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Algebraic Expressions

Algebraic Expressions

At a Glance - 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's a much shorter way: write 3x. When multiplying a number by a variable, we can write the number and the variable side by side. They get along swimmingly, so there's 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 × 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're 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're 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 3x, not x3. If you do write x3 people will probably know what you mean, but you probably won't be invited back to the convention.

Also, you should know 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 x17 in place of 17x, but it's 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 = x3. We can read x3 as "three copies of x," since x3 is an abbreviation for three copies of x multiplied together. Too bad we don't need 100 copies, 'cause then we'd get 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's safer to write division using fraction notation than it is to write division using the slash. Not that you'll be in any real physical danger if you do the latter,  but it isn't advisable and here's why. The expression 1/4x  is ambiguous, since it could mean either or . Avoid the problem by simply not writing 1/4x. No, your solution of avoiding the problem by skipping algebra altogether isn't a valid one. Nice try.

Exercise 1

Write y · y · y · y · y using exponent notation. 

Exercise 2

What are four different ways to write "q times r" in symbols? 

Exercise 3

What's wrong with abbreviating "y times 5" as y5? 

Exercise 4

How do we write "2 copies of y" in symbols?

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