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Unit: Number System
Chapter: Introduction & Conversion of Decimals
Reference: – Introduction to Decimals, Decimal Point and Place Value, Tenths, Hundredths, Thousandths, Like and Unlike Decimals, Converting Fractions to Decimals, Converting Decimals to Fractions, Converting Decimals to Percentages and Vice Versa, Terminating and Recurring Decimals
After studying this chapter, you should be able to understand:
- Introduction to Decimals & Their Need
- Decimal Point and Place Value
- Tenths, Hundredths, Thousandths
- Like and Unlike Decimals
- Converting Fractions to Decimals
- Converting Decimals to Fractions
- Converting Decimals to Percentages and Vice Versa
- Terminating and Recurring Decimals
Introduction
Decimals
Decimal numbers are special fractions with denominators as 10, 100, 1000, etc.
Suppose we divide one whole into 10 equal parts. Each part is called one-tenth of the whole and is written as 110
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or 0.1
When we further divide each of these units into 10 parts each, they become one-tenth of 110
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i.e. 110 of110 = 1100
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or 0.01
Decimals are used to represent numbers that are smaller than the unit 1, e.g. 0.47, 0.25, 0.73, etc.
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Let’s take this case, here in first diagram, 100 small blocks are adding up to form 1 single unit, means each block one-hundredth of a unit i.e. 1100
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units or 0.01 units.
In second figure, we have 47 blocks, means 0.47 units.
In third figure, we have 25 blocks, means 0.25 units.
In last figure, we have 73 blocks, means 0.73 units.
So, decimals are nothing but a way of writing whole numbers and fractional numbers together separated by a decimal point (dot).
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Place Value
When we write numbers, the position (or “place”) of each digit is important.
Consider the number 6357:
- the “7” is at the Ones position, meaning 7 ones (which is 7),
- the “5” is at the Tens position meaning 5 tens (which is fifty),
- the “3” is at the Hundreds position, meaning 3 hundred.
- The “6” is at the Thousands position, meaning 6 thousand.
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As we move towards left, each position is 10 times bigger!
Alternatively,
As we move towards right, each position is 10 times smaller
The number 6357 in terms of place value can be written as
6× 1000 + 3 × 100 + 5× 10 + 7 × 1 = 6000 + 300 + 50 + 7
Now, moving from right towards left ,7 is the unit digit, the second digit is 5 is the tens digit (represents 5 tens), third digit 3 is the hundreds digit and 6 is the thousands digit.
As we move from right to left the place value of a digit increases by 10 times.
If we extend this system to the right of units digit 7, the place value of the digits to the right of 7 will be tenths i.e. 110
Thus,the place of digits which are on the left of unit digits 7 are 10,100,1000 and the place of digits on the right of 7 are 110,1100,11000
We can continue with smaller and smaller values, from tenths, to hundredths, and so on, like in this example: |
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, next will be thousandths i.e. 11000
and so on.
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Let us see some example :
Example 1: Write the following numbers in the place value table: (a) 10.7 (b) 6.2
Solution :
Let us make a place value table and assign appropriate place value to the digits in the given numbers.
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Example 2 : Write the following as decimals :
(a) Three ones and six-tenths (b) Thirty-nine and one-tenth
Solution : (a) Three ones and six-tenths = 3 + 610
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= 3.6
(b) Thirty-nine and one-tenth = 39 + 110
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= 39.1
Example 3 : Write each of the following as decimals :
(a) 20 + 6 + 710
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(b) 100 + 2 + 910
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Solution :(a) 20 + 6 + 710
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We have 2 tens, 6 ones and 7 tenths. Therefore, the decimal representation is 26.7
(b) 100 + 2 + 910
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Here, we have 1 hundred, no tens, 2 ones and 9 tenths. Therefore, the decimal representation is 102.9
Life of Pi
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• The number Pi (the ratio of the circumference to the diameter of a circle) can’t be expressed as a fraction, this means it is an irrational number. When written as a decimal it never repeats and never ends.
• Here is Pi written to 50 decimal places: 3.14159265358979323846264338327950288419716939937510
• Want to remember the value of Pi (3.1415926) in easy way? You can do it by counting each word’s letters in ‘May I have a large container of coffee?’
