# Algebraic Expressions

# Variable Notations

Remember that mathematicians love to abbreviate things (RTMLTAT, for short). To write "3 multiplied by 4" in symbols, we could write 3 · 4, 3 × 4, or (3)(4). To write "3 multiplied by *x*" we could also write 3 · *x*, 3 × *x*, or (3)(*x*). However, there is a much shorter way: write 3*x*. When multiplying a number by a variable, we can write the number and the variable side by side. They get along swimmingly, so there is no need to separate them with a symbol. We can't do the same when multiplying numbers together, because if we write 2 next to 4, for example, we get 24. If you think that 2 times 4 is 24, then you may have taken a 2 x 4 to the back of the head.

When multiplying two (or more) variables, we also write the variables next to each other to show that they are being multiplied. For example, *xy* means "*x* times *y*." This is another reason that we go with such rarely used letters as our variables. If we used *a* and *b* most of the time, you might see *ab* and think we are talking about somebody's six-pack.

The mathematical convention (the usual way of doing things) is to write the number before the variable when multiplying numbers by variables. In other words, we write 3*x*, not *x*3. If you do write *x*3 people will probably know what you mean, but you likely won't be invited back to the convention.

Observe that *xy = yx* since multiplication of real numbers is *commutative.* When multiplying variables together, it can be helpful to write the variables in alphabetical order (*xy* or *xyz*), so we have a standard order in which to write them. Writing *yx* instead of *xy* isn't nearly as bad as writing *x*17 in place of 17*x*, but it is still frowned upon in certain circles. Generally the circles frequented by us math nerds. You scoff, but our frowns can be intimidating.

When we multiply a variable by itself several times—almost like cloning, but much less controversial—we can use *exponent* notation. For example, *x* ·* x* · *x* = *x ^{3}*. We may read

*x*as "three copies of

^{3}*x*," since

*x*is an abbreviation for three copies of

^{3}*x*multiplied together. It is too bad we don't need 100 copies, as then we receive a price break.

When dividing a variable by a number, there are a couple of different ways to write the division in symbols. Since

,

then and both mean "*x* divided by 4." In this expression, the *x* could not possibly stand for the United States of America, because our nation is indivisible. Pledge of allegiance, represent.

**Be careful**: It is safer to write division using fraction notation than it is to write division using the slash. Not that you will be in any real physical danger if you do the latter, unless you encounter a real fraction bully, but it isn't advisable and here's why. The expression 1/4*x* is ambiguous, as it could mean either or *.* Avoid the problem by simply not writing 1/4*x*. No, your solution of avoiding the problem by skipping algebra altogether is not a valid one. Nice try.