How to Divide by Zero?

Division by zero means that when any non-zero, positive, or negative number is divided by zero the result is always undefined. 

Related Topics

• How to Multiply and Divide Fractions
• How to Divide Mixed Numbers

A step-by-step guide to division by zero

Division by zero is considered as undefined where zero is the denominator or the division and is expressed as \(\frac{a}{0}\), \(a\) being a number or numerator or dividend. Dividing zero with any number will always give us a zero no matter with multiplication or division. Since multiplication and division go hand in hand, division by zero can also mean multiplication by zero. For example, \(\frac{7}{0} = x\) or \(0 × x = 5\). Here \(x\) has no value or number to complete the equation. Therefore, division by zero of any number is not defined.

One divided by zero

A division by zero is also considered undefined. According to mathematicians, any non-zero number, positive or negative divided by zero, is not defined because there is no value.

For example, let’s try to define \(\frac{1}{0}\). If we divide a non-zero number by a small positive number, we get a value. The small positive number is close to zero but not zero.

\(\frac{1}{0.1}= 10, \frac{1}{0.01} = 100, \frac{1}{0.000001} = 1,000,000\)

Hence, as we divide one by smaller and smaller positive numbers we get larger and larger positive numbers. Therefore, \(\frac{1}{0} = + infinity\).

When we divide negative numbers that are close to zero, we get:

\(\frac{1}{-0.1} = -10, \frac{1}{-0.01} = -100, \frac{1}{-0.000001} = -1,000,000\)

One divided by a negative number closer to zero brings in a different result that is closer to negative infinity. Therefore, \(\frac{1}{0} = -infinity\). Therefore, one divided by zero has no answer which results in being undefined.

Important notes about division and zero

Below are some facts about division and zero:

• Any number divided by \(1\) (the quotient equals the dividend), answers the same as the dividend. In other words, 1 is the divisor and the quotient will be equal to the dividend. For examples: \(23 ÷ 1 = 23\).
• A number cannot be divided by \(0\) and therefore the result is not defined. Example: \(60 ÷ 0 =\) undefined (but \(0 ÷ 60 = 0\)).
• When the dividend equals the divisor, which means the same numbers but not \(0\), then the answer is always \(1\). For examples: \(46 ÷ 46 = 1\).
• Zero is a real number, an integer, a rational number, and a whole number.
• Zero is always neutral i.e. zero is never written like \(+0\) or \(-0\).
• The power of any number that is raised by zero is always one.

Exercises for Division by Zero

Find each quotient.

1. \(\color{blue}{\frac{29}{0}}\)
2. \(\color{blue}{\frac{0}{125}}\)
3. \(\color{blue}{\frac{867}{0}}\)

1. \(\color{blue}{undefined}\)
2. \(\color{blue}{0}\)
3. \(\color{blue}{undefined}\)