RWM101: Foundations of Real World Math

Much like the identity property of addition (see section 1.3), the identity property of multiplication states that there is a number that serves as the multiplicative identity that does nothing when it is multiplied against any other number. What special number behaves in this way? Two will not work since, for example, $2\times 3=6$ and $3$ was not left alone in this multiplication. It multiplied to become $6$! Can you also see why $-1$ and $5$ fail to be multiplicative identities? Take a minute to explore and practice multiplying various numbers before reading on, and you will likely stumble across that one special, lonely number that works.

As you may have figured out, the number $1$ has this magical, do nothing property. For example, $1\times 3=3$ and $(-5)\times 1=-5$. In short, the multiplicative identity property states that if you multiply any number by $1$, the answer is simply the number you started with.

And as you may have also guessed, the reason for emphasizing $1$'s special status as a do-nothing multiplicative identity is the same as that for zero's special status as a do-nothing additive identity: it can be useful to pay attention to these special rules and objects, particularly when using other, more abstract or new mathematical operations.