Division By Zero

Division by zero is undefined in mathematics. When zero appears as the denominator, written as a/0, where ‘a’ is any number. Because no number multiplied by zero results in ‘a’, if a ≠ 0. Dividing zero by a non-zero number (like 0 ÷ 5) equals zero, but the reverse isn't true. For example, 8 ÷ 0 = x or x × 0 = 8. Here, x has no value that can make this equation true. Therefore, division by zero is undefined.

Here are some facts about division and zero:

• Dividing any number by 1 always results in the original number, since dividing by 1 doesn’t affect its value. For example, 20 ÷ 1 = 20.

• The result of any number divided by zero is undefined. For example, 50 ÷ 0 = undefined (but 0 ÷ 50 = 0).

• Zero can be considered a whole number, an integer, a real number, and a rational number.

• Dividing any number by itself (except zero) equals one. For example, 62 ÷ 62 = 

• Zero is neither positive nor negative, so it is simply written as +0 or -0. 

Any number divided by zero is undefined. For example, 1 ÷ 0 does not have an answer. If we divide 1 by a number close to zero, like 0.1, we get a large number, but division by zero itself is not possible, as it does not yield a valid result.

• 1 ÷ 0.1 = 10
   
• 1 ÷ 0.01 = 100
   
• 1 ÷ 0.000001 = 1,000,000

When you divide 1 by smaller and smaller positive numbers, the result increases and approaches positive infinity, whereas dividing 1 by smaller and smaller negative numbers gives results that decrease toward negative infinity.

• 1 ÷ -0.1 = -10
   
• 1 ÷ -0.01 = -100
   
• 1 ÷ -0.000001 = -1,000,000

When you divide 1 by a small negative number, the result becomes a large negative number and approaches negative infinity. However, dividing by zero itself has no answer, which makes it undefined.

In mathematics, when something is called undefined, it means there is no valid or meaningful answer.  When we do a mathematical calculation that defies logic, such as dividing by zero, we maintain that the result is undefined. For example, dividing by zero is undefined because it breaks the rules of arithmetic, where no number can make the equation work.

Some students may get confused while working with division by zero. Here are some tips and tricks that will be helpful for the students:

• Type any number ÷ 0 into a calculator, and it shows an error because division by zero is impossible.

• Dividing any number by zero, like 4 ÷ 0, has no value because you cannot break something into zero parts.

• If zero is divided by a non-zero number, like 0 ÷ 4 = 0. For example, zero cookies shared among 4 friends still means nothing for everyone.

• Every division problem can be checked by multiplication. For example, 10 ÷ 2 = 5 because 5 × 2 = 10. But for 10 ÷ 0 = x, no number x can be multiplied by 0 to make 10, so the division is undefined.