Multiplication by One

Throughout school we are taught that when something is multiplied by 1, it equals itself.

But as I am learning about higher level mathematics, I am understanding that not everything is as black and white as that (like how infinity multiplied by zero isn't as simple as it seems).

Is there anything in higher-level, more advanced, mathematics that when multiplied by one does not equal itself?

Solution 1:

Usually, if there is an operation called multiplication, it is defined as having an identity element. We call that element $1$. When we do that, we define $1\times x = x \times 1 = x$. Sometimes we only define one of the equalities because we have the power to derive the other one. If we don't have a $1$, we don't have a multiplicative identity.

Solution 2:

As Ross Millikan has pointed out, mathematicians like to call multiplicative identities $1$ because we like things to behave familiarly.

There are exceptions to this though, especially when we're working with sets of numbers that are endowed with structures other than the ones we usually use. The easiest example that comes to mind is the set $S=\{0,2,4,6,8\}$ with addition and multiplication defined modulo $10$ (in other words, if I multiply or add numbers and they exceed $10$, then I take the remainder dividing by $10$. So $6+8\equiv 4$ modulo $10$ since $14=10+4$, while $4\cdot 8\equiv 2$ modulo $10$ since $4\cdot 8=32=3\cdot 10+2.$) If you sit down and multiply each element of $S$ by $6$, you will find that $6$ is the multiplicative identity, not $1$ (which isn't even in the set).

This is kind of cheating though because in some sense $S$ with this structure is "the same" as $\{0,1,2,3,4\}$ with addition and multiplication defined modulo $5$, and in this case $1$ is is the multiplicative identity.