When we need to measure quantities that are not whole numbers, decimals are a convenient way to represent them. A decimal number has a whole number part and a fractional part separated by a decimal point. For example, in the number 5.7, the whole number part is 5 and the fractional part is 0.7. Decimals are used everywhere: in money (12.50 rupees), in measurements (1.5 metres) and in weights (2.5 kilograms).
The decimal system is an extension of the place value system. Just as the place values on the left of the decimal point are ones, tens, hundreds, on the right of the decimal point the place values are tenths, hundredths, thousandths and so on. The first digit after the decimal point represents tenths, the second digit represents hundredths and the third digit represents thousandths.
In this chapter, we learn to represent decimals on the number line, convert between fractions and decimals, compare decimals, and add, subtract, multiply and divide decimals. Decimals make calculations with parts of wholes much easier than fractions because we use the same rules of arithmetic we already know.
The first place after the decimal point is the tenths place. One tenth is written as 0.1, which equals the fraction 1/10. Two tenths is 0.2, which equals 2/10.
The second place after the decimal point is the hundredths place. One hundredth is written as 0.01, which equals the fraction 1/100. For example, 0.05 = 5/100 and 0.37 = 37/100.
The third place after the decimal point is the thousandths place. One thousandth is written as 0.001, which equals 1/1000. For example, 0.125 = 125/1000.
To convert a fraction with denominator 10, 100 or 1000 to a decimal, write the numerator and place the decimal point so that the number of decimal places matches the number of zeros in the denominator.
A number like 5.7 has a whole number part (5) and a decimal part (0.7). Such numbers are called mixed decimals. In expanded form, 5.7 = 5 + 7/10. Similarly, 12.53 = 12 + 5/10 + 3/100.
Decimals can be represented on the number line just like whole numbers. To show 0.1 to 0.9, we divide the space between 0 and 1 into 10 equal parts. To show hundredths, we divide the space between two tenths into 10 further equal parts.
For example, on a number line, 0.4 lies between 0 and 1, closer to 0.5. The number 1.3 lies between 1 and 2. The number 2.75 lies between 2.7 and 2.8.
To compare two decimal numbers:
Decimals with the same number of decimal places are called like decimals. To compare decimals with different numbers of decimal places, we first convert them to like decimals by adding zeros at the end. For example, 5.7 and 5.23 become 5.70 and 5.23.
To add or subtract decimals:
Multiply as with whole numbers, then place the decimal point so that the product has the same number of decimal places as the decimal factor.
Multiply the decimals as whole numbers, then count the total number of decimal places in both factors and place the decimal point in the product accordingly.
When we multiply a decimal by 10, 100 or 1000, the decimal point shifts to the right by as many places as there are zeros.
Divide as with whole numbers and place the decimal point in the quotient directly above the decimal point in the dividend.
Shift the decimal point of the divisor to the right to make it a whole number, and shift the decimal point of the dividend by the same number of places. Then divide.
When we divide a decimal by 10, 100 or 1000, the decimal point shifts to the left by as many places as there are zeros.
Decimals are used in everyday life for money, length, weight and capacity. The price of an item may be 45.50 rupees, a person's height may be 1.62 metres, and the weight of a bag may be 3.75 kilograms. When we convert rupees to paise, 1 rupee = 100 paise, so 5.40 rupees = 5 rupees 40 paise.
| Place | Value as fraction | Decimal |
|---|---|---|
| First place after decimal point | 1/10 | 0.1 |
| Second place after decimal point | 1/100 | 0.01 |
| Third place after decimal point | 1/1000 | 0.001 |
| Operation with 10, 100, 1000 | Rule | Example |
|---|---|---|
| Multiply by 10 | Shift decimal point 1 place right | 3.47 x 10 = 34.7 |
| Multiply by 100 | Shift decimal point 2 places right | 3.47 x 100 = 347 |
| Divide by 10 | Shift decimal point 1 place left | 3.47 divided by 10 = 0.347 |
| Divide by 100 | Shift decimal point 2 places left | 3.47 divided by 100 = 0.0347 |
Decimals give us a powerful and convenient way to represent parts of wholes. We learnt about tenths, hundredths and thousandths, how to compare and order decimals, and how to add, subtract, multiply and divide them. The rules of shifting the decimal point when multiplying or dividing by 10, 100 and 1000 make calculation fast. Decimals are used daily in money, measurement and weight, and they connect fractions to the real world.