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We have seen 0 as a number and its role in the place value system. We shall see that 0 also has some peculiar properties. Many of them can be derived only by using the logic of math.
This is an indicator of the power of arithmetic for answering questions about things which may not even exist in real life!
Is 0 Odd or Even?
Till now, we have used the imagery of sharing or pairing objects to understand the idea of odd & even numbers.
But what about 0? Is it an even or an odd number?
But 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.
Addition & Subtraction With 0
This is very easy to imagine. If we are talking about toffees, we can imagine 0 as a cup with no toffees in it.
Imagine that we have a jar of toffees. If we add a cup with 0 toffees or remove a cup with 0 toffees, the number of toffees in the jar will not change. Hence any number + 0 or any number – 0 will result in the same number.
In math we can say a + 0 = a & a -0 = x.
Multiplication With 0
We can imagine 8 X 0 as 8 baskets with 0 fruits in each of them. So 8 X 0 = 0.
What about 0 X 8? We cannot use the imagery of 0 baskets with 8 fruits in it. But we can use the logic of math.
As per commutative property of multiplication, 0 X 8 = 8 X 0.
Hence 8 X 0 = 0 X 8 = 0
In math we can say a X 0 = 0
Division With 0
We know that 8 ÷4 can be described as 8 apples shared equally among 4 baskets. Hence each basket would have 2 apples.
But what about 8 ÷0?
We cannot visualise 8 apples shared equally among 0 baskets.
Someone could say that it can be imagined as 0 apples per basket. Hence 8 ÷0 = 0
By the same argument even 9 ÷0 would be 0.
So these are not correct.
Another argument is that 8 ÷ 8 = 1, 8 ÷ 4 = 2, 8 ÷ 2 = 4 & 8 ÷ 1 = 8 & 8 ÷ ½ = 16.
We see that as we divide with smaller & smaller numbers, the quotient keeps on increasing.
We can think of 0 is a very very small number. Hence the answer would be a very very large number.
So 8 ÷ 0 = ∞ (infinity)!
But the problem is that this is true for any number.
So mathematicians have decided to call this an “undefined operation”.
By the same logic 0 ÷ 0 is also “undefined”.
In math we say “a ÷ 0” is undefined.
Exponentiating With 0
What about 2^0 ?
We know that 2^3 = 2 X 2 X 2 (3 times).
We cannot apply the same thing and say 2^0= 2 X 2……. 0 times. It does not make sense.
Hence we have to use a mathematical logic.
2^4 = 2^8 ÷ 2, 2^2 = 2^4 ÷ 2, 2^1 = 2^2 ÷ 2
Hence 2^0 = 2^1 ÷ 2 = 1
You can check that any number “raised” to 0 will be 1.
Hence we can say that a^0 = 1
What about 0^0
We use the exponentiation rule that 0^0 = 0^ (1-1)
0^0 = 0^1/ 0^1 = 0/0 which we have already found is "undefined".
Factorial of 0
4! Is defined as 4 X 3 X 2 X 1 & 3! = 3 X 2 X 1
We cannot use this fundamental definition to get 0! We use math logic.
We can write 4! = 4 X (3 X 2 X 1) = 4 X 3! = 4 X (4 – 1)!
So 3! = 3 X 2! = 3 X (3 – 1)!
2! = 2 X 1! = 2 X (2 -1)!
1! = 1 X (1 -1)! = 1 X 0!
Hence 1 = 1 X 0! (using the value of 1!)
Hence 0! = 1
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