Approximations

Approximations are linked to terms such as decimal places, significant figures, standard form and so on. It is a method of assuming precise values to figures. It is also a method of counting to the nearest whole numbers.

Common terms used in Approximations are decimal places (d.p) and significant figures (sigfig) or (s.f).

Note;

Before you can approximate any number, the following rule must be fully understood;

0	0
1	0
2	0
3	0
4	0
5	1
6	1
7	1
8	1
9	1

The table above simply suggest that during approximation, figures from 0-4 becomes 0 while figures from 5-9 becomes 1. Without knowing this rule it will be difficult for you to approximate any number.

Example 1; Correct 0.00630427 to 7 decimal places.

Solution

To do this, follow these steps;

Count 7 numbers after the decimal point. The seventh figure is ‘2’ and beside it is ‘7’ which falls between 5-9 so it becomes ‘1’ according to the rule in the table above. Add this ‘1’ to the seventh figure which is ‘2’, that gives 3.

The result is; 0.0063043. (Answer)

Example 2; Correct 15.300649 to 3 decimal places.

Solution

Count 3 numbers after the decimal point. The third number is ‘0’ and beside it is ‘6’ which falls between 5-9 so it becomes ‘1’ according to the rule above. Add this ‘1’ to the third number which is ‘0’, that gives 1.

The result is; 15.301000. This can also be written as 15.301. (Answer)

Note

All other numbers are rounded down and will become zero. It’s a rule!

Example 3; Express 0.006457 to 3 significant figures.

Solution

Applying the rule to significant figures is just a little bit different because here zeros are not counted unlike decimal places. In order to do this, follow the steps below;

Count 3 “non-zero” figures after the decimal point. The third figure is ‘5’ and beside it is ‘7’ which falls between 5-9 so it becomes ‘1’. Add this ‘1’ to the third figure which is ‘5’, that gives 6.

The result is; 0.00646. (Answer)

Example 4; Express 2.00584 to 4 sigfig.

Solution

A very important thing to note here is zeros that are found in between non-zero significant figures are counted as being significant.

Count 4 significant figures from the beginning of the number. The fourth figure will be ‘5’ and beside it is ‘8’ which falls between 5-9 so it becomes ‘1’. Add this ‘1’ to the fourth figure which is ‘5’, that gives 6.

The answer is 2.006.

Example 5; Round off 241.863 to the nearest whole number.

Solution

This is a little different from decimal places and significant figures but the same rule applies here as well.

To the nearest whole number means no decimal point should be found in the answer. This is how you do it;

Consider the figure after decimal point, which is ‘8’ and falls between 5-9 so it becomes ‘1’. Add this ‘1’ to the last figure before decimal point which is also 1 (1+1) that gives 2.

Therefore, the result is 242.

Example 6; Round off 2.2 hours to the nearest hours.

Solution

Follow the same step in example 5;

Consider the figure after decimal point which is ‘2’ and falls between 0-4 so it becomes ‘0’ according to the rule. Add this ‘0’ to the last figure before the decimal point which is 2 (0+2) that gives 2.

Answer is 2 hours.

Example 7; Take 173.245 to the nearest hundred.

Solution

In order to do this, you must follow place value counting system;

_{H T U}

173.245

H stands for hundred

T stands for tens

U stands for units.

To the nearest hundred means; Consider the hundredth value which is ‘1’ beside it is ‘7’ which falls between 5-9 so it becomes ‘1’. Add this ‘1’ to the hundredth value which is also 1 (1+1), that gives 2.

The answer is 200. (Remember all the other values becomes zero)

Example 8; Express 92,681 in the nearest thousand.

Solution

Apply place value counting system;

_{TTH TH H T U}

 92,681

To the nearest thousand means; Consider the thousandth value which is ‘2’ beside it is ‘6’ which falls between 5-9 so it becomes ‘1’. Add this ‘1’ to the thousandth value (2+1) that gives 3.

93,000. (Answer)

Practice these questions below;

1. Correct the following figures to the number of decimal places indicated.

(a) 3.0561 (2d.p)

(b) 0.008153 (1d.p)

(c) 0.09634 (3d.p)

(d) 0.0006005 (4d.p)