Basics of Decimal Numbers

Basics of Decimal Numbers

In this section you will explore the basics of decimal numbers. The objectives in this article are to look at different applications of decimal numbers, learn how to use place value, learn how to compare, order and round decimal numbers and how to add, subtract, divide and multiply decimal numbers.

Introducing Decimal Numbers

Decimal Numbers: A Decimal Number is a number which contains a Decimal Point

Example: Is 43.21 a Decimal Number?

Answer: Yes

Explanation: 43.21 is a Decimal Number because it contains a Decimal Point.

Example: Consider the following four numbers:

1. a) 123
2. b) 12.3
3. c) 1.23
4. d) 0.123

These four numbers have the same three digits, in the same order, yet they are all different! Which of these four numbers (if any) are decimal numbers?

Answer:

12.3, 1.23 and 0.123 are all decimal numbers, 123 is a whole number

Explanation:

A decimal number is defined as a number which contains a decimal point. As 12.3, 1.23 and 0.123 all have decimal points, they are all decimal numbers. As 123 has no decimal point it must be a whole number.

Now consider Figure1:

Figure 1

Whole Numbers: All digits on the left of the Decimal Point are Whole Numbers

Decimal Fractions: All digits on the right of the Decimal Point are Decimal Fractions (also known as fractions of a whole number)

Reading and Writing Decimal Numbers:

When you read a decimal number out loud, you must read the decimal fractions point one at a time. For example ‘43.21’ is read ‘forty three point two one’ NOT ‘forty three point twenty one’

Example: Read aloud or write down the following numbers:

4. a) 34.677
5. b) Eight hundred and thirty six point two five four
6. c) 0.123
7. d) two point one eight nine six three

Answer:

1. a) thirty four point six seven seven
2. b) 836.254
3. c) zero point one two three
4. d) 2.18963

Exception to the rule!

Although money is a decimal number, you do not read the numbers one at a time!

Example: Read the following numbers out loud

2. a) 12.34
3. b) £12.34

Answer:

1. a) twelve point three four
2. b) twelve pounds thirty four

 

Different Applications of Decimal Numbers

Decimals can be seen in measuring. Builders, engineers and cooks use decimal numbers to gain better perspective of extremely tiny measurements. The decimal point separates the Whole Numbers and the Decimal Fractions.

Length

1 metre = 100 centimetres

1 centimetres = 10 millimetres

Decimal points can be seen when measuring length. People use a decimal point to separate metres from centimetres and centimetres from millimetres.

Example

Write the following as decimal numbers:

1. a) 4 metres and 50 centimetres (answer should be in metres)
2. b) 4 centimetres and 5 millimetres (answer should be in centimetres)

Answer:

4. a) 4.5 metres
5. b) 4.5 centimetres

Explanation:

1. a) 1 metre = 100 centimetres

? metres = 50 centimetres

100/50 = 2.

1/2 = 0.5

Thus 0.5 metres = 50 centimetres

4 metres and 50 centimetres = 4 metres and 0.5 metres = 4.5 metres.

1. b) 1 centimetre = 10 millimetres

? centimetres = 5 millimetres

10/5 = 2

1/2 = 0.5

Thus 0.5 centimetres = 5 millimetres

4 centimetres and 5 millimetres = 4 centimetres and 0.5 centimetres = 4.5 centimetres .

Weight

1 kilogram = 1000 grams

Decimal points can be seen when measuring weight. People use the decimal point to separate kilograms from grams.

Example

Write 4 kilograms and 560 grams as a decimal number (answer should be in kilograms)

Answer

4.56 kg

Explanation:

1 kilogram = 1000 grams

? kilograms = 560 grams?

1000/560 = 100/56

1/ 100/56 = 56/100 = 0.56

Thus 560 grams = 0.56 kilograms.

4 kilograms and 560 grams = 4 kilograms and 0.56 kilograms = 4.56 kilograms.

Capacity

1 litre = 1000 millilitres

Decimal points can be seen when measuring capacity. People use the decimal point to separate litres from millilitres.

Example

Write 7 litres and 300 millilitres as a decimal number (answer should be in litres)

Answer

7.3l

Explanation:

1 litre = 1000 millilitres

? litres = 300 millilitres

1000/300 = 10/3

1 / 10/3 = 3/10 = 0.3

Thus 0.3 litres = 300 millilitres

7 litres and 300 millilitres = 7 litres and 0.3 litres = 7.3 litres = 7.3l

Money

1 pound = 100 pennies

Decimal numbers can be seen in currencies. For example, the UK Pound Sterling uses a decimal point to separate pounds from pennies.

Example

Write 2 pounds and 5 pence as a decimal number (answer should be in pounds)

Answer

£2.05

Explanation:

1 pound = 100 pennies

? pounds = 5 pennies

100/5 = 20

1/20 = 0.05

Thus £0.05 = 5 pennies

2 pounds and 5 pennies = 2 pounds and 0.05 pounds = £2.05.

 

Place Values of Decimal Numbers
Decimal means based on ten, i.e. numbers are orginised in a system based on multiples or sub-mulitples of ten. Thus the position or place of each digit is very important.

Figure 2 reveals what each place is called, using 234.567 as an example:

Figure 2

Example: Consider 234.567 Name the position of each digit:

Answer:

• 2 is in the hundreds position, this represents 2 hundreds
• 3 is in the tens postion, this represents 3 tens
• 4 is in the units postion, this represents 4 ones
• 5 is in the tenths position, this represents 5 tenths
• 6 is in the hundredths position, this represents 6 hundredths
• 7 is in the thousandths position, this represents 7 thousandths

Use of Zero

Zero is very important when it comes to decimal numbers. It can be used as a place holder:

e.g 1.024 or as a leading zero – to show that there are no whole numbers: e.g. 0.14

Place holders are very important, for example 1.024 is a different number to 1.24

Similarly, leading zeros are very important too. 0.124 is a different number to 124

 

Comparing and Ordering Decimal Numbers

Ordering whole numbers is easy, we can clearly see that 340 > 327, because whilst 300 = 300, 40 > 20.

Comparing decimal numbers is just as easy, though it seems tricky at first.

Consider 3.4 and 3.27 – which one is larger?

It may help to consider them in a table:

You can now clearly see that whilst 3=3 0.4 > 0.2. Thus 3.4 > 3.27

Example: Put these numbers in size order, from smallest to largest:

15.23, 15.7, 15.652

Answer: 15.23, 15.652, 15.7

Explanation:

Rounding Decimal Numbers

Rounding is an important skill used a lot with decimal numbers.

Rounding decimals to the nearest whole number

Whether we round up or round down depends on whether or not the digit in the first decimal place is more or less than 5.

If the digit in the first decimal place is less than 5 round DOWN

If the digit in the first decimal place is 5 or more round UP

Ignore digits after the first decimal place

In figure 5 we give an example to illustrate when you should round up and when you should round down:

figure 5

 

Example

Round the following numbers to the nearest metre:

3. a) 163.4m
4. b) 205.78m
5. c) 14.6609m
6. d) 109.01m

Answer

1. a) 163m
2. b) 206m
3. c) 15m
4. d) 109m

Explanation:

4. a) The digit in the first decimal place is 4. As 4 < 5 you must round DOWN. Rounding down gives 163m
5. b) The digit in the first decimal place is 7. As 7 > 5 you must round UP. Rounding up gives 206m (Note: you ignore 8 as it is after the first decimal place)
6. c) The digit in the first decimal place is 6. As 6 > 5 you must round UP. Rounding up gives 15m (Note: you ignore 609 as it is after the first decimal place)
7. d) The digit in the first decimal place is 0. As 0 < 5 you must round DOWN. Rounding down gives 109m

Rounding decimals to 2 decimal places

You may be asked to round to 2 decimal places, particularly if money is involved. The process is very similar to rounding to the nearest whole number, but now instead of considering the digit in the first decimal place, you must now consider the digit in the third decimal place

Whether we round up or round down depends on whether or not the digit in the third decimal place is more or less than 5.

If the digit in the third decimal place is less than 5 round DOWN

If the digit in the third decimal place is 5 or more round UP

Ignore digits after the third decimal place

Example

Tom is using a calculator to work out some prices, the answers are in pounds but some of the answers are to more than 2 decimal places. Round Tom’s answers to 2 decimal places

1. a) 1.29445
2. b) 2.33333
3. c) 0.125
4. d) 4.5

Answer

1. a) £1.29
2. b) £2.33
3. c) £0.13
4. d) £4.50

Explanation:

4. a) The digit in the third decimal place is 4. As 4 < 5 you must round DOWN. Rounding down gives £1.29
5. b) The digit in the third decimal place is 3. As 3 < 5 you must round DOWN. Rounding down gives £2.33
6. c) The digit in the third decimal place is 5. As 5 = 5 you must round UP. Rounding up gives £0.13
7. d) Because this is a money value, you must add a final 0 to give an answer to 2 decimal places.

 

Adding Decimal Numbers

Adding decimals is essentially the same as adding whole numbers, just be careful to keep the decimal points aligned.

Example

Perform the following calculations:

4. a) 4.3 + 3.4
5. b) 4.32 + 3.4
6. c) 4.3 + 3.45
7. d) 4.32 + 3.45

Answer

7. a) 7.7
8. b) 7.72
9. c) 7.75
10. d) 7.77

Subtracting Decimal Numbers

Subtracting decimals is essentially the same as subtracting whole numbers, just be careful to keep the decimal points aligned.

Example

Perform the following calculations:

4. a) 4.3 – 3.4
5. b) 4.32 – 3.4
6. c) 4.3 – 3.45
7. d) 4.32 – 3.45

Answer

1. a) 0.9
2. b) 0.92
3. c) 0.85
4. d) 0.87

Multiplying Decimal Numbers

Multiplying decimal numbers is a bit trickier than adding or subtracting decimal numbers.

Steps:

1. Multiply as if they were whole numbers
2. Put the decimal point in the appropriate place. The appropriate place is found by counting how many digits there were after the decimal point in the question (say x), then moving our answer from step one that x places to the right.

1. Check by using an estimation – i.e. round to your nearest number and check the estimation is in the same column as the answer

Example

Solve 24 x 7.8

Answer 187.2

Explanation

1. 24 is already a whole number so stays the same, 7.8 becomes 78. 24 x 78 = 1872
2. In this question there is just one digit after the decimal point (i.e. 8) So we must now move our answer 1 digit to the right. Hence 1872 becomes 187.2
3. Estimation: round 7.8 to 8. 24 x 8 = 192. As our estimation (192) is in the same column (hundreds) as our answer (187.2) you can be confident that the answer is the correct size.

Example

Solve 1.2 x 3.5

Answer 4.20

Explanation

1. First make 1.2 and 3.5 into whole numbers. 1.2 becomes 12 and 3.5 becomes 35. 12 x 35 = 420
2. In this question there were two digits after the decimal point (i.e. 2 and 5). So we must now move our answer 2 digits to the right. Hence 420 becomes 4.20
3. Estimation: If we round both our numbers to the nearest whole number we get 1 x 4 = 4. As our estimation (4) is in the same column (units) as our answer (4.2) you can be confident that the answer is the correct size.

Example

Solve 0.012 x 112

Answer 1.344

Explanation

1. 0.012 becomes 12. 12 x 112 = 1344
2. There were three digits after the decimal point ( 0, 1, 2) so we must now move our answer 3 digits to the right. Hence 1344 becomes 1.344
3. Estimation: 0.012 becomes 0.01 and 112 becomes 100. 0.01 x 100 = 1. As our estimation (1) is in the same column (units) as our answer (1.344) you can be confident that the answer is the correct size.

 

Dividing Decimal Numbers

 

Revision of division

First revise division:

Example: There are 4835 apples. They need to be put into packed of five. How many packets will there be?

Answer 967

Explanation

4835 / 5

5 goes into 4 zero times, so carry the 4 over and consider how many times 5 goes into 48.

5 goes into 48 nine times with 3 left over, so write 9 over the ’48’ and consider how many times 5 goes into 33.

5 goes into 33 six times with 3 left over, so write 6 over the ’33’ and consider how many times 5 goes into 35.

5 goes into 35 7 times with 0 left over. So write 7 above the ’35’ and we have our answer. 967.

 

Dividing Decimal Numbers

Dividing decimal numbers is easier than multiplying them.

1. Place a decimal point in the answer line at the end of the units column
2. Divide in the usual fashion
3. Check with an estimation

 

Example: Five people go out for a meal. The bill comes to £48.35. The people split the bill evenly – how much must each person pay?

Answer

9.67

Explanation:

1. In the question we have £48.35, hence 8 is in the units column. We must have two decimal points – one next to the 8 and one directly above the first decimal point, in the answers column
2. Divide as per usual, see figure 7:

figure 7

 

3. Check the answer with an estimation: 50/ 5 = 10. As our estimation (10) is very close to our answer (9.67) we know that our answer is an appropriate size.

 

Long Division and Decimal Numbers

Similarly, long division of decimal numbers is exactly the same as long division of whole number, but you still must keep the decimal point in the correct place.

Example

Divide £275.10 by 14 banks.

Answer: each bank gets £19.65

Explanation

Estimation: 280/ 14 = 20. This estimation is close to our answer so we can be sure the answer is of an appropriate size.