ZERO, odd or even?

By Suman Kandel

The number zero is beautiful and at the same time amazing and surprising number in modern mathematics. It has such properties and some wonders inside it that we all may want to know.

History

The word ‘zero’ came into existence as:
Śūnya →          ṣifr →         zefiro →      zero        → zero
(Sanskrit)    (Arabic)       (Italian)     (French)       (English)

A symbol for zero, a large dot, is used in the Bakhshali Manuscript, which contains problems of arithmetic, algebra and geometry. In 2017, three samples from the manuscript were shown by radiocarbon dating to come from three different dates, AD 224-383, AD 680-779, AD 885-993.

Even and Odd

Before defining even and odd numbers, let’s make us familiar to few mathematical terms:
Set of integers: The set of all positive and negative numbers along with zero are integers. It is denoted by ℤ and defined as, ℤ = {…,−3,−2,−1,0,1,2,3,…}
From above, we see that, the set of integers is an extended set of whole numbers where we duplicate each whole number by putting a minus (-) sign before it.

Odd numbers:

You already know what an odd number means. We all have heard that the number which is divisible by 1 and itself is an odd number and it is not divisible by 2. Examples are: 1, 3, 9, 217, etc.
That is an elementary definition. Let us try a mathematical one.
A number, p, is said to be odd if it can be expressed as;
p = 2q+1
where q ∈ ℤ.
Pick a number, let it be 1.
Now,
1 = 2*0 + 1
Where 0 ∈ ℤ
For 217,
217 = 2*108 +1
For 999997,
999997 = 2*499998 + 1
And so on.

Even numbers:

We are also familiar with primary level definition of even numbers. We were told that any number which is divisible by 2 is even. That’s true.
In mathematical form, it would be;
A number, p, is said to be even number if it can be expressed as;
p = 2q,
where q∈ ℤ.
Let’s try some numbers.
4 = 2*2, 2∈ ℤ.
Thus, 4 is even.
100 = 2*50, 50∈ ℤ.
Thus, 100 is even.
Now, take a big one, 25686.
25686 = 2* 12843, 12843∈ ℤ.
Thus, 25686 is even.

To zero again

Now, try to make 0 fit for the above definitions. Which one does it satisfy?

The odd? NO.

It satisfies the definition of an even number.
0 = 2*0
Now, recall from the definition of set of integers that, 0∈ ℤ.
Thus, from the definition of odd and even, we clearly see that 0 is even.

Some more properties:

(1) An even number lies between two odd numbers. For example: 4 lies between 3 and 5, 100 lies between 99 and 101. Here 3,5,99 and 101 are odd.
What for zero?


From the number line, it can be known that 0 lies between -1 and 1. -1 and 1 are odd. Thus, we can say that 0 should be even.

(2) The purity of evens.
Pick out an even, 26.
Now, 26/2 = 13.
13 is an odd number.
For 82,
82/2 = 41
41 is an odd number.
Here, the even numbers, 26 and 82 are said to be singly even.
There are many singly even, like, 34, 142, 1002, etc.
Let’s take an even, say, 4.
Now, 4 /2 = 2
2 is an even.
2/2 = 1.
4 can be divided by 2, two times.
Also, for 12,
12/2 = 6
6 is an even.
6/2 = 3
12 can be divided by 2, two times.
Here, 4 and 12 can be referred being as doubly even.
Other double evens are, 24, 264, 1024, etc.
Again, for 0.
0/2 = 0, 0 is even.
0/2 = 0, 0 is even
0/2 = 0, 0 is even
0/2 = 0, 0 is even
0/2 = 0, 0 is even
0/2 = 0, 0 is even
And so on.
Thus we observe that 0 is the purest even.

(3) ‘0’ is neither positive nor a negative number. The whole concept of positive and negative number is defined on the basis of 0. Numbers greater than 0 are positive while numbers less than 0 are negative. But 0 equals 0, which makes us absurd to define zero as positive or negative.

Source: wikipedia.org

 

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