Zero in Mathematics
Zero in Mathematics
Zero as a number is incredibly tricky to deal with. Though zero provides us with some useful mathematical tools, such as calculus, it presents some problems that if approached incorrectly, lead to a breakdown of mathematics as we know it.
Adding, subtracting and multiplying by zero are straightforward.
If c is a real number, c+0=c c-0=c c x 0=0
These facts are widely known and regarded to hold true in every situation.
However, division by zero is a far more complicated matter. With most divisions, for example,
10/5=2
We can infer that
2 x 5=10
But if we try to do this with zero,
10/0=a
0 x a=10
Can you think of a number that, when multiplied by 0, equals 10? There is no such number that we have ever encountered that will satisfy this equation.
Another example will emphasise the mysteriousness of dividing by zero.
One may assume that
(c x 0)∕0=c
The zeroes should cancel, as would be done with any other number. But since we know that c x 0=0 it follows that
(c x 0)/0=0/0=c
This does not seem to make sense. This also means that
1=0/0=2
1=2 since 1 and 2 are both real numbers. Actually, this means that 0/0 is equal to every real number!
In effect, there is no real answer to a division by zero. It cannot be done.
In fact, if we could divide by zero, it would be possible to prove anything that we could dream of. For example, imagine a student trying to prove to his teacher that he completed his homework, despite the fact that the teacher knows he has not done it. With some basic algebra skills and the power of zero, this can be easily done.
(do you think I should put the following in an appendix?)
The student would show the following ‘proof’ to the teacher.
Let a=1 and b=1. This means that b2 = ab (eq. 1) and a2 = a2 (eq. 2)
If we subtract eq. 1 from eq. 2 a 2 – b 2 = a 2 - ab
We can factorise the left hand side to (a+b)(a-b) and the right hand side to a(a-b)
(a + b)(a – b)
Zero as a number is incredibly tricky to deal with. Though zero provides us with some useful mathematical tools, such as calculus, it presents some problems that if approached incorrectly, lead to a breakdown of mathematics as we know it.
Adding, subtracting and multiplying by zero are straightforward.
If c is a real number, c+0=c c-0=c c x 0=0
These facts are widely known and regarded to hold true in every situation.
However, division by zero is a far more complicated matter. With most divisions, for example,
10/5=2
We can infer that
2 x 5=10
But if we try to do this with zero,
10/0=a
0 x a=10
Can you think of a number that, when multiplied by 0, equals 10? There is no such number that we have ever encountered that will satisfy this equation.
Another example will emphasise the mysteriousness of dividing by zero.
One may assume that
(c x 0)∕0=c
The zeroes should cancel, as would be done with any other number. But since we know that c x 0=0 it follows that
(c x 0)/0=0/0=c
This does not seem to make sense. This also means that
1=0/0=2
1=2 since 1 and 2 are both real numbers. Actually, this means that 0/0 is equal to every real number!
In effect, there is no real answer to a division by zero. It cannot be done.
In fact, if we could divide by zero, it would be possible to prove anything that we could dream of. For example, imagine a student trying to prove to his teacher that he completed his homework, despite the fact that the teacher knows he has not done it. With some basic algebra skills and the power of zero, this can be easily done.
(do you think I should put the following in an appendix?)
The student would show the following ‘proof’ to the teacher.
Let a=1 and b=1. This means that b2 = ab (eq. 1) and a2 = a2 (eq. 2)
If we subtract eq. 1 from eq. 2 a 2 – b 2 = a 2 - ab
We can factorise the left hand side to (a+b)(a-b) and the right hand side to a(a-b)
(a + b)(a – b)