# Rounding Numbers

Rounding number is the process of approximating the number to its closest value. It involves reducing the number of significant digits while retaining the general magnitude or size of the original number. The result of rounding is less accurate than the original number but simplifies calculations.

Suppose one of your friends covered 18 km while returning home from his office. But, when documenting the distance covered in paper, he estimated it as 20 km.

You may wonder where that extra 2 km comes from. We can explain it this way:

18 is closer to 20 than 10. So it is rounded off to 20. If the value would have been 14 km, the value would have approximated to 10 km.

But what if the value is 15 km? It would also go up and be approximated to 20 km. So, any value that equals or exceeds 15 will go up to 20, and values below 15 will round down to 10.

## Rules

We follow the following three steps while rounding a number:

1. Identifying the rounding digit
2. When rounding down, we keep it the same if the next digit is less than 5
3. When rounding up, we increase it by 1 if the next digit is 5 or more than 5

### Rounding to the Nearest Ten

Here, we need to check the digit to the right of the tens place, the ones place.

For example, rounding the number 147, the one’s digit 7 is ˃5, so we add 1 to the tens digit (4 + 1 = 5), and the digit becomes 150.

### Rounding to the Nearest Hundred

Here, we check the digit to the right of the hundreds place, the tens place.

For example, when rounding the number 568 to its nearest hundred, the tens digit is 6, which is ˃5, so we add 1 to the hundred digits (5 + 1 = 6). The remaining two digits at the ones and the tens placed will be 0. So, 568 becomes 600 after rounding.

### Rounding to the Nearest Thousands

Here, we check the digit to the right of the thousands place, that is, the hundreds place.

For example, when rounding the number 7892 to its nearest thousands, the tens hundreds is 8, which is ˃5, so we add 1 to the thousands digit (7 + 1 = 8), and the rest three digits at the ones, tens, and hundreds place will both be 0. So, 7892 becomes 8000 after rounding.

## Rounding Negative Numbers

But how can we round a negative number? The number -8.5 can be rounded to -9 or -8, but which one to follow?

It is again up to us which process we want to follow: rounding up or rounding down.

## Rounding Decimals

Like rounding whole numbers, we follow the same steps here.

### Rounding to the Nearest Tenths

Here, we need to check the digit to the left of the tens place, the Hundredth place. Thus, rounding to tenths leaves one number after the decimal point.

For example, when 4.57 is rounded to the nearest ten, it becomes 4.6, and 14.87 is rounded to 14.9

### Rounding to the Nearest Hundredths

Here, we need to check the digit to the left of the hundredth place, the thousandth place. Thus, rounding to hundredths leaves two numbers after the decimal point.

For example, 0.6278 is rounded to 0.63, and 1.8743 is rounded to 1.87

### Rounding to the Nearest Thousandths

Here, we need to check the digit to the left of the thousandth place. Thus, rounding to hundredths leaves three numbers after the decimal point.

For example, 0.5679 is rounded to 0.568and 6.3859is rounded to 6.386

## Methods

There are several methods of rounding numbers. While rounding, we either round up or round down to indicate whether the number has increased or decreased after rounding.

### Half Round Up

In this method of rounding, 0.5 goes up. Thus, 9.5 rounds up to 10.

Similarly,

• 4.5 rounds up to 5
• 9.9 rounds up to 10
• 5.3 rounds down to 5

Half Round Up with Negative Numbers

• -4.5 rounds up to -4
• -9.9 rounds down to -10
• -5.3 rounds up to -4

### Half Round Down

But we can round 0.5 down as well. In such a case, 9.5 will round down to 9.

Similarly,

• 2.5 rounds down to 2
• 4.4 rounds down to 4
• 9.6 rounds up to 10

Half Round Down with Negative Numbers

• -2.5 rounds down to -3
• -4.4 rounds up to -4
• -9.6 rounds down to -10

All the above rounding methods are symmetric. As we can see, 4.5 rounds up to 5, so -4.5 should be rounded to -5.

However, some methods may need to be more symmetrical.

### Round Half Away from Zero

It means rounding half the values away from zero. Here, the method has no biases towards positive or negative numbers but is biased away from zero. It involves rounding a half value towards the next integer, which is closer to positive or negative infinity on the basis of its value.

For example,

• 7.5 rounds to 8
• -7.5 rounds to -8

### Round Half towards Zero

It is similar to rounding half away from zero but in the opposite direction. It has no bias towards a positive or a negative number. Still, it has a tendency towards 0, which means half the values will be rounded to the next integer closer to zero.

For example,

• 7.5 rounds to 7
• -7.5 rounds to -7

### Round Half to Even

Rounding half to even is used as a tie-breaking rule as it has no biases towards a positive or a negative number or even towards or away from zero. Here, the half values are rounded to the nearest even integer.

For example,

• 5.5 rounds up to 6 (since 6 is an even number)
• 6.5 rounds down to 6 (since 6 is an even number)
• -6.5 rounds up to 7
• -8.5 rounds up to 8 (since 8 is an even number)

### Round Half to Odd

It is also a tie-breaking rule, like rounding half to even. However, the number is rounded to the nearest odd integer for rounding half to odd.

For example,

• 5.5 rounds down to 5 (since 5 is an odd number)
• 7.5 rounds down to 7 (since 7 is an odd number)
• -8.5 rounds down to -9 (since 9 is an odd number)

### Floor and Ceiling Method

These methods give the nearest integer down or up for floor and ceiling.

The number remains the same after rounding an integer.

So the floor of 6 is 6, and the ceiling of 6 is also 6.

However, depending on the method, the digit attains the nearest integer up or down for a decimal number.

For example, 2.7 is rounded down to 2 in the floor method, while it is rounded up to 3 in the ceiling method.

## Rounding Fractions

Rounding to fractions is the same as rounding decimals. We first convert the fraction into decimal and then round the number to tenths, hundredths, thousandths, and so on.

### Rounding to Nearest Tenths

${\dfrac{14}{25}}$ is equal to 0.56, which, when rounded to the nearest tenths, gives 0.6

### Rounding to Nearest Hundredths

${\dfrac{78}{48}}$ is equal to 1.625, which, when rounded to the nearest hundredths, gives 1.63

### Rounding to Nearest Thousandths

${\dfrac{1823}{2000}}$ is equal to 0.9115, which, when rounded to the nearest thousandths, gives 0.912

## Rounding to Significant Digits

A significant figure is each digit in a number that helps to express the number with the required degree of accuracy. For example, the number 15.6 has three significant figures.

There are specific rules for deciding the number of significant figures.

1) All nonzero digits are significant.

For example, 1.7 has 2 significant figures, and 2.678 has 4 significant figures.

2) Zeroes between nonzero digits are significant.

For example, 3006 has 4 significant figures, and 10.3 has 3 significant figures.

3) Zeroes to the left of the first nonzero digit are not significant.

For example, 0.002 has 1 significant figure, and 0.045 has 2 significant figures.

4) Zeroes to the right of a decimal point in a number are always significant.

For example, 0.045 has 2 significant figures, and 0.700 g has 3 significant figures.

5) For non-decimal numbers, the trailing zeros are not necessarily significant.

However, 280 can be written as 2 or 3 significant figures, and 60,700 may be 3, 4, or 5 significant figures. This ambiguity is removed by expressing the number in scientific notation. Depending on the number of significant figures, which is correct, 60,700 is written as:

• 6.07 × 104 calories (for 3 significant figures)
• 6.070 × 104 calories (for 4 significant figures)
• 6.0700 × 104 calories (for 5 significant figures)

## Solved Examples

Round i) 767 ii) 34.68 to the nearest ten.

Solution:

i) 767 on rounding to the nearest ten, give 770
ii) 34.68on rounding to the nearest ten, give 30

Round i) 8974 ii) 7646.89 to the nearest hundred.

Solution:

i) 8974on rounding to the nearest hundred gives 9000
ii) 7646.89 on rounding to the nearest hundred gives 7,600