The Four Operations with Numbers and Fractions

The four operations

The four operations are addition, subtraction, multiplication and division. They act on many forms of number: integers, decimals, proper and improper fractions, and mixed numbers. The mathematical meaning of an operation does not change when the numbers are negative or written in another form, although the most convenient calculation method may change.

Addition combines quantities. Subtraction finds a difference or removes one quantity from another. Multiplication can represent equal groups or scaling, while division can represent sharing or finding how many groups fit into a quantity.

The inverse connection between addition and subtraction provides a useful check: if $27+36=63$, then $63-36=27$. Multiplication and division similarly undo one another, provided division by zero is not attempted.

Adding and subtracting signed numbers

A plus or minus sign can describe an operation, but it can also be part of a number. Brackets make the distinction clear. In $8+(-3)$, the first sign means “add”, while the second tells us that $3$ is negative.

Adding a positive number moves the value upwards; adding a negative number moves it downwards:

$8+(-3)=8-3=5.$

Subtracting a number means adding its opposite. This single rule handles positive and negative values:

$a-b=a+(-b).$

For example,

$5-(-7)=5+7=12,$

because the opposite of $-7$ is $+7$. By contrast,

$-5-7=-5+(-7)=-12.$

When two numbers have the same sign, add their magnitudes and retain that sign: $-6+(-9)=-15$. When their signs differ, subtract the smaller magnitude from the larger and use the sign of the number with the larger magnitude: $-11+4=-7$, whereas $11+($.

Formal column addition and subtraction

Column methods depend on place value. Write units beneath units, tens beneath tens and so on. For decimal numbers, align the decimal points rather than merely aligning the final digits. Placeholder zeros can expose the correct columns.

For example, $23.47+8.9$ is set out as:

  23.47
+  8.90
-------
  32.37

Add from right to left, carrying into the next column whenever a column totals ten or more.

Subtraction uses the same alignment. If the upper digit in a column is too small, exchange one unit from the column to its left. Thus $15.6-7.83$ becomes:

  15.60
-  7.83
-------
   7.77

For a calculation involving signed numbers, first use the sign rules to decide whether magnitudes should be added or subtracted. For instance, $-23.6+8.9$ has different signs. Since $23.6>8.9$, calculate $23.6-8.9=14.7$ and use the negative sign, giving $$.

Multiplying and dividing signed numbers

For multiplication and division, first calculate using the magnitudes. Then determine the sign:

• equal signs give a positive answer;
• different signs give a negative answer.

Therefore,

$(-6)\times(-4)=24,\qquad (-6)\times4=-24,$

and

$(-35)\div(-7)=5,\qquad 35\div(-7)=-5.$

These rules also apply to negative decimals and fractions. Zero multiplied by any number is zero. Division by zero is undefined, so zero must never be used as a divisor.

Formal multiplication with integers and decimals

In long multiplication, multiply by each digit of the second number in turn. Each new row must begin in the place-value column belonging to that digit. For $347\times26$:

    347
×    26
-------
   2082   (347 × 6)
  6940    (347 × 20)
-------
   9022

For decimal multiplication, first multiply as though the factors were integers. Then restore the decimal point by counting the total number of decimal places in the original factors. For example, $3.2\times1.5$ uses $32\times15=480$. The original factors contain two decimal places altogether, so the answer is $4.80=4.8$.

This rule follows from place value: $3.2=\frac{32}{10}$ and $1.5=\frac{15}{10}$, so their product is .

Formal division with integers and decimals

Short division records how often the divisor fits into each part of the dividend, carrying any remainder into the next place-value column. For example, $936\div4=234$: four fits into nine twice with one left over; that remainder joins the next digit to make thirteen, and the process continues.

When dividing a decimal by a whole number, keep the decimal point in the quotient directly above the decimal point in the dividend. Thus $14.7\div3=4.9$.

When the divisor is a decimal, multiply both the dividend and divisor by the same power of ten until the divisor is a whole number. This does not change the quotient because both quantities have been scaled equally:

$3.75\div0.15=375\div15=25.$

After completing the magnitude calculation, apply the sign rule if either number was negative.

Understanding fractions before calculating

In $\frac{a}{b}$, the denominator $b$ gives the size of the parts and the numerator counts them. Fractions can be combined by addition or subtraction only when they describe parts of the same size.

A mixed number can be converted to an improper fraction by multiplying the whole number by the denominator and then adding the numerator:

$2\frac{3}{5}=\frac{2\times5+3}{5}=\frac{13}{5}.$

To reverse the process, divide the numerator by the denominator. Since $13\div5$ is $2$ remainder $3$, $\frac{13}{5}=2\frac{3}{5}$.

A negative mixed number such as $-2\frac{3}{5}$ means the negative of the whole mixed number, so it equals $-\frac{13}{5}$.

Adding and subtracting fractions

Fractions with the same denominator already have equal-sized parts, so combine only their numerators:

$\frac{3}{8}+\frac{2}{8}=\frac{5}{8}.$

With different denominators, find a common denominator and form equivalent fractions. For example,

$\frac{2}{3}+\frac{5}{6}=\frac{4}{6}+\frac{5}{6}=\frac{9}{6}=\frac{3}{2}=1\frac{1}{2}.$

The denominators are not added because sixths added to sixths remain sixths. A lowest common denominator usually keeps the arithmetic smaller, although any common multiple works if the result is simplified afterwards.

Mixed numbers may be converted to improper fractions first. For example,

$3\frac{1}{4}-1\frac{5}{8}=\frac{13}{4}-\frac{13}{8}=\frac{26}{8}-\frac{13}{8}=\frac{13}{8}=1\frac{5}{8}.$

The signed-number rules still apply. For instance,

$-\frac{3}{4}+\frac{1}{2}=-\frac{3}{4}+\frac{2}{4}=-\frac{1}{4}.$

Multiplying fractions

Convert mixed numbers to improper fractions, then multiply the and multiply the :

$\frac{2}{5}\times\frac{3}{7}=\frac{6}{35}.$

Common factors can be cancelled before multiplication because cancelling divides a numerator and a denominator by the same non-zero factor. This often makes the calculation easier:

$\frac{3}{8}\times\frac{4}{9}=\frac{1}{2}\times\frac{1}{3}=\frac{1}{6}.$

For mixed numbers,

$2\frac{2}{3}\times1\frac{1}{4}=\frac{8}{3}\times\frac{5}{4}=\frac{10}{3}=3\frac{1}{3}.$

Apply the multiplication sign rule separately. Thus $-\frac{2}{3}\times-\frac{9}{5}=\frac{6}{5}$.

Dividing fractions

Dividing by a non-zero number is equivalent to multiplying by its reciprocal. If

$\frac{3}{4}\div\frac{2}{5}=x,$

then $x\times\frac{2}{5}=\frac{3}{4}$. Multiplying by undoes multiplication by , so

$\frac{3}{4}\div\frac{2}{5}=\frac{3}{4}\times\frac{5}{2}=\frac{15}{8}=1\frac{7}{8}.$

Keep the first fraction, replace division with multiplication and take the reciprocal of the second fraction. Convert mixed numbers before doing this:

$2\frac{1}{3}\div1\frac{3}{4}=\frac{7}{3}\div\frac{7}{4}=\frac{7}{3}\times\frac{4}{7}=\frac{4}{3}=1\frac{1}{3}.$

The divisor must not be zero because zero has no reciprocal.

Choosing and checking a method

In a contextual calculation, translate words into the correct operation before calculating. “Altogether” may indicate addition, “difference” subtraction, “of” multiplication and “shared equally” division, but the whole situation determines the operation.

Preserve exact fractions while calculating unless a decimal answer is requested. Afterwards, check the sign, size and form of the result. For example, multiplying by $\frac{1}{2}$ should halve a positive quantity, while dividing it by $\frac{1}{2}$ should double it. Use the inverse operation where possible, simplify fractions, and attach any required unit.