Exact arithmetic with fractions and pi

An exact value contains all the information in a number. Fractions and expressions involving $\pi$ often give exact values:

$\frac{2}{3},\qquad \frac{7\pi}{4},\qquad 3+\frac{\pi}{2}.$

A rounded decimal is an approximation. For example,

$\frac{2}{3}=0.666\ldots\approx 0.667,$

where $=$ means equal and $\approx$ means approximately equal. Exact arithmetic prevents rounding error, exposes mathematical structure and is expected throughout A-level Mathematics.

Prerequisites

You should be able to:

• recall multiplication tables and integer factors;
• use negative numbers and the order of operations;
• find a highest common factor and a lowest common multiple;
• distinguish significant figures from decimal places.

Review standard form, estimation and accuracy if rounding and significant figures are unfamiliar. This page supports later work on algebraic fractions, circle geometry and indices, roots and surds.

What makes a value exact?

Integers, fractions of integers and finite combinations of exact constants are exact. The symbol $\pi$ represents one particular irrational number, not the rounded decimal $3.14$.

$\pi=3.14159265\ldots$

The decimal continues without terminating or recurring. Therefore $3\pi$ is exact, while $9.42$ is only an approximation to it.

Terminating and recurring decimals can also represent exact values:

$0.125=\frac18,\qquad 0.\overline{3}=\frac13.$

The issue is not whether a number is written as a decimal. The issue is whether digits have been discarded.

Self-check 1

Classify each value as exact or approximate.

1. $\dfrac{\pi}{6}$
2. $0.25$
3. $\sqrt{2}$
4. $1.414$ used for $\sqrt2$

Answers

1. Exact. No rounding has taken place.
2. Exact, because $0.25=\frac14$.
3. Exact. The root symbol specifies the number completely.
4. Approximate, because digits of $\sqrt2=1.414213\ldots$ have been discarded.

Equivalent fractions and simplest form

Multiplying or dividing the numerator and denominator by the same non-zero number does not change a fraction:

$\frac{a}{b}=\frac{ka}{kb}\qquad (b\ne0,\ k\ne0).$

This works because $\frac{k}{k}=1$. To simplify $\frac{42}{63}$, divide both parts by their highest common factor, $21$:

$\frac{42}{63}=\frac{42\div21}{63\div21}=\frac23.$

A fraction is in simplest form when its numerator and denominator have no common factor greater than $1$. Conventionally, the denominator is positive:

$\frac{6}{-15}=-\frac{6}{15}=-\frac25.$

Never cancel terms joined by addition. Cancellation removes a common factor, not a common term:

$\frac{3\times 5}{3\times 7}=\frac57,$

but

$\frac{3+5}{3+7}=\frac8{10}=\frac45\ne\frac57.$

Adding and subtracting fractions

Fractions can be added only when they describe parts of the same size. This is why a common denominator is required.

Worked example 1: unlike denominators

Calculate $\frac34+\frac56$ exactly.

The lowest common multiple of $4$ and $6$ is $12$:

$$\begin{aligned} \frac34+\frac56 &=\frac9{12}+\frac{10}{12}\\ &=\frac{19}{12}. \end{aligned}$$

The improper fraction $\frac{19}{12}$ is usually more useful in algebra than the mixed number $1\frac7{12}$.

Worked example 2: subtraction and signs

Calculate $-\frac7{10}-\left(-\frac23\right)$.

Subtracting a negative becomes addition:

$$\begin{aligned} -\frac7{10}-\left(-\frac23\right) &=-\frac7{10}+\frac23\\ &=-\frac{21}{30}+\frac{20}{30}\\ &=-\frac1{30}. \end{aligned}$$

A quick size check supports the answer: $-0.7+0.666\ldots$ should be a small negative number.

A useful general rule

For $b\ne0$ and $d\ne0$,

$\frac{a}{b}+\frac{c}{d}=\frac{ad+bc}{bd}.$

This always gives a common denominator, although $bd$ is not always the lowest one. Simplify afterwards.

Misconception: adding denominators

In general,

$\frac ab+\frac cd\ne\frac{a+c}{b+d}.$

For instance, $\frac12+\frac12=1$, not $\frac24=\frac12$. Denominators name the size of the parts, so they are not added.

Multiplying fractions

Multiply numerators together and denominators together:

$\frac ab\times\frac cd=\frac{ac}{bd}.$

Cancel common factors before multiplying to keep the numbers small.

Worked example 3: cross-cancellation

Calculate $\frac{14}{15}\times\frac{25}{21}$.

Factor or cancel across the product:

$$\begin{aligned} \frac{14}{15}\times\frac{25}{21} &=\frac{\cancel{14}^{\,2}}{\cancel{15}^{\,3}} \times\frac{\cancel{25}^{\,5}}{\cancel{21}^{\,3}}\\ &=\frac{2\times5}{3\times3}\\ &=\frac{10}{9}. \end{aligned}$$

Cancellation is valid here because the whole numerator and denominator are products.

Dividing fractions

Dividing by a non-zero fraction is equivalent to multiplying by its reciprocal:

$\frac ab\div\frac cd=\frac ab\times\frac dc=\frac{ad}{bc}.$

This follows from asking how many lots of $\frac cd$ fit into $\frac ab$. Multiplication by $\frac dc$ scales $\frac cd$ to $1$.

Worked example 4: division by a negative fraction

Calculate $\frac58\div\left(-\frac{15}{16}\right)$.

$$\begin{aligned} \frac58\div\left(-\frac{15}{16}\right) &=\frac58\times\left(-\frac{16}{15}\right)\\ &=-\frac{5\times16}{8\times15}\\ &=-\frac23. \end{aligned}$$

Only the fraction being divided by is inverted. Also, division by zero is undefined, so the divisor must not be zero.

Mixed calculations and fraction bars

A fraction bar acts as a grouping symbol. Complete the entire numerator and denominator before dividing.

Worked example 5: order of operations

Evaluate

$\frac{\frac34-\frac16}{1+\frac25}.$

First simplify the numerator:

$\frac34-\frac16=\frac9{12}-\frac2{12}=\frac7{12}.$

Then simplify the denominator:

$1+\frac25=\frac75.$

Now divide:

$$\frac{\frac7{12}}{\frac75} =\frac7{12}\times\frac57 =\frac5{12}.$$

The two $7$s cancel because they are factors after the numerator and denominator have each been simplified.

Worked example 6: powers and negatives

Evaluate $\left(-\frac23\right)^2-\frac12\div\frac34$.

Powers and division come before subtraction:

$$\begin{aligned} \left(-\frac23\right)^2-\frac12\div\frac34 &=\frac49-\frac12\times\frac43\\ &=\frac49-\frac23\\ &=\frac49-\frac69\\ &=-\frac29. \end{aligned}$$

The brackets matter: $\left(-\frac23\right)^2=\frac49$, whereas $-\left(\frac23\right)^2=-\frac49$.

Exact arithmetic with pi

Treat $\pi$ as an exact constant and manipulate it like an algebraic factor.

$3\pi+5\pi=8\pi,\qquad \frac{6\pi}{9}=\frac{2\pi}{3}.$

Unlike terms cannot be combined:

$3+2\pi$

is already in simplest exact form. Since $3$ and $2\pi$ are different types of term, $3+2\pi\ne5\pi$.

Worked example 7: circumference and arc length

A circle has radius $6$ cm. Find exactly the length of an arc subtending $75^\circ$ at the centre.

The whole circumference is

$2\pi r=12\pi\text{ cm}.$

The arc is $\frac{75}{360}=\frac5{24}$ of the full circle:

$$\begin{aligned} \text{arc length} &=\frac5{24}\times12\pi\\ &=\frac{5\pi}{2}\text{ cm}. \end{aligned}$$

Only convert to a decimal if requested:

$\frac{5\pi}{2}\approx7.85\text{ cm}\qquad\text{(3 significant figures).}$

Worked example 8: a composite exact area

A semicircle of radius $4$ cm is attached to a rectangle measuring $8$ cm by $3$ cm. Find the total area exactly.

$$\begin{aligned} A &=8\times3+\frac12\pi(4^2)\\ &=24+8\pi\text{ cm}^2. \end{aligned}$$

Do not combine $24$ and $8\pi$. The exact answer $24+8\pi$ preserves both contributions.

Exact first, approximate last

Keep exact values throughout a calculation and round once, at the end. Early rounding can change the final answer.

Worked example 9: avoiding premature rounding

Evaluate $\frac{7\pi}{3}-\frac{2\pi}{5}$, giving an exact answer and then a decimal to $3$ significant figures.

Work exactly:

$$\begin{aligned} \frac{7\pi}{3}-\frac{2\pi}{5} &=\frac{35\pi}{15}-\frac{6\pi}{15}\\ &=\frac{29\pi}{15}. \end{aligned}$$

Then evaluate:

$\frac{29\pi}{15}=6.073745\ldots\approx6.07\qquad\text{(3 significant figures).}$

Using $\pi=3.14$ at the first line throws away information for no benefit.

Write enough calculator digits before rounding, and include the requested accuracy. Never write $\frac{29\pi}{15}=6.07$, because the two values are not equal.

Self-check 2

Try these without converting to decimals.

1. Simplify $\dfrac{84}{126}$.
2. Calculate $\dfrac79-\dfrac5{12}$.
3. Calculate $\dfrac{18}{35}\times\dfrac{14}{27}$.
4. Calculate $-\dfrac56\div\dfrac{10}{9}$.
5. Evaluate $\dfrac{1-\frac38}{\frac54}$.
6. Simplify $4\pi-\dfrac{3\pi}{2}$.
7. A circle has diameter $11$ cm. Give its circumference exactly and to $3$ significant figures.

Answers

1. $\dfrac23$, after dividing numerator and denominator by $42$.
2. $\dfrac{28}{36}-\dfrac{15}{36}=\dfrac{13}{36}$.
3. $\dfrac{18}{35}\times\dfrac{14}{27}=\dfrac4{15}$.
4. $-\dfrac56\times\dfrac9{10}=-\dfrac34$.
5. $1-\dfrac38=\dfrac58$, so $\dfrac58\div\dfrac54=\dfrac12$.
6. $\dfrac{8\pi}{2}-\dfrac{3\pi}{2}=\dfrac{5\pi}{2}$.
7. $C=\pi d=11\pi$ cm exactly, and $11\pi=34.557\ldots\approx34.6$ cm to $3$ significant figures.

Common errors to eliminate

• Changing only one part of a fraction: equivalent fractions require the same non-zero multiplier in numerator and denominator.
• Adding denominators: use a common denominator, then add or subtract numerators.
• Cancelling across addition: cancellation applies to factors of an entire product, not separate terms in a sum.
• Inverting the wrong fraction: in $a\div b$, replace only $b$ by its reciprocal.
• Treating $\pi$ as $3.14$: retain $\pi$ unless a decimal approximation is requested.
• Using $=$ before a rounded answer: use $\approx$ when digits have been discarded.
• Rounding intermediate values: store exact expressions or full calculator precision, then round once.

What to learn next

You are ready to move on when you can perform all four fraction operations, simplify a compound fraction, retain $\pi$ exactly and explain the difference between $=$ and $\approx$.

Next, study indices, roots and surds for other exact forms, then algebraic fractions to apply the same principles to expressions containing variables. For geometric applications, continue to circle geometry and Pythagoras and trigonometry.