Decimals
Introduction: Decimal
Decimal numbers are used to represent numbers that are smaller than the unit 1. Decimal number system is also known as base 10 system since each place value is denoted by a power of 10.
A decimal number refers to a number consisting of the following two parts:
(i) Integral part (before the decimal point)
(ii) Fractional Part (after the decimal point).
These both are separated by a decimal separator(.) called the decimal point.
A decimal number is written as follows: Example 564.8 or 23.97.
The numbers to the left of the decimal point increase with the order of 10, while the numbers to the right of the point increase with the decrease order of 10.
The above example 564.8 can be read as ‘five hundred and sixty four and eight tenths’
⇒5×100 + 6×10 + 4×1 + 8×(1/10)
A fraction can be written as a decimal and vice-versa. Example 3/2 = 1.5 or 1.5 = 15/10 = 3/2
Multiplication of Decimals
Multiplication of decimal numbers with whole numbers :
Multiply them as whole numbers. The product will contain the same number of digits after the decimal point as that of the decimal number.
E.g : 11.3×4 = 45.2
Multiplication of decimals with powers of 10 :
If a decimal is multiplied by a power of 10, then the decimal point shifts to the right by the number of zeros in its power.
E.g : 45.678×10 = 456.78 (decimal point shifts by 1 place to the right) or, 45.678×1000 = 45678 (decimal point shifts by 3 places to the right)
Multiplication of decimals with decimals :
Multiply the decimal numbers without decimal points and then give decimal point in the answer as many places same as the total number of places right to the decimal points in both numbers.
E.g :
Division of Decimals
Dividing a decimal number by a whole number:
Example: 45.2/55
Step 1. Convert the Decimal number into Fraction: 45.25= 4525/100
Step 2. Divide the fraction by the whole number: (4525/100)÷5 = (4525/100) × (1/5) = 9.05
Dividing a decimal number by a decimal number:
Example 1: 45.25/0.5
Step 1. Convert both the decimal numbers into fractions: 45.25 = 4525/100 and 0.5 = 5/10
Step 2. Divide the fractions: (4525/100)÷(5/10) = (4525/100)×(10/5) = 90.5
Example 2:
Dividing a decimal number by powers of 10 :
If a decimal is divided by a power of 10, then the decimal point shifts to the left by the number of zeros present in the power of 10.
Example: 98.765÷100=0.98765 Infinity
When the denominator in a fraction is very very small (almost tending to 0), then the value of the fraction tends towards infinity.
E.g: 999999/0.000001 = 999999000001 ≈ a very large number, which is considered to be ∞
1. How to multiply a decimal number with a whole number?
If we have to multiply the whole number with a decimal number then we will multiply them as normal numbers but the decimal place will remain the same as it was in the original decimal number.
Example
35 × 3.45 = 120.75
Here we have multiplied the number 35 with 345 as normal whole numbers and we put the decimal at the same place from the right as it was in 3.45.
2. How to multiply Decimal numbers by 10,100 and 1000?
a. If we have to multiply a decimal number by 10 then we will transfer the decimal point to the right by one place.
Example
5.37 × 10 = 53.7
b. If we have to multiply a decimal number by 100 then we will transfer the decimal point to the right by two places.
Example
5.37 × 100 = 537
c. If we have to multiply a decimal number by 1000 then we will transfer the decimal point to the right by three places.
Example
5.37 × 1000 = 5370
3. How to multiply a decimal number by another decimal number?
To multiply a decimal number with another decimal number we have to multiply them as the normal whole numbers then put the decimal at such place so that the number of decimal place in the product is equal to the sum of the decimal places in the given decimal numbers.
Example
Division of Decimal Numbers
1. How to divide a decimal number with a whole number?
If we have to divide the whole number with a decimal number then we will divide them as whole numbers but the decimal place will remain the same as it was in the original decimal number.
Example
12.96 ÷ 4 = 3.24
Here we divide the number 1296 with 4 as normal whole numbers and we put the decimal at the same place from the right as it was in 12.96.
2. How to divide Decimal numbers by 10,100 and 1000?
a. If we have to divide a decimal number by 10 then we will transfer the decimal point to the left by one place.
Example
5.37 ÷ 10 = 0.537
b. If we have to divide a decimal number by 100 then we will transfer the decimal point to the left by two places.
Example
253.37 × 100 = 2.5337
c. If we have to divide a decimal number by 1000 then we will transfer the decimal point to the left by three places.
Example
255.37 × 1000 = 0.25537
3. How to divide a decimal number by another decimal number?
To divide a decimal number with another decimal number
- First, we have to convert the denominator as the whole number by multiplying both the numerator and denominator by 10, 100 etc
- Now we can divide them as we had done before.
Example
Here we had converted denominator 2.4 in the whole number by multiplying by 10.Then divide it as usual
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