What are decimals?

We explain what decimals are and how the concept is explained to primary-school children, as well as how they are taught to relate decimals to money and measurement, the equivalence between fractions and decimals, partitioning, rounding and ordering decimals and adding, subtracting, multiplying and dividing decimals. Login or Register to add to your saved resources

A decimal is a number expressed in the scale of tens. Commonly speaking we talk about decimals when numbers include a decimal point to represent a whole number plus a fraction of a whole number (tenths, hundredths, etc.).

A decimal point is a point or dot used to separate the whole part of a number from the fractional part of a number.What are Decimals?

Explaining decimals to children

One of the best ways to describe decimals, is to show a child a blank hundred number square or number chart and explain that this represents 'one':What are Decimals?Children need to know that when we talk about decimals, it is as if we were splitting one whole up into smaller parts.

What are Decimals?What are Decimals?The place value of decimal numbers can be shown the following table:What are Decimals?What are Decimals?What are Decimals?

Learning about decimals in KS2

In Year 4, children need to understand the concept of a 'hundredth' and a 'tenth'.

They need to be able to write decimal equivalents of any number of tenths and hundredths, for example: 3/10 = 0.3 and 7/100 = 0.07. 

They also need to know decimal equivalents to 1/4, 1/2 and 3/4. This diagram is a good way of making this concept clear to them:What are Decimals?Children in Year 4 also need to know the effect of multiplying and dividing numbers one-digit and two-digit numbers by 10 and 100 (teachers will talk about digits sliding to the left and right). 

  • 5 multiplied by 10 is 50 (the five slides one place to the left)
  • 5 divided by 10 is 0.5 (the five slides one place to the right)
  • 5 multiplied by 100 is 500 (the five slides two places to the left)
  • 5 divided by 100 is 0.05 (the five slides two places to the right)
  • Two-digit number examples: 35 ÷ 10 = 3.5, 741 ÷ 100 = 7.41

Partitioning, rounding and ordering decimals

In Year 5, children need to count from any given number in decimal steps, so they may be given the following sequence and asked to continue it:

0.3, 0.6, 0.9, ___, ___, ___

(The next decimals would be 1.2, 1.5, 1.8 – it helps to think of counting in 3s here, but instead of 3, 6, 9, 12, 15, 18 you are using decimal numbers.)

They also need to partition, round, compare and order decimals, for example:

Partition 8.49      (8 + 0.4 + 0.09)

Round 7.4 to the nearest whole number  (7, because the 4 is smaller than 5, so the number gets rounded down)

Round 1.38 to one decimal place   (1.4, because the 3 is rounded up to 4 because the 8 is bigger than 5)

Put these decimals in order: 0.8, 0.14, 1.8, 0.4 (0.14, 0.4, 0.8, 1.8)

Introduction to Decimals

What are Decimals?

Problem: Order these numbers from least to greatest:

What are Decimals?

  • Analysis: We know that  is a mixed number: it consists of a whole number and a fraction. Let's use place value to help us compare these numbers
  • 57= (5 x 10) + (7 x 1)
  • = (5 x 10) + ( 7 x 1) + (4 x ) + (9 x )
  • 58= (5 x 10) + (8 x 1)
  • Answer: Ordering these numbers from least to greatest, we get:

What are Decimals?

That's a lot of writing! We can use decimals to write  more easily.

Definition: A decimal is any number in our base-ten number system. Specifically, we will be using numbers that have one or more digits to the right of the decimal point in this unit of lessons.

The decimal point is used to separate the ones place from the tenths place in decimals. (It is also used to separate dollars from cents in money.

)  As we move to the right of the decimal point, each number place is divided by 10.

Below we have expressed the number  in expanded form and in decimal form.

Mixed Number – Expanded  Form – Decimal Form
= (5 x 10) + ( 7 x 1)  + (4 x ) + (9 x ) = 57.49

As you can see, it is easier to write  in decimal form. Let's look at this decimal number in a place-value chart to better understand how decimals work.

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PLACE VALUE AND DECIMALS
5 7 . 4 9

As we move to the right in the place value chart, each number place is divided by 10. For example,  thousands divided by 10 gives you hundreds. This is also true for digits to the right of the decimal point. For example, tenths divided by 10 gives you hundredths.

When reading decimals, the decimal point should be read as “and.” Thus, we read the decimal 57.49 as “fifty-seven and forty-nine hundredths.” Note that in daily life it is common to read the decimal point as “point” instead of “and.” Thus, 57.49 would be read as “fifty-seven point four nine.

” This usage is not considered mathematically correct.

Example 1: Write each phrase as a fraction and as a decimal.

phrase fraction decimal
six tenths .6
five hundredths .05
thirty-two hundredths .32
two hundred sixty-seven thousandths .267

So why do we use decimals?

Decimals are used in situations which require more precision than whole numbers can provide. A good example of this is money: Three and one-fourth dollars is an amount between 3 dollars and 4 dollars. We use decimals to write this amount as $3.25.

A decimal may have both a whole-number part and a fractional part. The whole-number part of a decimal are those digits to the left of the decimal point. The fractional part of a decimal is represented by the digits to the right of the decimal point. The decimal point is used to separate these parts. Let's look at some examples of this.

decimal whole-number part fractional part
    3.25 3 25
    4.172 4 172
  25.03 25 03
    0.168 168
132.7 132 7

Let's examine these decimals in our place-value chart.

PLACE VALUE AND DECIMALS
3 . 2 5
4 . 1 7 2
2 5 . 3
. 1 6 8
1 3 2 . 7

Note that 0.168 has the same value as .168. However, the zero in the ones place helps us remember that 0.168 is a number less than one. From this point on, when writing a decimal that is less than one, we will always include a zero in the ones place. Let's look at some more examples of decimals.

Example 2: Write each phrase as a decimal.

Phrase Decimal
fifty-six hundredths 0.560
nine tenths 0.900
thirteen and four hundredths 13.040
twenty-five and eighty-one hundredths 25.810
nineteen and seventy-eight thousandths 19.078

Example 3: Write each decimal using words.

Decimal Phrase
0.0050 five thousandths
100.6000 one hundred and six tenths
2.2800 two and twenty-eight hundredths
71.0620 seventy-one and sixty-two thousandths
3.0589 three and five hundred eighty-nine ten-thousandths

Expanded Form

We can write the whole number 159 in expanded form as follows: 159 = (1 x 100) + (5 x 10) + (9 x 1). Decimals can also be written in expanded form. Expanded form is a way to write numbers by showing the value of each digit. This is shown in the example below.

Example 4: Write each decimal in expanded form.

Decimal Expanded form
4.1200 = (4 x 1) + (1 x ) + (2 x )
0.9000 = (0 x 1) + (9 x )
9.7350 = (9 x 1) + (7 x ) + (3 x ) + (5 x )
1.0827 = (1 x 1) + (0 x )  + (8 x ) + (2 x ) + (7 x )

Decimal Digits

In the decimal number 1.0827, the digits 0, 8, 2 and 7 are called decimal digits.

Definition: In a decimal number, the digits to the right of the decimal point that name the fractional part of that number, are called decimal digits.

Example 5: Identify the decimal digits in each decimal number below.

Decimal Number Decimal Digits
1.40000 1.40000
359.62000 359.62000
54.00170 54.00170
0.72900 0.72900
63.10148 63.10148

Writing whole numbers as decimals

A decimal is any number, including whole numbers, in our base-ten number system. The decimal point is usually not written in whole numbers, but it is implied. For example, the whole number 4 is equivalent to the decimals 4. and 4.0. The whole number 326 is equivalent to the decimals 326. and 326.0. This important concept will be used throughout this unit.

Example 6: Write each whole number as a decimal.

Whole Number Decimal Decimal with 0
17 17. 17.0
459 459. 459.0
8 8. 8.0
1,024 1,024. 1,024.0
519 519. 519.0
63,836 63,836. 63,836.0
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Often, extra zeros are written to the right of the last digit of a decimal number. These extra zeros are place holders and do not change the value of the decimal. For example:

7.5 =  7.50 =  7.500 =  7.5000 and so on.
9 =  9. =  9.0 =  9.00 and so on.

Note that the decimals listed above are equivalent decimals.

So how long can a decimal get?

A decimal can have any number of decimal places to the right of the decimal point. An example of a decimal number with many decimal places is the numerical value of Pi, shortened to 50 decimal digits, as shown below:

3.14159 26535 89793 23846 26433 83279 50288 41971 69399 37510

Summary: A decimal is any number in our base ten number system. In this lesson we used numbers that have one or more digits to the right of the decimal point.

The decimal point is used to separate the whole number part from the fractional part; it is handy separator. Decimal numbers are used in situations which require more precision than whole numbers can provide.

As we move to the right of the decimal point, each number place is divided by 10. 

Exercises

Directions: Read each question below. Select your answer by clicking on its button. Feedback to your answer is provided in the RESULTS BOX. If you make a mistake, choose a different button.

1. Which of the following is equal to ?
2. Which of the following is equal to ?
3. Which decimal represents seven hundred sixty-two and three tenths?
4. Which of the following is the expanded form of 0.546?
5. Which of the following is equal to twenty-five and sixty-nine thousandths?

What Are Decimal Numbers?

Decimal numbers are used to represent numbers that are smaller than 1 unit. Decimals are written to the right of the units place separated by a period. That is the say:

Hundreds Tens Units; Tenths Hundredths Thousandths

In the image that appears below, the first square represents the unit 1. If we divide the unit into 10 equal parts (the second square), the smaller parts are tenths. If we divide the tenths into 10 equal parts or the unit 1 into 100 equal parts (the third square), the resulting parts are hundredths.

What are Decimals?

Let’s look at some examples:

What are Decimals?

  • First example: If we divide one unit into 10 equal parts, we will have tenths. And we have colored in 7 of these parts. The way to write this is 0 units.7 tenths = 0.7.
  • Second example: In the second example we also have tenths and we have colored in 1 tenth. We write this in the following form: 0 units.1 tenth = 0.1.
  • Third example: In the third example we have represented hundredths, of which we have colored 6 tenths and 4 hundredths. So we write 0 units.6 tenths.4 hundredths = 0.64.
  • Fourth example: In the fourth example we have represented hundredths (one unit divided by 100), of which we have colored 3 tenths and 5 hundredths. We write 0 units.3 tenths.5 hundredths = 0.35.

Let’s see another example:

What are Decimals?

  • Fifth example: We have 2 whole units colored and for the third unit, which is divided into hundredths, we have 8 tenths colored and 1 hundredth colored. So we write: 2 units.8 tenths 1 hundredth = 2.81

What is the relationship between decimal numbers and fractions?

A unit is represented by 1.

Tenths are 1 unit divided into 10 equal parts = 1/10 = 0.1.

Hundredths are 1 unit divided into 100 equal parts = 1/100 = 0.01.

Thousandths are 1 unit divided into 1,000 equal parts = 1/1,000 = 0.001.

  • Example of converting from a decimal to a fraction:

7.508

We focus on the last number, the 8, which occupies the thousandths place, so the denominator will have to be 1,000. And in the numerator, we write the complete number without the decimal point. So, 7.508 = 7508/1000

  • Example for converting from a fraction to a decimal:

402/100

Since the denominator is 100, the last number of the numerator (2), has to be the hundredths, the previous number (0) has to be the tenths and the first number (4) has to be the units, putting the decimal point after the units. So, 402/100 = 4.02

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In Smartick we offer you lots of exercises to practice decimal numbers in a fun way. We hope that you enjoy them. Register now and try it free!

Learn More:

  • Estimations: Rounding Decimals to The Nearest Whole Number
  • Understand What a Fraction Is and When It Is Used
  • Using Mixed Numbers to Represent Improper Fractions
  • Working with Decimals: Addition and Subtraction
  • Homogeneous and Heterogeneous Fractions

Decimals

Fractions and decimals are two different ways to represent parts of a whole number. Decimals are a way to express tenths, hundredths, thousandths (and beyond) of a unit.

Working with decimals may look a bit complex to start with but, don’t worry, they’re only numbers and they obey rules like other numbers.

Working with Decimals

Adding and Subtracting Decimals

Decimals extend the number system beyond the simple ‘hundreds, tens, units’ into ‘tenths of units’, ‘hundredths of units’ and so on.

Working with decimals is therefore essentially the same as working with any other number.

After looking at our pages on Numbers, Addition and Subtraction, you would have no concerns about adding thousands to the mix, so why worry about tenths and hundredths?

If you were adding numbers without decimals, you would start with the units, and move along to tens, then thousands and so on. The same rule applies if there are decimals. Add them first, then units, then tens and so on.

The most important rule to remember is to line up the decimal points in your calculation, ensuring that the decimal point in the answer also lines up with the decimal points above it.

If you're confused about 'carrying over' when adding or subtracting see our pages Addition and Subtraction for help.

When multiplying and dividing decimals, the calculation works in the same way as with whole numbers. We multiply the numbers as if there was no decimal point at all. At the end of the calculation, we make sure that we have the decimal point in the correct place in our answer:

Starting with the answer that you have obtained by multiplying the numbers, move the decimal point the same number of places to the left as there are numbers after the decimal point in the two factors.

Dividing Decimals

Multiplying and dividing by 10

Multiplying by 10 moves the decimal point one place to the right (increasing the original number by a factor of 10). Dividing by 10 moves it one place to the left (decreasing the original number by a factor of 10).

You can use this fact to make dividing decimals a whole lot easier. Multiply by 10 the number that you are dividing by (the denominator) until it is a whole number. Multiply by 10 the number that you are dividing (the numerator) the same number of times. Then do the calculation.

Top Tip

If you’ve done a multiplication or division involving decimals, then check to see if the answer looks about right. In other words, if you took away the numbers after the decimal point, and rounded up or down to a whole number, would it still be about right?

If your answer looks much too big or too small, then check the position of your decimal point. It may well be a position out in either direction.

Converting Between Fractions and Decimals

Converting from decimals to fractions is fairly straightforward. Any number can be expressed as a fraction by simply putting it over one.

The same rule applies to decimals.

Put the decimal over one, and then multiply both top and bottom by 10 until you no longer have a decimal point. Then, if possible, convert your fraction to a mixed number and/or reduce it down to its smallest form.

See our page on Fractions for more.

Converting from Fractions to Decimals

Converting from fractions to decimals is slightly harder, but gets easier once you realise that a fraction is actually a division calculation.

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