# Rounding and Significant Figures

In this Math tutorial, you will learn:

• What is rounding? Why do we use it?
• How many types of rounding are there? What are the corresponding digits that are associated to each of them?
• How do we round numbers to the nearest ten?
• How do we round numbers to the nearest hundred/thousand and more.
• How to perform rounding in a number axis?
• How to round non-whole numbers?
• What are significant figures?
• How to find the number of significant figures in a given value?
• How are significant figures used in practice?

## Introduction

Do you always write the exact number of items in a collection? Why?

How do we express the number of items in a collection if we are not sure about the exact value.

A calculator gives a value of 6.66666 cm when we want to find the side of an equilateral triangle if the perimeter is 20 cm. Is it correct to write the result in so many digits after the decimal place? Why?

We do not always need to know the exact value of something. In these situations we can use an approximate value to provide an idea of how large or small the quantity involved is. Usually, we use values ending with zero to represent these approximate values; we call these rounding. Let's take a closer look at this new operation.

## What is Rounding?

Rounding means replacing a number with an approximate value that has a shorter and simpler representation. Numbers are rounded when the accuracy is not very relevant but there is room for approximate values to express a quantity so that it can be more easily understood.

For example, we more frequently say "I will be there in about half an hour" rather than saying "I will be there in 30 minutes". This is because 30 minutes is quite exact where are half an hour is quite approximate, this allows us to have a few minutes of delay and still be considered on time.

The value of a rounded number is slightly different from the original number. A rounded integer always ends with zero. We use the symbol (≈) to represent rounding in expressions.

There are two types of rounding: (1) rounding up, and (2) rounding down. When a number is rounded up, its value increases to become the nearest number ending with zero to the original number. On the other hand, when a number is rounded down, its value decreases to become the nearest number ending with zero to the original number."

Rule: "A number is rounded down when the digit that becomes zero in the original number was 1, 2, 3 and 4, while a number is rounded up when the digit that becomes zero after rounding was previously 5, 6, 7, 8 and 9.

For example, when 78 is rounded to the nearest ten, we obtain 80 because this number is the nearest ten to the original number.

## Rounding to the Nearest Ten

When a number is rounded to the nearest ten it has at least one zero at the end. For example, 7 ≈ 10; 29 ≈ 30; 61 ≈ 60, etc.

We use the rounding to the nearest ten rule when estimating the approximate value of a small number of items. For example, we say "there are about 30 people in the bus", "there are about 40 walnuts in the box" and so on.

When a number ends with 0, it has no need to be rounded to the nearest ten. When a number ends with 1, 2, 3, 4, it becomes smaller after rounded to the nearest ten. For example, 1 becomes 0; 34 becomes 30; 51 becomes 50; 143 becomes 140; 15,472 becomes 15,470; etc.

On the other hand, when a number ends with 5, 6, 7, 8 and 9, it becomes bigger after it is rounded to the nearest ten. For example, 8 becomes 10; 67 becomes 70; 115 becomes 120; 1,479 becomes 1,480; etc. Let's look at a couple of examples:

A number rounded to the nearest ten, usually has one zero at the end. However, there are some cases when a number may have more than one zero after being rounded to the nearest ten. This occurs when the number is close to hundreds, thousands, etc. For example, 99 becomes 100; 204 becomes 200; 997 becomes 1,000; 10,002 becomes 10,000; etc.

The range of numbers that give the same rounding to the nearest ten is 10 numbers. For example, all numbers from 25 to 34 become 30 after being rounded to the nearest ten. These numbers are: 25, 26, 27, 28, 29, 30, 31, 32, 33 and 34.

## Rounding to the Nearest Hundred

When rounding to the nearest hundred, the number will have at least two zeroes at the end after being rounded to the nearest hundred. We have to check the value of tens to know whether the number must be rounded up or down. The same rules are applied, i.e. when the tens digit ends with 0, 1, 2, 3 and 4, the number is rounded down to the nearest hundred, so its value decreases. On the other hand, if the value of tens digit is 5, 6, 7, 8 and 9, the number increases as it takes the value of the upper nearest hundred. In this case, the value of units' digit is not important; it will become zero in any case. The number rounded to the nearest hundred has at least two zeroes at the end. Lets's look at a couple of examples:

For example, 49 becomes 0 when rounded to the nearest hundred because the value of tens is 4, while 50 becomes 100 as the value of tens is 5. On the other hand, 149 also becomes 100 when rounded to the nearest hundred.

The number rounded to the nearest hundred has at least two zeroes at the end. However, there are some cases where the new number contains three zeroes at the end after being rounded to the nearest 100. This occurs when the number to be rounded is close to thousands. For example, 978 ≈ 1,000; 2017 ≈ 2000; etc.

## Rounding to the Nearest Thousand

The same procedure is also used when rounding a number to the nearest thousand. In this case, we look at the hundreds digit (the third from the right). If it is 0, 1, 2, 3 and 4, then the number is rounded down to the nearest thousand. This means the rounded number becomes smaller than the original. On the other hand, when the hundreds digit is 5, 6, 7, 8 and 9, the number is rounded up to the nearest thousand and the number therefore becomes bigger. Obviously, the number will have at least 3 zeroes at the end.

For example, when rounded to the nearest thousand, 472 becomes 0; 597 becomes 1000; 1946 becomes 2000; and so on.

As you see, a number rounded to the nearest thousand usually has three zeroes at the end. However, a number may end with more than three zeroes after being rounded to the nearest thousand. This occurs when the number is close to ten thousand, hundred thousand and so on. For example, when rounded to the nearest thousand, 9758 becomes 10,000; 99673 becomes 100,000; and so on.

We can continue using the same logic for further rounding such as to the nearest ten thousand, hundred thousand, one million, etc. In such cases, we must consider the fourth, fifth and sixth digit from the right respectively. They must obey the aforementioned rules regarding to the possible rounding up or down.

### Example 1

Round the number 437,891 to the nearest 10, 100, 1000 and 10,000.

### Solution 1

We must consider the value of the units' digit when rounding to the nearest ten. Since this value is 1, we have a rounding down. Hence, 437,891 ≈ 437,890.

We must consider the value of the tens digit when rounding to the nearest hundred. Since this value is 9, we have a rounding up. Hence, 437,891 ≈ 437,900.

We must consider the value of the hundreds digit when rounding to the nearest thousand. Since this value is 8, we have a rounding up. Hence, 437,891 ≈ 438,000.

We must consider the value of the thousands digit when rounding to the nearest thousand. Since this value is 7, we have a rounding up. Hence, 437,891 ≈ 440,000.

## Explaining Rounding in the Number Axis

The rounding process can be more easily understood when numbers are shown in the number axis. For example, the number 7 is shown in the number axis as follows

From the figure, it is clear that the number 7 is closer to 10 than 0. Hence, when rounded to the nearest ten, the number 7 becomes 10.

When rounding to the nearest 10, 100 or 1000 is shown in the number line, it is not necessary to show all numbers from 0 to several thousands, we can simply show the part of number axis around the number involved. For example, if we have to round 543 to the nearest 10, 100 and 1000, we can use the following figures to clarify this process, where the original number is indicated by arrow while the new number after rounding is indicated by a circle:

## Other Types of Rounding

When the numbers involved are not integers, we may need to round them to the nearest integer. In order to do this, the original number must have at least one digit after the decimal point. Then, the known rules discussed earlier in this guide are used. For example, when we want to round the number 4.7 to the nearest integer, we get 5, as this is the closest integer to 4.7. The other integer around 4.7 is 4, which is farther than 5 from the original number.

It is obvious that in rounding to the nearest integer we must consider the first digit after the decimal place (we call it the tenth value). When this digit is 0, 1, 2, 3 and 4, the number is rounded down to the nearest integer, i.e. its value decreases to the nearest integer. On the other hand, when the value of the first digit after the decimal place is 5 or more, the value after rounding to the nearest integer increases. This is illustrated in the above example.

In addition, we may need to round a number to the nearest tenth, i.e. to write it with one decimal place. This occurs when the original number has 2 or more decimal places. In this case, we have to consider the value of the first number after the decimal point (the value of tenth therefore) to know whether the original number has to be rounded up or down.

For example, when 8.35 is rounded to the nearest tenth it becomes 8.4, while when the same number is rounded to the nearest integer, it becomes 8; when 12.62 is rounded to the nearest tenth it becomes 12.6 and when the same number is rounded to the nearest integer, it becomes 13, and so on.

### Example 2

Round the number 739.46 to the nearest thousand, hundred, ten, unit (integer), and tenth.

### Solution 2

When rounded to the nearest thousand, 739.46 becomes 1000, as it is closer to 1000 than 0.

When rounded to the nearest hundred, 739.46 becomes 700, as it is closer to 700 than 800.

When rounded to the nearest ten, 739.46 becomes 740, as it is closer to 740 than 730.

When rounded to the nearest integer, 739.46 becomes 739, as it is closer to 739 than 740.

When rounded to the nearest tenth, 739.46 becomes 739.5, as it is closer to 739.5 than 739.4.

We may extend this logic to rounding to the nearest hundredth, thousandth, etc. However, we will discuss these situations when dealing with decimal numbers later in this tutorial.

## Significant Figures

Another way of rounding numbers is to count only the first few digits (we call them "figures") that have a value attached to them. This method of rounding is called significant figures and it's often used with larger numbers and very small numbers.

For example, when we have the number 0.00278 and rounding to the nearest unit is required, the result is 0. The same thing is true when asked to round the number to the nearest tenth or hundredth. The results will be 0.0 and 0.00 respectively. Therefore, it is more appropriate to take a look at the structure of number and see which rounding is meaningful to apply. To do this, we have to look at the digits (figures) that are not zero. In the specific case, we must start looking from the digit 2 and onwards for any possible rounding, as this operation (rounding) is meaningful only when non-zero digits are involved. Such digits that are important for any possible rounding are known as "significant figures".

For example, the number above has 3 significant figures: 2, 7 and 8 because only these numbers can be used for rounding.

## Rules of Significant Figures

The following rules are applied to find the number of significant figures for a given number:

1. The zeroes at the end of whole numbers are not counted as significant figures.
For example, 3900 has only two significant figures: 3 and 9. This is because only these two numbers are considered for any possible rounding. In other words, we can round this number only to the nearest thousand or ten-thousand, as the other digits are zero and it is not necessary to do any rounding with them.
2. The zeroes at the beginning of any decimal number are not counted as significant figures.
For example, the number 0.0036 has only two significant figures (3 and 6) as we can obtain a meaningful rounding only to the nearest hundredth and thousandth, in which the two above non-zero digits are considered. This is not the case of zeroes at the end of a decimal number. These zeroes are all counted, as they determine the precision of a result. In other words, in practice is not the same thing to write 0.040 m and 0.04 m. in the first case, the value is measured with a precision up to 0.001 m (1 mm therefore), while in the second case, the precision of measurement taken is up to 0.01 m (1 cm therefore).
3. Any zero in-between two non-zero digits (both in whole number as well as in decimals) are always counted as significant figures.
For example, the number 0.0407 has three significant figures (4, 0 and 7). The other two zeroes are at the beginning of a decimal, so they are not counted.

### Example 3

How many significant figures do the following numbers have?

1. 403500
2. 3.0040
3. 0.00150

### Solution 3

1. 403500 is a whole number. In such numbers, only the zeroes at the end are not counted as significant. Therefore, this number has four significant figures: 4, 0, 3 and 5.
2. 3.0040 is a decimal number. In such numbers, only zeroes at the beginning are not counted. Therefore, this number has five significant figures (all digits are significant).
3. 0.00150 is a decimal number too. However, in this case there are some zeroes at the beginning, which are not counted as significant. Therefore, this number has only three significant figures: 1, 5 and 0.

Finding the Number of Significant Figures by Expressing the Number in Decomposed (Standard) Form

We can identify the number of significant figures by writing the number in the standard form. In such cases, only the part of the number that comes before the powers of ten is considered as significant. Let's consider again the numbers of the previous question to explain this point. We have:

1. 403500 = 4.035 × 105. In this case, the number has four significant figures: 4, 0, 3 and 5, as they come before the powers of ten.
2. 3.0040 = 3.0040 × 100. Nothing changes to the number, so there are five significant figures in it (all digits are significant).
3. 0.00150 = 1.50 × 10-3. This number has three digits before the powers of ten, so it has three significant figures.

All these results were obtained earlier, when we calculated the number of significant figures of the above number by means of the first method.

## Applications of Significant Figures

Significant figures have a wide range of applications in practice. For example, if the perimeter P of an equilateral triangle is 8.00 cm and we want to calculate the side length L, we write

L = P/3 = 8.00 cm/3 = 2.6666..cm

However, it is clear that we cannot express the result in this way, as we cannot be that precise in measurements. This means we have to express the result at only two decimal places as this is the tiniest division of units. Therefore, e must write the result at three significant figures: one before the decimal point and two after it. Hence, we must write the result as L = 2.67 cm.

### Example 4

The sides of a quadrilateral are 3.2 dm, 5 dm, 4.32 dm and 7.29 dm. Calculate the perimeter of this quadrilateral and express the result in the correct number of significant figures.

### Solution 4

The perimeter of a quadrilateral is the sum of all sides. Hence, we have

P = 3.2 dm + 5 dm + 4.32 dm + 7.29 dm = 19.89 dm

However, it makes no sense to write the result like this, as one of the sides (the 5 cm one) is measured in whole decimetres, so we must write the result in decimetres as well to ensure it fits the least precise measurement. Therefore, we must round the result to the nearest whole number, i.e. P ≈ 20 dm. Hence, it is written in one significant figure, as the number taken as a reference to determine the precision of result (the number 5) has only one significant figure.

Apparently, the rounded result seems less accurate than the original. However, in science we prefer to be sure in the correctness of our findings. Therefore, it is better to round the result and give an approximate value to the correct result than to make groundless assumptions.

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