A Guide on Decimals

Decimals numbers have two parts, a whole number and a fractional, separated by a dot. The number system is infinite and is thus divided into various types, namely, real numbers, natural numbers, whole numbers, rational numbers, decimal numbers, and many more.

Definition of decimals 

As we discussed, decimal numbers have two parts, a whole number and a fractional, separated by a dot. The dot that separates the two parts is called the “decimal point.” 

A decimal represents numbers in integer and non-integer forms. 

Every rational number can be expressed in the decimal form. However, the reverse does not hold true since non-terminating and non-recurring decimal points cannot be expressed in p/q form. 

For example:

67.80

Here, 67 is the “whole number” part, and 80 is the “fractional number” part. 

“.” denotes the decimal point.

78.39

Here, 78 is the “whole number” part, and 39 is the “fractional number” part. 

Types of decimal numbers

There are mainly four types of decimal numbers.

1. Terminating decimal numbers

These have a finite number of digits after a decimal point and are also known as terminating decimal points.  They are also named as exact decimal numbers. The number of digits is countable after the decimal point.

Example:  34.900

                  78.90

                   6.69 

         These numbers can be written in p/q form.

1. Non-terminating decimals

These numbers with infinite numbers after a decimal point are known as non-terminating decimal points. The number repeats endlessly. 

      They are further categorised into two categories: recurring and non-recurring decimal numbers. 

• Recurring decimal numbers- decimal numbers having infinite digits after a decimal point are called recurring decimal numbers, and a specific set of digits keep repeating. 

Example: 5.909090…, 78.050505…, etc. Here, in the first example, the set ‘90’ keeps repeating in 5.909090, whereas ‘05’ keeps recurring in 78.0500505…

You can also put a bar sign over the digits that are repeating. They are rational numbers and can be written in p/q form. 

• Non-recurring decimal numbers- these decimal numbers are: non-terminating and non-repeating. There is an infinite number of digits after a decimal point, and they do not even repeat or show any pattern in their decimal values. 

Example: 89.09897…, 78967.4567…, 3.1428…(pi), etc.

These numbers cannot be written in p/q form and are called irrational numbers.

Properties of decimals 

• The product remains the same if we multiply two decimal numbers in any order. Example: 0.9 * 0.89 = 0.801 and 0.89 * 0.9 = 0.801
• The product remains the same if we multiply a whole number and a decimal number. Example: 56 * 3 = 168 and 5.6 * 3 =16.8, only a decimal is placed
• If we multiply a decimal fraction by 1, the product itself is the decimal fraction. Example: 1 * ⅚ = ⅚
• The product is zero if we multiply the decimal fraction by zero. 0 * ⅞ = ⅞ 
• If we divide the decimal number by the same number, the quotient obtained is 1
• The quotient is zero if we divide zero by any decimal. Example: 8.9 * 0 = 0
• We can not divide the decimal numbers by 0 as there is no reciprocal of 0

Decimal to fraction conversion

The digits after the decimal point show the tenths, hundreds, thousandths, and so on, starting from the left-most digit on the decimal point’s right. 

For converting decimals into fractions, we express the decimal numbers in the expanded form and then simplify the values. 

Example: 0.98 = 98 * (1/100) = 98/100 = 49/50. Since the decimal point has two digits to the right of the decimal point, the expansion will go only to the hundredths.

If there had been four digits, let’s say, 0.4545, the expansion would have gone to 10000.

4545*(1/10000) = 4545/10000=909/2000.

Fraction to decimal conversion

We use the long division method to convert fractions into decimals to get to the decimal counterpart.  

Example: 25/2 is a fraction. 

After dividing it, we get 12.5.

Conclusion

This topic is a guide on decimal numbers with two parts, a whole number part, and a fractional part, separated by a decimal point.

We discussed the definition of decimals. We also studied the types of decimal numbers and properties of decimals.

This is a crucial topic for students interested in or/and pursuing Mathematics and those appearing in board exams. This topic is also a game-changer for students or aspirants appearing in SSC exams.