Multiplication

With the trend towards more and earlier calculator usage, some people have lost a comfort with basic arithmetic operations like $\;5\cdot 7 = 35\;$ and $\;8 + 6 = 14\;.$ It's a waste of valuable time to use your calculator for problems such as these.

In this section, your basic multiplication skills are brought ‘up to speed’ so you won't be wasting mental energy on arithmetic and will be able to concentrate on higher-level ideas.

In mathematics, a vertically centered dot ‘$\;\cdot\;$’ is often used to denote multiplication. Thus, $\;3\cdot 5\;$ denotes ‘$\,3\,$ times $\,5\,$’.

Consider the multiplication problem $\;3\cdot 5\;$:

One interpretation is $3\,$ groups of $\,5\;,$ so:

$3\cdot 5 = 5 + 5 + 5 = 15$

Another interpretation is $5\,$ groups of $\,3\;,$ so:

$ 3\cdot 5 = 3 + 3 + 3 + 3 + 3 = 15 $

Thus, multiplication is a shorthand for repeated addition.

Algebra uses letters to represent numbers. (LOTS more on this later on!)

The expression $\;2x\;$ means ‘$\,2\,$ times $\,x\,$’. This is a shorthand for the addition problem $\;x + x\;.$

Similarly, the expression $\;3y\;$ means ‘$\,3\,$ times $\,y\,$’. This is a shorthand for the addition problem $\;y + y + y\;.$

Thus, for example, $\;5\cdot 3x\;$ means $\;3x + 3x + 3x + 3x + 3x\;,$ which is $\;15x\;.$ It's much easier to think in terms of multiplication rather than repeated addition, so you immediately get $\;5 \cdot 3x = 15x\;.$ Similarly, $\;6\cdot 7t = 42t\;.$

To commute means to change places.

The Commutative Property of Multiplication states that for all numbers $\;x\;$ and $\;y\;,$   $x \cdot y = y\cdot x\;.$ That is, you can change the places of the numbers in a multiplication problem, and this does not affect the result.

The expression $\;x\cdot y\;$ can more simply be written as $\;xy\;.$ That is, juxtaposition (placing two things side-by-side) can be used to denote multiplication. Of course, this notation can't be used with numbers like $\,2\,$ and $\,3\,,$ because ‘$\,23\,$’ means the number twenty-three, not $\,2\,$ times $\,3\,$!

So, we can restate the Commutative Property of Multiplication:    For all numbers $\;x\;$ and $\;y\;,$   $xy = yx\;.$

If you're a sociable person, then you probably like being in groups; i.e., you like to associate with other people. In mathematics, associative laws have to do with grouping.

The Associative Property of Multiplication states that for all numbers $\,x\,,$ $\,y\,,$ and $\,z\,,$   $(x\cdot y)\cdot z = x\cdot(y\cdot z)\;.$

Notice that the order in which the numbers are listed on both sides of the equation is exactly the same; only the grouping has changed. The Associative Property of Multiplication states that in a multiplication problem, the grouping of the numbers does not affect the result.

Thanks to the associative property, we can write things like $\;2\cdot 3\cdot 4\;$ without ambiguity! Think about this—if the grouping mattered, then $\;(2\cdot 3)\cdot 4\;$ and $\;2\cdot (3\cdot 4)\;$ would give different results, so you'd always have to use parentheses to specify which way it should be done.

Multiplying a number by $\;1\;$ does not change it. In other words, multiplying by $\;1\;$ preserves the identity of the original number.

More precisely:   for all numbers $\,x\,,$   $x\cdot 1 = 1\cdot x = x\;.$ For this reason, the number $\;1\;$ is called the multiplicative identity.

Multiplying a number by $\;0\;$ always gives $\;0\;.$ More precisely:   for all numbers $\,x\,,$   $x\cdot 0 = 0\cdot x = 0\;.$ This fact is sometimes called the Multiplication Property of Zero.

Practice