successive odd numbers, each multiplied by x, which makes sense, because this is in effect what we did to get them. If you think about it, it stands to reason, that if a certain sum of numbers adds to a square, then multiplying that whole sum, or each member of that sum individually by the number you have squared, will then give you its cube, since as noted before, x³ = x² × x.
The above illustration shows how the cubes of successive numbers can be arranged into Gnomons, where each colour represents a sum adding to the cube of the lowest number in the sum, known as the Base. This is owing to the fact that each cube is a multiple of its own base for a start.
Now each sum has a different formula. Take the fact that 27 = 3 + 6 + 9 + 6 + 3. If we take x as equal to 3 then this ends up being x + 2x + 3x + 2x + x, so it is that we get a kind of triangular sum of x’s, but we can also note that it could be seen in this specific case as x³ = x² + 6x.
What this means graphically is that when we draw that function, then x³ = x² + 6x implies also that x³ - x² - 6x = 0, and on the graph this would occur when x = 3 and y = 0
There shall indeed be more of this to come, with relevant explanations of these value specific functions.
We continue this journey in the next Hub, Moving on to Higher Powers - a First look at Exponents, 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, The Very Next Step - Squares and the Power of Two , 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 :
Most of the material here are my own discoveries, so anything that is Copyright and or a Registered Trademark will be acknowledged as such. Thus, 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 as Mathematics is a living language, and it was Ralph Waldo Emerson who described the demise of any language that does not keep growing. Some information has been referenced in a number of publications, most in the public domain, as well as on Wikipedia ( copyright 2013 Wikimedia Foundation ).