Who can live without numbers?
Better still, what is a number?
The idea of numbers has fascinated humans for centuries and will continue to do so for ages to come. Numbers, at least in symbolic form, are akin to clothing that we put on magnitudes or quantities. Quantity is contemporarily taken to denote things that can be increased or decreased. The temperature of today can be represented using quanities. But the other fluid idea that enables our comprehension of the increase or decrease of quantities in question are nothing but an embodiment of our conception and manipulation of numbers. At deeper levels, perhaps, the metaphysical, and depending on whom you ask, numbers are a fabric upon which our whole cosmos is erected. An ancient Greek sect, known as the Pythagoreans, worshipped numbers.
The applications of numbers are evident in a wide range of domains. Distance, displacement, brokeness, richness, temperature, age, and an immeasurable complex of other notions are expressed using numbers. Added to their ubiquity is the immense wealth of knowledge that humans have accumulated and will continue to accumulate in designated and related fields, the sciences in particular, that use numbers as currency. If arithmetic were perceived as a boardgame, say like chess or checkers, then numbers will be the pieces, and the operations called addition, subtraction, multipication and division will be akin to the various moves that the game pieces, numbers in this case, can engage in. We, as players of the game, will move these pieces and convert them into other types of pieces with the assistance of the said operations.
In chess, for example, there are certain moves that oversee the promotion of a pawn to higher ranks like queen, knight, rook or bishop. Perhaps, adding 1 to 2 oversees a promotion of one of them to 3. In effect, the arithmetic operations permit us, the players, to reconfigure numbers, create new ones from old ones, dismantle others as in the case of division, combine some as is the case of multiplication and perhaps make some disappear as in the case of subtraction. Kids and those who’ve been kids before may have experienced some good moments while learning to count using sticks, pebbles, and other toys that are suitable for such. In using these objects to learn how to count and operate on numbers, there was the chance to rearrange these pebbles into rows and/or columns.
It’s time for a silly game. Suppose that we can represent numbers using discrete objects that we shall symbolize by dots, asterisks and perhaps some interesting characters. One dot or asterisk would symbolize the number 1. Two dots will symbolize the number 3, and so on. With that in place, what happens if we try to arrange these dots(numbers) in different orientations as we perform some playful operations of addition on them.
The table below shows the first 10 numbers as arrangements of a sequence/line of big dots.
| Number | Dot Representation |
|---|---|
| 1 | o |
| 2 | o o |
| 3 | o o o |
| 4 | o o o o |
| 5 | o o o o o |
| 6 | o o o o o o |
| 7 | o o o o o o o |
| 8 | o o o o o o o o |
| 9 | o o o o o o o o o |
| 10 | o o o o o o o o o o |
The shape traced by the table above reveals the form of a triangle. This form may be more evident if we erase the table’s headers and first column.
*
* *
* * *
* * * *
* * * * *
* * * * * *
* * * * * * *
* * * * * * * *
* * * * * * * * *
* * * * * * * * * *
The form assumed by the arrangement of asterisks above traces a triangle. In effect, the numbers from one to ten have been lined up such that each number, say the number 2, is on the second line and the number 3 on the third line, and so on. This reveals some interesting properties of numbers that can be explored further, alongside some additionally awe-inspiring operations.
The table and triangle that was observed above can be considered to represent the sum of numbers from a given number that’s designated as the start number, to the another given number that marks the end. For example, suppose we want to add all numbers from 1 to 5. It’s easy to use the asterisk symbol and the technique shown above. One can write 1 asterisk for the number 1, and then add 2 to 1, by writing 2 asterisks on the next line etc,.
In the light of the adding consecutive numbers as lines represented by a collection of asterisks the size of the magnitude of the number in question, we can shape these sums and reveal interesting geometric objects. In the same manner, we may say that we are shaping numbers.
As an exercises of this concept, consider summing 1 to 5. The arithmetic representation of this operation is: ( 1 + 2 + 3 + 4 + 5 = 15 )
The asterisk representation will be:
( * + * * + * * * + * * * * + * * * * * = * * * * * * * * * * * * * * * )
Suppose further that the plus(+) symbol is used in this game to signify the beginning of a newline and the numbers(asterisks) after any plus symbol should be placed on the next line. Doing such will leave us with:
*
* *
* * *
* * * *
* * * * *
And the sum total of these asterisks is 15.
This is an example of a more general operation called finding the sum of the first n numbers. So far, we’ve obtained the sum of the first 10 numbers, in first table above, and now the sum of the first 5 numbers, as shown in the results above. The sum in this case is the total number of asterisks that we can count from the game of representing and summing numbers using asterisks and newlines for plus symbols.
The stupid games above involved summing the positive whole numbers in increasing order.
What if one were to examine the numbers in increasing order, but eliminate all the even numbers. Perhaps, we’d obtain a numberline as shown below:
1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, …
That is, we’ve eliminated:
2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, …
It’s also evident that the odd numbers above are generated by continuously adding 2 to a given predecessor of a number, where the first number, and original predecessor, is 1. As such, 1 + 2 = 3, and adding 2 to 3 yields 5, which then consumes another 2 to form 7, and so on.
Now, what happens if we sum this lineup of numbers?
1 + 3 = 4 4 + 5 = 9 9 + 7 = 16
The first 3 sums above show something interesting. The sums are the squares of the numbers 2, 3, and 4.
If we add the next odd number, that is 9, to the last sum above, which is 16, we obtain 25 and 25 is the square of 5.
Hence we can obtain the squares of the first n numbers simply by playing with the first n odd numbers. Now, let’s use the asterisks notation to draw of display these numbers.
| Number | Sum | Presentation |
|---|---|---|
| 1 | 0 + 1 | o |
| 4 | 1 + 3 | o o |
| o o | ||
| 9 | 4 + 5 | o o o |
| o o o | ||
| o o o | ||
| 16 | 9 + 7 | o o o o |
| o o o o | ||
| o o o o | ||
| o o o o | ||
| 25 | 16 + 9 | o o o o o |
| o o o o o | ||
| o o o o o | ||
| o o o o o | ||
| o o o o o | ||
| 36 | 25 + 11 | o o o o o o |
| o o o o o o | ||
| o o o o o o | ||
| o o o o o o | ||
| o o o o o o | ||
| o o o o o o |
Again, if the table is erased, and if we view them individually, the form becomes more evident.
Representation for number 1
*
Representation for number 4
* *
* *
Representation for number 9
* * *
* * *
* * *
Representation for number 16
* * * *
* * * *
* * * *
* * * *
Representation for number 25
* * * * *
* * * * *
* * * * *
* * * * *
* * * * *
Representation for number 36
* * * * * *
* * * * * *
* * * * * *
* * * * * *
* * * * * *
* * * * * *
In section 2 above, the sum of the first n numbers was displayed as a right angled triangle. This section is just a redisplay of those triangles but in form of isoleces triangles. That is, we shall make them look like triangles which have 2 sides that are equal.
The pictures below are displays starting with 0 as the first number.
Representation for the sum of the first 1, number(s)[ 0 ]
Representation for the sum of the first 2, number(s)[ 1 ]
*
Representation for the sum of the first 3, number(s)[ 3 ]
*
* *
Representation for the sum of the first 4, number(s)[ 6 ]
*
* *
* * *
Representation for the sum of the first 5, number(s)[ 10 ]
*
* *
* * *
* * * *
Representation for the sum of the first 6, number(s)[ 15 ]
*
* *
* * *
* * * *
* * * * *
Representation for the sum of the first 7, number(s)[ 21 ]
*
* *
* * *
* * * *
* * * * *
* * * * * *
Representation for the sum of the first 8, number(s)[ 28 ]
*
* *
* * *
* * * *
* * * * *
* * * * * *
* * * * * * *
Representation for the sum of the first 9, number(s)[ 36 ]
*
* *
* * *
* * * *
* * * * *
* * * * * *
* * * * * * *
* * * * * * * *
As shown above, numbers, in concert with operations like addition, can be configured to assume interesting geometric shapes. More interesting shapes in higher dimensions like cubes can be obtained from such simple operations on numbers.
In general, such numbers that are obtained from adding classes of other numbers and arriving at interesting geometric shapes are called polygonal numbers.