Numbers and Operations Study Guide for the Math Basics

Division

Division is the opposite of multiplication and as such, could be looked at as repeated subtraction. For example, if you wanted to find out what 12 divided by 3 is, you could just see how many times you can subtract 3 from 12. 12 - 3 = 9, 9 - 3 = 6, 6 - 3 = 3, and finally, 3 - 3 = 0. You can subtract 3 four times, so \(12\div3=4\). In this division equation, 12 is called the dividend, 3 is the divisor, and 4 is the quotient. Here’s another way to represent the problem.

\[\require{enclose} \begin{array}{r} 4 \\[-3pt] 3 \enclose{longdiv}{12} \\[-3pt] \end{array}\]

Of course, if you know your multiplication tables, this process can be sped up. Rephrase the question “how many times can 3 go into 12” as “ how many groups of 3 make up 12” which is multiplication problem “3 times what = 12?” The answer to this is “four groups of 3 make up 12”:

\[3 \cdot 4 = 12\]

Sometimes a number might not evenly divide into another. In this case, a remainder will be left over. Try this:

\[14 \div 3\]

In this case, 3 can go evenly into 12 or 15, but not 14. Choose the multiple of 3 that gives you just less than 14 (12) and , so put 4 on top (over the last place of the number you are presently dividing into).

\[\require{enclose} \begin{array}{r} 4 \\[-3pt] 3 \enclose{longdiv}{14} \\[-3pt] \end{array}\]

Now, multiply 4 and 3, put the answer below 14, and subtract to find the remainder.

\[\require{enclose} \begin{array}{r} 4 \\[-3pt] 3 \enclose{longdiv}{14} \\[-3pt] \underline{-12}\\[-3pt] 2 \\[-3pt] \end{array}\]

So the answer to \(14 \div 3\) is \(4\) remainder \(2\), or equivalently \(14\div 3 = 4\;R\;2\).

This process is crucial to understand if you want to divide numbers with multiple digits. Try this one:

\[168 \div 12\]

Instead of looking at multiples of 12 (you probably don’t have those memorized), look to see how many times 12 goes into just the first 2 digits of the dividend: 16. In this case, 12 goes into 16 only 1 time, so put 1 on top of the 6 and do the process above (remainders).

\[\require{enclose} \begin{array}{r} 1 \;\;\\ 12 \enclose{longdiv}{168} \\[-3pt] \underline{-12}\phantom{8} \\[-3pt] 4 \phantom{8} \\[-3pt] \end{array}\]

Now, drop down the next digit of the dividend (8) and place it behind the (pseudo)remainder. Repeat the process with 48. In this case, \(12 \cdot 4 = 48\), so place a 4 above the 8.

\[\require{enclose} \begin{array}{r} 1\;4 \\ 12 \enclose{longdiv}{1\,6\,8} \\[-3pt] -\;\underline{\;1\,2\downarrow} \\[-3pt] 4\;8 \\[-3pt] \end{array}\]

Now multiply that 4 by the 12 and place it underneath, subtracting like you did before to find the remainder.

\[\require{enclose} \begin{array}{r} 14 \\ 12 \enclose{longdiv}{168} \\[-3pt] \underline{-12}\phantom{8} \\[-3pt] 48 \\[-3pt] \underline{-48} \\[-3pt] 0 \\ \end{array}\]

The remainder is \(0\), so the answer to \(168 \div 12\) is \(14\).

Here’s another example just in case you need it:

\(868 \div 7\).

part 1:

\[\require{enclose} \begin{array}{r} 1 \quad\\ 7 \enclose{longdiv}{868} \\[-3pt] \underline{-7}\phantom{68} \\[-3pt] 1 \phantom{68} \\[-3pt] \end{array}\]

part 2:

\[\require{enclose} \begin{array}{r} 12 \;\;\\ 7 \enclose{longdiv}{868} \\[-3pt] \;\underline{-7\!\!\downarrow}\phantom{8} \\[-3pt] \quad 16 \phantom{8} \\[-3pt] \underline{-14} \phantom{8}\\ 2 \phantom{8}\\ \end{array}\]

part 3:

\[\require{enclose} \begin{array}{r} 124 \\ 7 \enclose{longdiv}{868} \\[-3pt] \;\underline{-7}\phantom{68} \\[-3pt] \quad 16 \phantom{8} \\[-3pt] \underline{-14}\!\! \downarrow\\ 28\\ \;\; \underline{28}\\ \;\; 0 \end{array}\]

So, \(868 \div 7 = 124\).

Order of Operations (PEMDAS)

The order of operations has been devised organically by mathematicians of the past as an agreed upon way to simplify expressions. It helps assure that each expression has only one possible simplified value.

Remember to follow the order of operations (PEMDAS):

Do everything in parentheses (P), left to right.
Evaluate any exponents (E), left to right.
Do all multiplication and division (MD), in order, left to right.
Then do all addition and subtraction (AS), in order, from left to right.

(Good way to remember: Please Excuse My Dear Aunt Sally)

Here’s how it works when using the order of operations correctly:

\[15 - (3 + 6) + 12 \div 4 \cdot 2^3\] \[15 - \quad 9 \quad + 12 \div 4 \cdot 2^3\] \[15 - \quad 9 \quad + 12 \div 4 \cdot 8\] \[15 - \quad 9 \quad + \quad 3 \;\, \cdot \;8\] \[15 - \quad 9 \quad + \quad 24\] \[\quad 6 \quad \;\;\;+ \quad24\] \[\quad\;30\]

Properties of Real Numbers

Real numbers (the combined set of all rational and irrational numbers) have certain properties that enable you to manipulate expressions and equations.

Commutative

The first of these properties, commutative property, allows for changing the order of terms in any addition or multiplication problem. Here’s an example:

\(2 + 3 = 3 + 2\) and \(3 \cdot 5 = 5 \cdot 3\)

It’s important to know that this (and many of the following properties) don’t work for subtraction or division. For example,

\(4 - 1 = 3\), whereas \(1 - 4 = -3\). Therefore, \(4 - 1 \ne 1 - 4\).

Associative

This property allows you to group (associate) terms any way you want in a multiplication or addition problem. For example:

\[2 + (3+5) = (2 + 3) + 5\]

and

\[6 \cdot (2 \cdot 4) = (6 \cdot 2) \cdot 4\]

Distributive

When combining multiplication and addition, the distributive property might be useful. Here are examples:

\(a(b + c) = ab+ac\) or \(3(5x + 4) =15x + 12\)

Identity Properties

The number 1 is called the multiplicative identity because anything times 1 is itself:

\[a \cdot 1 = a\]

Similarly, 0 is called the additive identity because anything increased by 0 is itself:

\[a + 0 = a\]

Inverse Properties

Two numbers are additive inverses if they add together to create the additive identity (0). For example, 3 and -3 are additive inverses because

\[3+(-3)=0\]

In general, a and -a are additive inverses.

Two numbers are multiplicative inverses if they multiply together to create the multiplicative identity (1). For example:

\(4\) and \(\frac{1}{4}\)are multiplicative inverses because \(4 \cdot \frac{1}{4} = 1\).

In general, \(a\) and \(\frac{1}{a}\) are multiplicative inverses, as are \(\frac{a}{b}\) and \(\frac{b}{a}\).