# GRE Arithmetic | Decimals

The value of a digit can be determined by using its digit, position of digit in the number and base of the given number system.

**Decimal number system** has base 10 as it uses 10 digits from 0 to 9 and the successive positions to the left of the decimal point are as: 10^{0}, 10^{1}, 10^{2}, 10^{3}, … and so on. Successive positions to the right of the decimal point are as: 10^{-1}, 10^{-2}, 10^{-3}, … and so on.

**Example:**

For example, the digits of the number 4321.1234 have following place values:

4321.1234 can be written as,

= (4 x 1000)+(3 x 100)+(2 x 10)+(1 x l) + (1 x 0.1)+(2 x 0.01)+(3 x 0.001)+(4 x 0.0001) = (4 x 10^{3})+(3 x 10^{2})+(2 x 10^{1})+(1 x 10^{0}) + (1 x 10^{-1})+(2 x 10^{-2})+(3 x 10^{-3})+(4 x 10^{-4}) = 4000 + 300 + 20 + 1 + 0.1 + 0.02 + 0.003 + 0.0004 = 4321.1234

Since each place value is a power of 10, every decimal can be converted to an integer divided by a power of 10. We can convert a fractional decimal number to its equivalent fraction with integers in numerator and denominator.

**Example-1:**

4.5 = 4 + 5/10 = 40/10 + 5/10 = 45/10

**Example-2:**

75.18 = 75 + 18/100 = 7500/100 + 18/100 = 7518/100

**Example-3:**

0.369 = 0 + 369/1000 = 369/1000

We can also convert an Integer in numerator and denominator to its equivalent decimal fraction by dividing numerator by denominator. The result will either **terminate** or **repeat** without end. Repeating part of a decimal can be indicate using **bar** over repeating digits.

**Example-4:**

3/4 = 0.75

**Example-5:**

134/20 = 6.7

**Example-6:**

`30/28 = 1.0714285`

**Note:**

- Every
**Rational number**can be expressed as a terminating or repeating decimal and converse is also true. **Irrational numbers**can not be expressed in numerator/denominatr form because these numbers neither have repeating fraction (i.e., √2 = 1.41421356237) nor has finite fraction (0.03033033303333…).