Study Guide: Division
Terms to Understand
Dividend
Divisor
Quotient
Remainder
Bring Down
Place Value
Definition of Division
Division is one of the four basic arithmetic operations (+, -, $\times$, $\div$). It is the opposite of multiplication. This means that division undoes multiplication, and multiplication undoes division. Division can be thought of as a method for distributing a large group into several equal and smaller groups (or parts).
In multiplication, ${\color{blue}3}$ groups of ${\color{green}4}$ parts can be combined to become one group of ${\color{red}12}$ parts:
\[ {\color{blue}3}\times{\color{green}4}={\color{red}12} \].
In division, a large group of ${\color{red}12}$ parts can be divided into ${\color{blue}3}$ equal and smaller groups, each containing ${\color{green}4}$ parts:
\[ {\color{red}12}\div{\color{blue}3}={\color{green}4} \]
Notation for Division
Symbols for division include a slash, ($\text{a/b}$), a fraction line, $(\dfrac{a}{b})$, and the division sign ($\text{a}\div \text{b}$). Each of these expressions is read, “a divided by b”.
Dividends, Divisors, Remainders and Quotients
The first number (or numurator) is the Dividend, and the second number (or denominator) is the Divisor. The result (or answer) is the Quotient, and any left-over amount is the Remainder.
In the below example, the dividend is ${\color{red}14}$, the divisor is ${\color{blue}4}$, the remainder reduces to $\frac{1}{2}$, and the simplified quotient (or answer) is ${\color{teal}3\frac{1}{2}}$.
$$ {\color{red}14} \div {\color{blue}4} = 3\frac{2}{4} = {\color{teal}3\frac{1}{2}} $$
Remainders
Not all numbers divide evenly into other numbers. Sometimes something is left over. That last bit is always less than the divisor, and is called the remainder.
For example, dividing 4 into 6, or $6 \div 4$, results in a quotient of 1 and a remainder of 2. This is sometimes written as, $6 \div 4 = 1 \text{ R}2$. But it’s much better to write remainders as either fractions or decimals.
Written as a fraction, the remainder is the numerator, and the divisor is the denominator. It this example, the remainder is written: $\frac{2}{4}$
\[ 6 \div 4 = 1\frac{2}{4} \]
Of course this fraction can be simlified. That’s one of the advantages of using fractions in remainders. They make it easy to see the relationship between the remainder and the divisor.
\begin{array}{rl} 6 \div 4 &= 1\frac{2}{4} \\\ &= 1\frac{1}{2} &Simplified \\\ &= 1.5 & Decimal; form \end{array}
Division of a Fraction
\begin{array}{rll} 14 \div 4 &= \frac{14}{4} &\text{Fraction form} \\\ &= 3 \frac{2}{4} &\text{14 contains 3 groups of 4, with a remainder of 2} \\\ &= 3 \frac{1}{2} &\text{Fraction reduced } \\\ &= 3.5 &\text{Decimal form} \\\ \end{array}
Long Division
Long division is a method used to break harder division problems into a sequence of easier steps.
$$ \require{enclose} \begin{array}{rll} {\color{green}1}{\color{darkorange}2}{\color{maroon}5} && \hbox{Quotient (or answer)} \\[-3pt] {\color{red}4} \enclose{longdiv}{500}\kern-.2ex && \hbox{Original problem ($500 \div {\color{red}4}$)} \\[-3pt] \underline{{\color{blue}-4}\phantom{00,}} && \hbox{Multiply: ${\color{red}4} \times {\color{green}1} = {\color{blue}4}$} \\[-3pt] 10\phantom{0} && \hbox{Subtract $5, – 4 = 1$. Bring down the 0 in the Tens Column.} \\[-3pt] \underline{\phantom{0}-8\phantom{0}} && \hbox{Multiply: ${\color{red}4} \times {\color{darkorange}2} = 8$} \\[-3pt] \phantom{0}20 && \hbox{Subtract $10, – 8 = 2$. Bring down the 0 in the Ones Column.} \\[-3pt] \underline{\phantom{0}-20} && \hbox{Multiply: ${\color{red}4} \times {\color{maroon}5} = 20$} \\[-3pt] \phantom{00}0 && \hbox{Subtract. No remainder and nothing more to bring down.} \end{array} $$
$$ \textit{Thus: } 500 \div 4 = 125 $$
Note that because there are so many steps in long division, you should always check your work by using multiplication (the inverse operation).
$$ 125 \times 4 = $$
Division by Zero
Because division by zero does not make logical sense, it is considered “undefined”. On most computers and electronic calculators, dividing by zero results in an error.
For any value of $x$:
\[ \frac{x}{0}=\text{Undefined} \]