4 Rounding to $n$ significant figures

This process is similar to rounding to decimal places but there are some subtle differences.

To round a number to $n$ significant figures we look at the ${\left(n+1\right)}^{th}$ digit in the decimal expansion of the number.

• If the ${\left(n+1\right)}^{th}$ digit is 0, 1, 2, 3 or 4 then we round down : that is, we simply chop to $n$ places, inserting zeros if necessary before the decimal point. (In other words we neglect the ${\left(n+1\right)}^{th}$ digit and any digits to its right.)
• If the ${\left(n+1\right)}^{th}$ digit is 5, 6, 7, 8 or 9 then we round up : we add $1$ to the $n^{th}$ decimal place and then chop to $n$ places, inserting zeros if necessary before the decimal point.

Examples are given on the next page.

$\begin{array}{ccccc}\hfill \frac{1}{3}& \hfill =\hfill & 0.3333\hfill & \hfill & \text{rounded to 4 significant figures}\hfill \\ \hfill & \hfill \hfill & \hfill & \hfill & \hfill \\ \hfill \frac{8}{3}& \hfill =\hfill & 2.66667\hfill & \hfill & \text{rounded to 6 significant figures}\hfill \\ \hfill & \hfill \hfill & \hfill & \hfill & \hfill \\ \hfill \pi & \hfill =\hfill & 3.142\hfill & \hfill & \text{rounded to 4 significant figures}\hfill \\ \hfill & \hfill \hfill & \hfill & \hfill & \hfill \\ \hfill 2136& \hfill =\hfill & 2000\hfill & \hfill & \text{rounded to 1 significant figure}\hfill \\ \hfill & \hfill \hfill & \hfill & \hfill & \hfill \\ \hfill 36.78& \hfill =\hfill & 37\hfill & \hfill & \text{rounded to 2 significant figures}\hfill \\ \hfill & \hfill \hfill & \hfill & \hfill & \hfill \\ \hfill 6.2399& \hfill =\hfill & 6.240\hfill & \hfill & \text{rounded to 4 significant figures}\hfill \end{array}$

Sometimes the phrase “significant figures" is abbreviated as “s.f." or “sig.fig."

Example 3

Write down each of these numbers, rounding them to 4 significant figures:

$0.12345$ , $-0.44444$ , $0.5555555$ , $0.000127351$ , $25679$ , $123.456789,$ $3456543$

Solution

$0.1235$ , $-0.4444$ , $0.5556$ , $0.0001274$ , $25680$ , $123.5$ , $3457000$

Task!

Write down each of these numbers rounded to 3 significant figures:

$0.87264$ , $0.1543$ , $0.889412$ , $-0.5555$ , $2.346$ , $12343.21$ , $4245321$

Answer

4.1 Arithmetical expressions

A quantity made up of numbers and one or more of the operations $+$ , $-$ , $\times$ and $∕$ is called an arithmetical expression . Frequent use is also made of brackets, or parentheses , $\left(\right)$ , to separate different parts of an expression. When evaluating an expression it is conventional to evaluate quantities within brackets first. Often a division line implies bracketed quantities. For example in the expression $\frac{3+4}{7+9}$ there is implied bracketing of the numerator and denominator i.e. the expression is $\frac{\left(3+4\right)}{\left(7+9\right)}$ and the bracketed quantities would be evaluated first resulting in the number $\frac{7}{16}$ .

4.2 The BODMAS rule

When several arithmetical operations are combined in one expression we need to know in which order to perform the calculation. This order is found by applying rules known as precedence rules which specify which operation has priority. The convention is that bracketed expressions are evaluated first. Any multiplications and divisions are then performed, and finally any additions and subtractions. For short, this is called the BODMAS rule.

If an expression contains only multiplication and division we evaluate by working from left to right. Similarly, if an expression contains only addition and subtraction we evaluate by working from left to right. In Section 1.2 we will meet another operation called exponentiation, or raising to a power. We shall see that, in the simplest case, this operation is repeated multiplication and it is usually carried out once any brackets have been evaluated.

Example 4

Evaluate $4-3+7\times 2$

Solution

The BODMAS rule tells us to perform the multiplication before the addition and subtraction. Thus

$4-3+7\times 2=4-3+14$

Finally, because the resulting expression contains just addition and subtraction we work from the left to the right, that is

$4-3+14=1+14=15$

Task!

Evaluate   $4+3\times 7$ using the BODMAS rule to decide which operation to carry out first.

Answer

Task!

Evaluate   $\left(4-2\right)\times 5$ .

Answer

Example 5

Evaluate $8÷2-\left(4-5\right)$

Solution

The bracketed expression is evaluated first:

$8÷2-\left(4-5\right)=8÷2-\left(-1\right)$

Division has higher priority than subtraction and so this is carried out next giving

$8÷2-\left(-1\right)=4-\left(-1\right)$

Subtracting a negative number is equivalent to adding a positive number. Thus

$4-\left(-1\right)=4+1=5$

Task!

Evaluate $\frac{9-4}{25-5}$ .

(Remember that the dividing line implies that brackets are present around the numerator and around the denominator.)

Answer

Exercises

1. Draw a number line and on it label points to represent $-5$ , $-3.8$ , $-\pi$ , $-\frac{5}{6}$ , $-\frac{1}{2}$ , 0, $\sqrt{2}$ , $\pi$ , 5.
2. Simplify without using a calculator
   1. $-5\times -3$ ,
   2. $-5\times 3$ ,
   3. $5\times -3$ ,
   4. $15\times -4$ ,
      
      
   5. $-14\times -3$ ,
   6. $\frac{18}{-3}$ ,
   7. $\frac{-21}{7}$ ,
   8. $\frac{-36}{-12}$ .
3. Evaluate
   1. $3+2\times 6$ ,
   2. $3-2-6$ ,
   3. $3+2-6$ ,
   4. $15-3\times 2$ ,
   5. $15\times 3-2$ ,
   6. $\left(15÷3\right)+2$ ,
   7. $15÷3+2$ ,
   8. $7+4-11-2$ ,
   9. $7\times 4+11\times 2$ ,
   10. $-\left(-9\right)$ ,
   11. $7-\left(-9\right)$ ,
   12. $-19-\left(-7\right)$ ,
   13. $-19+\left(-7\right)$ .
4. Evaluate
   1. $\left|-18\right|$ ,
   2. $\left|4\right|$ ,
   3. $\left|-0.001\right|$ ,
   4. $\left|0.25\right|$ ,
   5. $\left|0.01-0.001\right|$ ,
   6. $2!$ ,
   7. $8!-3!$ ,
   8. $\frac{9!}{8!}$ .
5. Evaluate
   1. $8+\left(-9\right)$ ,
   2. $18-\left(-8\right)$ ,
   3. $-18+\left(-2\right)$ ,
   4. $-11-\left(-3\right)$
6. State the reciprocal of
   1. 8,
   2. $\frac{9}{13}$ .
7. Evaluate
   1. $7\pm 3$ ,
   2. $16\pm 7$ ,
   3. $-15\pm \frac{1}{2}$ ,
   4. $-16\pm 0.05$ ,
   5. $|-8|\pm 13$ ,
   6. $|-2|\pm 8$ .
8. Which of the following statements are true ?
   1. $-8\le 8$ ,
   2. $-8\le -8$ ,
   3. $-8\le \left|8\right|$ ,
   4. $|-8|<8$ ,
   5. $|-8|\le -8$ ,
      
      
   6. $9!\le 8!$ ,
   7. $8!\le 10!$ .
9. Explain what is meant by saying that addition of numbers is
   1. associative,
   2. commutative. Give examples.
10. Explain what is meant by saying that multiplication of numbers is
    1. associative,
    2. commutative. Give examples.

Answer