Decimal Notation
The decimal number system is the most common system of counting in the world. It is a base 10 system, which means that it is based on a 10 number cycle, the numbers 0 - 9. Each number is also divided into 10 sections and so on and so on. These sections are notated by a decimal point, which is a dot placed after the number in the one's place and before the number representing the number of sections of the next number. The number after the decimal point represents a fraction of a number with the base of 10.
For example, 2.2 is a number written in decimal notation depicting 2 whole numbers and 2/10's of another whole number. And 37.55 is 37 whole numbers and 55/100's of another whole number.
Addition and Subtraction with Decimal Notation.
When adding or subtracting whole numbers, you just line them up on the right and add (or subtract).
For example:
25 + 13 = 38
or
59 - 22 = 37.
Numbers that include decimals are different. You can't just line them up and add. If you do that, you end up with the following:
41.78 + 13.2 is equal to 43
It's easy to see how this is not right. When adding or subtracting with decimals, the numbers need to line up along the decimals. It doesn't matter how many numbers are on either side of the decimal. That is where they need to line up to get the correct answer. Let's do the problem from above correctly. If you need to, you can add zeros at the end of a number to make sure there are two numbers in each column.
41.78 + 13.20 is equal to 54.98.
This way, you will get the right answer every time without trouble.
Thus we can conclude that
Basic operations on decimals are performed in the same way as basic operations are performed on the whole numbers.
- To add or subtract decimals, we align the decimal points of given numbers and perform the operation. In the sum or difference, we mark decimal point in the column of decimal points.
- If the decimals are of different decimal places, we convert them into same decimal places by putting zeros at the right of the decimals wherever necessary.
Multiplication and Division with Decimal Notation.
- To multiply two decimals, we multiply them as whole numbers ignoring the decimal points. We insert the decimal point in the product by counting as many places from right to left as the sum of number of decimal places of the given decimals.
- To multiply a decimal by 10, 100 or 1000, we simply shift the decimal point in the given number from left to right.
- one place, if it is multiplied by 10
- two places, if it is multiplied by 100
- three places, if it is multiplied by 1000 and so on.
- In division of decimals we proceed as:
- Decimal ÷ Whole Number (other than zero):
We first divide the whole number part of the dividend and then move towards decimal part of dividend. Correspondingly, we mark decimal point in the quotient also and then continue division as usual. - Whole Number ÷ Decimal:
We first remove the decimal point from the denominator by multiplying both numerator and denominator with 10, 100 or 1000 etc. We perform the operation of division as usual. - Decimal ÷ Decimal:
We first remove the decimal point from the denominator and perform division operation as usual. - Whole Number ÷ Whole Number (other than zero):
When the divisor is greater than the dividend we mark a decimal point in both the quotient and dividend and continue the division. - Decimal ÷ 10 or 100 or 1000: We simply move the decimal point in the dividend to the left by:
- one place if it is divided by 10
- two places if it is divided by 100
- three places if it is divided by 1000 and so on.
- Decimal ÷ Whole Number (other than zero):