# Approximations

Approximation or approximate numbers can be defined as a value that is similar to the other value but not exactly equal to it. The approximation is used when the results of the numeric calculations are uncertain. They mostly come into use when the decimal numbers with larger place values come as a result.

## Rounding numbers to the nearest 10, 100, 1,000

To approximate to the nearest ten, look at the digit in the **tens column**.

To approximate to the nearest hundred, look at the digit in the **hundreds column**.

For the nearest thousand, look at the digit in the **thousands column**.

Then do the following:

- draw a vertical line to the right of the place value digit that is required
- look at the next digit
**if it's 5 or more**, increase the previous digit by one**if it's 4 or less**, keep the previous digit the same- fill any spaces to the
**right**of the line with zeros.

### Examples

Round 4,853 to the nearest 10, 100 and 1,000.

- 485|
**3**to the nearest 10 is 4,850 - 48|
**5**3 to the nearest 100 is 4,900 - 4|
**8**53 to the nearest 1,000 is 5,000

Round 76,982 to the nearest 10, 100 and 1,000.

- 7698|
**2**to the nearest 10 is 76,980 - 769|
**8**2 to the nearest 100 is 77,000 - 76|
**9**82 to the nearest 1,000 is 77,000

# Rounding to decimal places

When **rounding using** decimal places(dp), the degree of accuracy that is required is usually given. However, there are certain calculations where the degree of accuracy may be more obvious. For example, calculations involving money should be given to two decimal places to represent the pence.

To round to a decimal place:

- look at the first digit after the decimal point if rounding to one decimal place or the second digit for two decimal places
- draw a vertical line to the right of the place value digit that is required
- look at the next digit
**if the next digit is 5 or more**, increase the previous digit by one**if it's 4 or less**, keep the previous digit the same- remove any numbers to the
**right**of the line

### Examples

Round 248.561 to 1 decimal place, then round it to 2 decimal places:

- 248.5|
**6**1 to 1 decimal place is 248.6 - 248.56|
**1**to 2 decimal places is 248.56

Notice that your answer should have the same number of decimal places as the approximation asked for.

Round 0.08513 to 1 decimal place and then to 2 decimal places:

- 0.0|
**8**513 to 1 decimal place is 0.1 - 0.08|
**5**13 to 2 decimal places is 0.09

# Rounding to significant figures

The method of **rounding to a significant figure** is often used as it can be applied to any kind of number, regardless of how big or small it is. When a newspaper reports a lottery winner has won £3 million, this has been rounded to one significant figure. It rounds to the most important figure in the number.

To round to a significant figure:

- look at the first non-zero digit if rounding to one significant figure
- look at the digit after the first non-zero digit if rounding to two significant figures
- draw a vertical line after the place value digit that is required
- look at the next digit
**if the next digit is 5 or more**, increase the previous digit by one**if it is 4 or less**, keep the previous digit the same- fill any spaces to the
**right**of the line with zeros, stopping at the decimal point if there is one

### Examples

**Round 53,879 to 1 significant figure, then 2 significant figures.**

- 5|
**3**879 to 1 significant figure is 50,000 - 53|
**8**79 to 2 significant figures is 54,000

Notice that the number of significant figures in the question is the maximum number of non-zero digits in your answer.

**Round 0.005089 to 1 significant figures, then 2 significant figures.**

- 0.005|
**0**89 to 1 significant figure is 0.005 - 0.0050|
**8**9 to 2 significant figures is 0.0051

# Truncation

When we truncate a number, we find an estimate for the number without doing any rounding. To truncate a number, we miss off digits past a certain point in the number, filling-in zeros if necessary to make the truncated number approximately the same size as the original number.

To truncate a number to 1 decimal place, miss off all the digits after the first decimal place.

To truncate a number to 2 decimal places, miss off all the digits after the second decimal place.

To truncate a number to 3 significant figures, miss off all the digits after the first 3 significant figures (the first non-zero digit and the next two digits). Fill in any spaces with zeros to make the number approximately the same size as the original value.

### Examples

Truncate 3.784 to 1 decimal place and then to 2 decimal places.

**3.7**|84 truncated to 1 decimal place is 3.7

**3.78**|4 truncated to 2 decimal places is 3.78

Truncate 63,854 and 0.04988 to 3 significant figures.

**63,8**|54 truncated to 3 significant figures is 63800

**0.0498**|8 truncated to 3 significant figures is 0.0498

# Estimating calculations

Sometimes it is a good idea to estimate the value of a calculation rather than work it out exactly. In this situation, round the numbers in the question before performing the calculation. Usually, numbers are rounded to one significant figure. The 'approximately equal to' sign, ≈, is used to show that values have been rounded.