DECIMALS

Let’s start this article with a simple question: are all numbers integers?

Well, the obvious answer to that question is a NO. All numbers are not integers.

Consider the case of 0.333333. What is this number? An integer? Well, it is a decimal.

But what are decimals?

Decimals are nothing else but the values lying between two integers on the number line.

Relating decimals to fraction: When we solve a fraction of the form p/q where q is non-zero, it is not necessary that it would reduce to an integral value. When we are left with a remainder, we ultimately convert it into decimal form. Some examples of decimals are 4.5, 9.6, 6.78, and 99.98, these all are decimal numbers.

Decimals can be subdivided in three parts:

Terminating decimals: A decimal number that terminates at a point is known as a terminating decimal e.g. 4.25, 6.5432, 2.3, etc. Here, there is nothing coming after: 25 in 4.25; 5432 in 6.5432 and 3 in 2.3.

It has an end or terminating point; hence, it is called a terminating decimal.

Recurring decimals: It is a type of decimal in which a digit or a set of digits is repeated continuously. Recurring decimals are written in abridged form, the digits that are repeated being marked by a bar placed over it e.g. 3.666666… = , or 2.347347347 = ,
Non-recurring non-terminating decimals: Those decimals, which don’t form any pattern or are irrational in pattern, are called non-recurring non-terminating decimals e.g.√2, √3, √7, π, e, etc.

Conversion of Recurring Decimal to Fraction:

All recurring numbers are rational numbers.

Hence, we can represent a recurring decimal by a fraction.

Pure recurring decimals: These are the decimals in which recurring starts just after the decimal.

Example 1: Convert 0.333333…… in fraction.

Solution:

Let x = 0.33333….. …(i)

So, 10x = 3.333333…. …(ii)

Subtract equation (i) from equation (ii) we will get

9x = 3 or

Example 2: Convert 0.67676767….. into fraction

Solution:

Let x = 0.67676767…………….……(i)

Or 100x = 67.67 ….……(ii)

Subtracting equation (i) from equation (ii) we will get

99x = 67, or

Example 3: Convert in fraction

Solution:

We know that = 0.267267267267…….

Let x = 0.267267267…… ….(i)

Or 1000x = 267.267267267…… ….(ii)

Subtracting equation (i) from equation (ii) we will get

999x = 267,

Mixed recurring decimals: These are the type of decimals in which recurring does not occur just after the decimal but after appearance of certain number of digits for e.g.

Example 4: Convert in fraction.

Solution:

Let x = = 0.26767676767……

So 1000x = 267.67676767……. ….(i)

And 10x = 2.676767676767…… ….(ii)

Subtracting equation (ii) from equation (i) we will get

990x = 265,

Example 5: Convert in fraction

Solution:

Let x = or 0.234545454545…….

So 10000x = 2345.4545454545…… ….(i)

And 100x = 23.454545….. ….(ii)

Subtracting equation (ii) from equation (i) we will get.

9900x = 2345 – 23 = 2322,

Some of the short tricks shown below can be used to save time:

Short Trick 1: A pure recurring decimal can be converted into fraction by writing down the number which is recurring as numerator and in denominator put as many number of 9’s as the number of digits on which recurring occurs.

=> etc.

Short Trick 2: To convert a mixed or impure recurring decimal into fraction, follow these steps.

Step 1: In the numerator write down the entire number formed by non–recurring and recurring numbers and subtract from it the part of the decimal that is not recurring.

Step 2: In the denominator write as many 9’s as the number of digits having recurring on it and then next to it write as many zeroes as there are digits without recurring in the given decimal.

Now take a look at some interesting tooltips which will help you solve your calculations easily.

Tooltip 1: Forming Decimals from Fractions

How we can write 4/5 in decimal form?

Since the number 4 is smaller than the 5, so the decimal value will be less than one.

Multiple and divide both numbers by 10. We have:

40/ (10×5) = 8/10

Thus, the final result is 0.8

So 4/5 = 0.8 and 0.8 is the decimal form of 4/5.

Tooltip 2: Adding Decimals

While adding decimals, you should always write the decimals in a vertical column with the decimal points aligned vertically.

e.g. Add all these numbers 0.567 +78 +5.06+56

Here we can write the decimals as follows:

78.000

+ 56.000

+ 5.060

+ 0.567

139.627

Tooltip 3: Subtracting Decimal

While performing addition, we can write the numbers in any order. But while subtracting, we should preferably write the numbers in descending order and the vertical column with decimal points should be aligned to the same decimal points.

Let’s take an example: we have to subtract 0.567 from 5.06. If we write it as:

5.06

– 0.567

4.507

Then this result is wrong because in case of subtraction we need equal digits in both the decimals, so these blank spaces are filled with 0

So this can be done like as

5.060

– 0.567

4.493

This is the right approach for the question.

Tooltip 4: Multiplying Decimals

As a first step, multiply the given integers in the normal form keeping the decimals aside. The number of decimal places in the product is then equal to the total of the decimal places in the two decimals. It is as simple as that.

Consider the following example:

5.060

×0.567

2.869020

First multiply 5060 with 567 and get the result as 2869020 and then move the decimal point to 6 places from the right i.e. between 2 and 8. Effectively, we move it to six decimal places which is the sum total of the decimal places in the two numbers.