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We are aware only of the books of Elements ( 1 to 6) which deal with Plane Geometry. Books 7 to 9 deal with numbers & their properties.
Greeks explored properties of numbers by considering them as geometrical figures.
Greek is a Cultural rather than a Geographical Marker
We need to clarify here that the term "Greek" is used in a cultural rather than geographical sense. Greek mathematicians lived in cities spread over the entire Eastern Mediterranean from Italy to North Africa but were united by Greek culture and the Greek language. For example Euclid lived in Alexandria which is in today's Egypt.
What is 1?
1 is considered the root idea from which all numbers emerged. The idea of 1 can have several interpretations which are dependent on the context.
Anything which can be perceived as a "unified whole" can be called 1. A finger, or the head or the entire body of an individual can be represented as 1. Any set can be called 1.
A student or an entire class or an entire school or a country or the Universe itself can also be considered as 1. This is the starting point for the idea of a fraction or a part of a whole.
Numbers Have Several Properties
Each number has certain properties and it also has arithmetic relations with other numbers. Exploring the properties and relations of numbers would give students a good foundation of Number Sense and fluency in operations. In this chapter we would explore properties.
But most of these properties are taught only as abstract concepts. Primary children would understand and remember them if presented and explored visually through activities using concrete materials like tokens.
Before we proceed, we need to understand what an array is. An array is an arrangement of tokens in columns & rows as shown below. Children are familiar with this idea through assemblies & PT classes. The following array has 3 rows with 4 tokens in each row. For example this array can be described as a 3 by 4 or 4 by 3 array.
Even & Odd
One of the most basic properties of numbers is that they are either odd or even.
The given number of tokens should be arranged in rows, with each row having 2 tokens. Each subsequent row should be below the top one.
If all the tokens are used up, the number is Even. If there is a token left, then the number is Odd. It would also become clear that for any Odd number, of any magnitude, only 1 token would be left.
Usually in a 2 digit number such as 56, it is taught that the number is even because the number in the Units Place, in this case 6, is even. No one asks about why the number in the Tens place, in this case 5, which is odd, is not considered. Representing the number in bundles and sticks will reveal that all bundles (in the decimal system) are even (as they can be shared equally among 2 persons) and hence bundles need not be taken into account. Only the number in the Units place needs to be considered.
Parity & Humans
We have already seen that many animals seem to possess an elementary number sense. But recent research is revealing that honey bees are able to distinguish between odd & even. Obviously they must be using a kind of pattern identification.
This is considered the first instance of a non-human being displaying knowledge of a higher math concept. In higher math, the idea of odd & even is seen as modified into that of "parity".
The idea of odd & even seems to be built into our brain. Humans demonstrate accuracy, speed, language and spatial relationship biases when categorising numbers as odd or even.
For example, we tend to respond faster to even numbers with actions performed by our right hand, and to odd numbers with actions performed by our left hand.
We are also faster, and more accurate, when categorising numbers as even compared to odd. And research has found children typically associate the word “even” with “right” and “odd” with “left”.
Hence there seems to be an evolutionary basis for parity in our brains.
Possibly more research into animal brains may revel traces of parity even among non-humans.
Is Zero Odd or Even?
0 is not a quantity. It indicates absence of a quantity. Hence, we cannot use the imagery of a collection of objects to decide this question.
We need to use the logic of arithmetic, related to odd & even numbers, to decide if 0 is an even or odd number.
0 is considered as an even number. There are three ways to justify this.
1. In arithmetic, the idea of sharing in 2 equal parts is represented by the idea of dividing by 2 and 0 is divisible by 2. Hence 0 is an even number.
2. 0 + odd = odd & 0 + even = even because adding by 0 does not change a number. These equations would be true only if 0 is considered even.
3. One more than 0 is 1 which is odd. Two more than 0 is 2 which is even. Hence 0 is an even number.
This is an indicator of the power of arithmetic for answering questions about things which may not even exist in real life!
Composite (Rectangle) Numbers
If a given number can be arranged, if necessary, after several tries, into a rectangular array, like the above, then the number is Composite. Its factors would be the Number of Rows and the Number of tokens in a Row. Initially, they can be called Rectangle Numbers.
Any child can, after some trial & error, can find that 12 can be arranged in a 3X4 and 2X6 array. IN later years, she can find out that these are the factors of 12.
If a rectangular array has 3 tokens in a row, adding more tokens in the rows (in sets of 3) would give multiples of 3 & thus lead to multiplication facts of 3 and the multiplication table for 3!
Prime (Line) Numbers
If a given number cannot be arranged, even after several tries, into a rectangular array, then the number is Prime. The only way it can be arranged is in One Row with all the tokens in that row or One column. It resembles a Line and can initially be called Line Numbers.
7 can only be arranged in a line. In later years they can be introduced to the term ‘prime’.
Properties of Individual Numbers
Apart from these general properties, individual numbers have a number of interesting properties on which a huge number of books have been written.
For example 2 is the only even prime. 6 is the smallest Perfect Number.
4 is the only number which can be represented with the same two numbers in 3 different ways. 2 + 2, 2 X 2, 2 ^ 2.
16 is the only number which can be expressed as exponential relations between 2 unique numbers in two ways. 2^4 & 4^2
2520 is the LCM of all numbers from 1 to 10. It is also 7X12X30; the product of days of the week, months of the year and days of the month.
Perfect Numbers
A perfect number is a number the total of whose factors (excluding the number itself) is the same as the number itself.
The first perfect number is 6 since 6 = 1 + 2 + 3.
Other perfect numbers are 28, 496, and 8128.
Odd Perfect Numbers
All known Perfect numbers have been found to be even. Whether any of the Perfect numbers are odd is the oldest unsolved problem in math. Some of history's greatest mathematicians, such as Euclid, Euler, and Descartes, explored this problem but were unable to solve it completely.
No odd perfect number has been found till now.
Abundant Numbers
Numbers which have a large number of factors are useful in math.
This quality was defined as "abundance". An abundant number is defined as one which is less than the sum of its factors.
12 is the first Abundant number. The sum of its factors, 1,2,3,4,6 is 16. Its "abundance" is 16 - 12 = 4.
The first 10 abundant numbers are 12, 18, 20, 24, 30, 36, 40, 42, 48, 54.
There is even an opinion that it would have been more convenient for our place value system if it had been a Duodecimal (based on twelve) rather than Decimal.
Dudeney Numbers
Dudeney numbers are those where the sum of the digits of its cube are equal to the number itself.
One example is 18. 18^3 = 5832 and 5+8+3+3=18.
Other Dudeney numbers are 8 (512), 17 (4913), 26 (17576) & 27 (19683)
Automorphic Numbers
Automorphic numbers are those whose squares end with them. 76 & 25 are examples.
76^2 = 5776 & 25^2 = 625
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