Understanding Decimals

A decimal fraction (or, a decimal number) is a fraction whose denominator is 10 or a higher power of 10.

Thus , etc. are all decimal fractions.

In order to express a given decimal fraction in shorter form, the denominator is not written ,but it’s absence is shown by a dot (called a decimal point) , inserted in a proper place.

Example:

1. When there is no number to the left of the decimal point, generally, a zero is written, i.e, .72 is written as 0.72.
2. 2.4 means (2+0.4). Here 2 is the integral part and 0.4 is the decimal part of the number 2.4.
3. Any extra zero (or zeroes) written after the decimal part of a number does not change it’s value.

Example: Value of 3.5 is the same as 3.50 or 3.500 or 3.5000 and so on.

Reading Decimal Numbers

The integral part is read according to its value and decimal part is read by naming each digit, in order, separately.

Example: 21.45 will be read as twenty one point four-five.

Converting Decimal fraction to Vulgar fraction

Remove decimal point from the given decimal number . And in its denominator write as many zeroes , as the number of digits in the decimal part, to the right of 1.

Thus, and so on

In a decimal system, the first place on the right of the decimal point is called tenths place; the second place to the right of decimal is called hundredths place and so on.

Similarly, the first place on the left of decimal is the units place; the second place on the left of decimal is the tens place and so on.

Converting fraction to Decimal fraction:

1. When the denominator of the given fraction is 10, 100, 1000 etc.:

Counting from extreme right to left, mark the decimal point after as many digits of the numerator as there are zeroes in the denominator.

etc.

2. When the denominator of the given fraction is other than 10 or higher power of 10:

Divide in an ordinary way and mark the decimal point in the quotient just after the division of unit digit is completed. After this, any number of zeroes can be borrowed to complete the division.

Decimal Places

The number of figures that follow the decimal point is called the number of decimal places. Thus, 28.497 has 3 decimal places, 153.46 has 2 decimal places and so on.

Decimal Addition

Write the given decimals in such a way that the decimal points of all the numbers fall in the same vertical line. Digits with the same place value are placed one below the other that is units are below units, tens below tens and so on.

Addition is started from the right side, as done in the usual addition(empty places may be filled up by zeroes). In the result (total), the decimal point is placed under decimal points of the numbers added.

Note:

A whole number can also be expressed as a decimal number by putting a decimal after its last (unit) digit and after it as many zeroes as are required.

Example:  etc.

Decimal Subtraction

In subtraction also, the numbers are written in such a way that their decimals are in the same vertical line. Digits with the same place value are placed one below the other ( empty places may be filled by zeroes).

Subtraction is started from the right side, as in the case of normal subtraction.

In the result, decimal point is placed just under the other decimal points.

Decimal Multiplication

1. Multiplication by 10, 100, 1000, etc

Shift the decimal point, in the multiplicand, to the right by as many digits as there are zeroes in the multiplier.

Example:

2. Multiplication by a whole number

Multiply in an ordinary way without considering the decimal point.

In the product, the decimal point should be fixed by counting as many digits from the right as there are decimal places in the multiplicand.

Example: 

3. Multiplication of a decimal number by another decimal number.

Multiply the two numbers in a normal way, ignoring their decimals.

In the product, decimal point is fixed counting from right, the digits equal to the sum of decimal places in the multiplicand and the multiplier.

Example:

Since the multiplicand (32.5) has one decimal place and multiplier (0.07) has two decimal places, their product will have 1+2= 3 decimal places.

Decimal Division

1. Division by 10, 100 ,1000, etc.

Shift the decimal point to the left by as many digits as there are zeroes in the divisor.

Example: 

2. Division by a whole number:

Divide in the normal manner, ignoring the decimal, and mark the decimal point in the quotient, while just crossing over the decimal point in the dividend.

Example:

3. Division of a decimal number by another decimal number:

Shift the decimal points of the dividend and the divisor both by as many equal number of digits, which reduces the divisor to a whole number.

The division is then carried out as in Case 2 described above

Now consider

Here, the division will not be exact that is all the digits in the dividend will be exhausted but still there will be some remainder left. So, we go on writing zeroes (one by one) with the remainder and continue the division process. We can continue writing zeroes, because adding zero at the extreme right of a decimal number does not change the number. Therefore, division can be continued for as many decimal places as we like to.

(upto 3 decimal places)

Terminating Decimals

Sometimes in a decimal division, the dividend is exactly divisible and no remainder is left after certain number of steps. Such answers in the quotient are called terminating decimals.

Example:  is a terminating decimal.

Non-terminating Decimals

In a decimal division, sometimes the remainder does not become zero(does not terminate) no matter how long the division is continued.

In such cases the quotient is called non-terminating decimal.

which is a non-terminating decimal.

The fact, that it is a non- terminating decimal; is shown by writing the digits of the quotient till the division is carried out. After that few dots are put to show that this division continues.

Recurring Decimals

On performing a division, sometimes we find that the same remainder is left, no matter how long we continue the division process.

Consider 2/3. Here, the remainder is always 2. For this reason, same digit 6 appears again and again in the quotient.

This fact is shown by putting a dot or a bar over the repeating digit or digits in the quotient.

The above which is a non-terminating repeating decimal is called a recurring decimal .The dot over 6 shows that 6 repeats infinitely.

Rounding off Decimal Numbers

Sometimes, answers have a large number of decimal places, for example, 8.6672843, 5.36592 etc .

But we need answers only upto a few number of decimal places. In such cases, the answers are rounded off to the required number of decimal places.

Rounding off:

1. If the answer required is correct to two decimal places, we retain digits upto three decimal places.
2. If the digit in the third decimal place is five or more than five ,the the digit in the second decimal place is increased by one and, if the digit in the third decimal place is less than five, then the digit in the second decimal place is not altered.
3. The third digit which was retained is now omitted.

Example: To get 3.946824 correct to three decimal places first write it as 3.9468

Then according to the rule, the digit in the third place changes from 6 to 7.

Therefore 3.946824= 3.947 correct to three decimal places.

Significant Figures

Significant figures are the total number of digits present in anumber except the zeroes preceding the first numeral.

In counting the number of significant digits, it should be noted that:

1. The position of the decimal is disregarded.
2. All zeroes in between the numerals are counted
3. All zeroes after the last numeral are counted.
4. The zeroes preceding the first numeral are not counted.

Exercise

1. 1.Convert the following into vulgar fractions in their lowest terms
   1. 2.04
   2. 0.085
   3. 8.025
2. Convert into decimal fractions:
   • 
3. Write the number of decimal places in
   1. 8235.456
   2. 0.000879
4. Write the following decimals as word statements
   1. 1.9, 4.4, 7.5
   2. 0.005, 0.20, 111.519
5. Find the difference between 6.85 and 0.685
6. Take out the sum of 19.38 and 56.025 from 200.111.
7. Add 13.95 and 1.003 and from the result, subtract the sum of 2.794 and 6.2
8. What should be added to 39.587 to give 80.375?
9. What is the excess of 584.29 over 213.95?
10. Evaluate:
    1. (6.25+ 0.36) – (17.2 – 8.97)
    2. 879.4 – (87.94 – 8.794)
11. Evaluate:
    1. 
    2. 
    3. 
    4. 
    5. 
12. Divide:
    • 
    • 
13. Find whether the given division forms a terminating decimal or a non-terminating decimal:
    1. 2. 3.
14. Express as recurring decimals:
    1. 2. 
15. Round off: 0.62, 100.479, 0.065 and 0.024 to the nearest hundredths.
16. Write the number of significant figures in:
    1. 
    2.