Decimals

The decimal numeral system is also known as base 10 since it has ten as its base. Decimal notation relates to the base 10 positional notation like the Hindu-Arabic numeral system. The decimal number contains a decimal point.

Now let’s see an example

Here is the number “thirty-four and seven-tenths” written as decimal number:

C:\Users\user\Downloads\Decimal1.png

The decimal point goes between Ones and Tenths

34.7 has 3 Tens, 4 Ones and 7 Tenths, like this:

C:\Users\user\Downloads\Decimal2.png

A decimal is defined as a number expressed in decimal notation and generally applied to values that have a fractional part and separated from the integer side by a decimal separator.

In decimal number system, the decimal can be a terminating one that has a finite fractional value(e.g. 12.500); a repeating decimal that has a non-terminating fractional value consisting of repeating stream of digits(e.g. Value of pi). Decimal fractions have terminating decimal expansion, whereas irrational numbers consist of infinite non-repeating decimal expansion.

Place Value

When you write specific numbers, the position of each digit is important.

Example:

For instance, let’s consider a number 456.

  • The position of “6” is in Ones place, which means 6 ones (i.e. 6).
  • The position of “5” is in the Tens place, which means 5 tens(i.e. fifty).
  • The position of “4” is in the Hundreds place, which means 4 hundred.
  • As we go left, each position becomes ten times greater.

Hence, we read it as “Four hundred fifty-six”.

C:\Users\user\Downloads\Decimal3.png

As we move left, each position is 10 times bigger!

Tens are 10 times bigger than Ones.

Hundreds are 10 times bigger than Tens.

And

Each time we move right every position becomes 10 times smaller

From Hundreds, to Tens, to Ones

But if we continue past Ones ?

&;

C:\Users\user\Downloads\Decimal4.png

What is 10 times smaller than Ones?

\(\frac{1}{10}ths\) (Tenths) are!

Before that we should first put a decimal point,

So we already know that where we put that decimal point.

C:\Users\user\Downloads\Decimal5.png

We say the above example as four hundred and fifty-six and eight-tenths but

We usually just say four hundred and fifty-six point eight.

Types of Decimal Numbers:

Decimal Numbers may be of different kinds:

Recurring Decimal Numbers (Repeating or Non-Terminating Decimals)

Example-

3.125125 (Finite)

3.121212121212….. (Infinite)

&;

Non Recurring Decimal Numbers (Non Repeating or Terminating Decimals):

Example:

3.2376 (Finite)

3.137654….(Infinite)

&;

Decimal Fraction:

It represents the fraction whose denominator in powers of ten.

Example:

81.75 = 8175/100

32.425 = 32425/1000

Converting the Decimal Number into Decimal Fraction:

For the decimal point place a “1” in the denominator and remove the decimal point.

“1” is followed by a number of zeros equal to the number of digits following the decimal point.

For Example:

8  1 . 7  5

↓ ↓  ↓

     1 0 0

81.75 = 8175/100

&;

8 represents the power of 10^1 that is the tenths position.

1 represents the power of 10^0 that is the units position.

7 represents the power of 10^(-1) that is the one-tenths position.

5 represents the power of 10^(-2) that is the one-hundredths position.

So that is how each digit is represented by a particular power of ten in the decimal number.

&;

Place Value of Decimal Numbers:

The place value is obtained by multiplication of the digit in the decimal number with its power of ten that the digit holds at its position.

The power of ten can be found using the following Place Value Chart:

Decimal Number: Power of Tens

The digits to the left of the decimal point are multiplied with the positive powers of ten in an increasing order from right to left.

The digits to the right of the decimal point are multiplied with the negative powers of 10 in an increasing order from left to right.

Following the same example 81.75

The decimal expansion of this is :

{(8*10)+(1*1)} + {(7*0.1)+(5*0.01)}

Where each number is multiplied by its associated power of ten.

To learn more on decimals, division of decimals and operations of converting fractions to decimals, visit www.byjus.com.


Practise This Question

420 as a decimal is :