Using the Properties of Zero
Using the Properties of Zero
We have already learned that zero is the additive identity, since it can be added to any number without changing the number’s identity. But zero also has some special properties when it comes to multiplication and division.
Multiplication by Zero
What happens when you multiply a number by \(0?\) Multiplying by \(0\) makes the product equal zero. The product of any real number and \(0\) is \(0.\)
Definition: Multiplication by Zero
For any real number \(a,\)
\(a·0=0\phantom{\rule{2em}{0ex}}0·a=0\)
Example
Simplify:
(a) \(\phantom{\rule{0.2em}{0ex}}-8·0\)
(b) \(\phantom{\rule{0.2em}{0ex}}\frac{5}{12}·0\)
(c) \(\phantom{\rule{0.2em}{0ex}}0\left(2.94\right)\).
Solution
| (a) | |
| \(-8\cdot 0\) | |
| The product of any real number and 0 is 0. | \(0\) |
| (b) | |
| \(\frac{5}{12}·0\) | |
| The product of any real number and 0 is 0. | \(0\) |
| (c) | |
| \(0\left(2.94\right)\) | |
| The product of any real number and 0 is 0. | \(0\) |
Dividing with Zero
What about dividing with \(0?\) Think about a real example: if there are no cookies in the cookie jar and three people want to share them, how many cookies would each person get? There are \(0\) cookies to share, so each person gets \(0\) cookies.
Remember that we can always check division with the related multiplication fact. So, we know that
Definition: Division of Zero
For any real number \(a,\) except \(0,\frac{0}{a}=0\) and \(0÷a=0.\)
Zero divided by any real number except zero is zero.
Example
Simplify:
(a) \(\phantom{\rule{0.2em}{0ex}}0÷5\phantom{\rule{0.2em}{0ex}}\)
(b) \(\frac{0}{-2}\)
(c) \(\phantom{\rule{0.2em}{0ex}}0÷\frac{7}{8}\).
Solution
| (a) | |
| \(0÷5\) | |
| Zero divided by any real number, except 0, is zero. | \(0\) |
| (b) | |
| \(\frac{0}{-2}\) | |
| Zero divided by any real number, except 0, is zero. | \(0\) |
| (c) | |
| \(0÷\frac{7}{8}\) | |
| Zero divided by any real number, except 0, is zero. | \(0\) |
Now let’s think about dividing a number by zero. What is the result of dividing \(4\) by \(0?\) Think about the related multiplication fact. Is there a number that multiplied by \(0\) gives \(4?\)
Since any real number multiplied by \(0\) equals \(0,\) there is no real number that can be multiplied by \(0\) to obtain \(4.\) We can conclude that there is no answer to \(4÷0,\) and so we say that division by zero is undefined.
Definition: Division by Zero
For any real number \(a,\phantom{\rule{0.2em}{0ex}}\frac{a}{0},\) and \(a÷0\) are undefined.
Division by zero is undefined.
Example
Simplify:
(a) \(\phantom{\rule{0.2em}{0ex}}7.5÷0\)
(b) \(\phantom{\rule{0.2em}{0ex}}\frac{-32}{0}\)
(c) \(\phantom{\rule{0.2em}{0ex}}\frac{4}{9}÷0\).
Solution
| (a) | |
| \(7.5÷0\) | |
| Division by zero is undefined. | undefined |
| (b) | |
| \(\frac{-32}{0}\) | |
| Division by zero is undefined. | undefined |
| (c) | |
| \(\frac{4}{9}÷0\) | |
| Division by zero is undefined. | undefined |
We summarize the properties of zero.
Definition: Properties of Zero
Multiplication by Zero: For any real number \(a,\)
\(\phantom{\rule{4em}{0ex}}\begin{array}{c}a·0=0\phantom{\rule{2em}{0ex}}0·a=0\phantom{\rule{2em}{0ex}}\text{The product of any number and 0 is 0.}\hfill \end{array}\)
Division by Zero: For any real number \(a,\phantom{\rule{0.2em}{0ex}}a\ne 0\)
\(\phantom{\rule{4em}{0ex}}\frac{0}{a}=0\) Zero divided by any real number, except itself, is zero.
\(\phantom{\rule{4em}{0ex}}\frac{a}{0}\) is undefined. Division by zero is undefined.
Optional Video: Multiplying and Dividing Involving Zero
This lesson is part of:
Properties of Real Numbers