To understand the inverse property of multiplication, let’s go through the given examples and then deduce the property.
### Examples
1. [tex]\(\frac{1}{5} \cdot 5 = 1\)[/tex]
- Here, [tex]\(\frac{1}{5}\)[/tex] is the multiplicative inverse of 5.
2. [tex]\(\sqrt{2} \cdot \frac{1}{\sqrt{2}} = 1\)[/tex]
- Here, [tex]\(\frac{1}{\sqrt{2}}\)[/tex] is the multiplicative inverse of [tex]\(\sqrt{2}\)[/tex].
### Explanation
The inverse property of multiplication can be stated as:
> For each real number [tex]\(a\)[/tex], there exists another real number [tex]\(b\)[/tex] such that:
[tex]\[ a \cdot b = 1 \][/tex]
where [tex]\(b = \frac{1}{a}\)[/tex], which is known as the "multiplicative inverse" or "reciprocal" of [tex]\(a\)[/tex].
This property holds for all real numbers except one specific number. Which number is it?
#### Zero as a Special Case
- The number [tex]\(0\)[/tex] does not have a multiplicative inverse. This is because there is no number that can be multiplied by [tex]\(0\)[/tex] to result in [tex]\(1\)[/tex]. Mathematically, [tex]\(\frac{1}{0}\)[/tex] is undefined.
### Conclusion
The inverse property of multiplication holds for all real numbers except [tex]\(0\)[/tex], because the multiplicative inverse of zero is not defined.
### Final Answer
The inverse property of multiplication: For all real numbers except [tex]\[\boxed{0}\][/tex] [tex]\[\boxed{a \cdot \frac{1}{a} = 1}\][/tex]
