maths > decimals

Decimal Division

    what you'll learn...


Decimals are fractions with standard place values. In this page, the following for decimals are explained.

 •  Division in first principles -- splitting a quantity into a number of parts and measuring one part

 •  simplified procedure : Sign property of division and long division method for Decimals.


"Division" can be described as

dividend split into divisor number of parts.

11÷4= 2 quotient and 3 remainder. Considering the figure with 11 cars and 4 dotted-boxes.11 is split into 4 equal parts with each assigned 2 cars. There are 3 cars remaining. The remaining cars are not enough to assign equally to each dotted-box.

One part is counted as 2 and remaining amount is 3.

division is inverse of multiplication. And division of fractions are explained with that.

  =34×32 (multiply by inverse)


Decimals are fractions with standardized place-values.

Consider 0.6÷3=0.2.

By first principles, the dividend 0.6=610 is split into divisor 3 number of equal parts : 210;210;210. And one part is 210, or equals 0.2 in decimals.

Consider 0.6÷0.3=2".

By first principles, the dividend 0.6=610 is split into divisor 0.3=310 number of equal parts :

This is done in two steps:
first step: 610 is split into 110 (denominator of 310) times is 6010.
Second step: the result from the first step 6010 is split into 3 (numerator of 310) times is 2010

The result is 2010, or equals 2 in decimals.

aligned or opposed to direction

Decimals are directed numbers, that is decimals are either positive or negative.

0.2aligned in direction is "+0.2".

0.2opposed in direction is "-0.2".

Directed numbers, positive and negative, are explained as "aligned in direction" and "opposed in direction" respectively.

Consider 0.6÷(-0.3)=-2

By first principles, the dividend 0.6=610aligned in direction is split into the divisor 0.3=310opposed in direction number of equal parts :

This is done in two steps,
first step: 610aligned in direction is split into 110aligned in direction (denominator of 310) times is 6010aligned in direction.
Second step: the result from the first step 6010aligned in direction is split into 3 (numerator of 310)opposed in direction times is 2010opposed in direction.

The result is 2010opposed in direction, or equals -2 in decimal number form.

decimal division

Decimal Division by first principle : Decimal division is splitting the dividend, into divisor number of parts with sign of the numbers (direction) handled appropriately.

decimal division

In whole numbers, we have studied Division by Place-value as illustrated in the figure. This procedure is used in decimals in a later step.

In Integers, we have studied Sign-property of Division.
 •  +ve ÷ +ve = +ve
 •  +ve ÷ -ve = -ve
 •  -ve ÷ +ve = -ve
 •  -ve ÷ -ve = +ve
This is applicable to decimals.

In Fractions, we have studied that division is inverse of multiplication.
For example, to divide 45÷32, it is modified to 45×23

Decimals are divided keeping in mind the place-value representation, which is a form of fractions.
For example, to divide 1.8÷.06, it is equivalently thought as 1810÷6100 and modified to multiplication 1810×1006.


Consider division of 0.002÷0.05

This is equivalently 21000÷5100

Understanding the above, a simplified procedure to divide the decimals is devised.

The decimal point of dividend and divisor are removed and the numbers are divided as integers.
eg: 0.002 is modified to the integer form 2.
0.05 is modified to the integer form 5.

The number of decimal-places in the dividend and divisor are counted.
eg: 0.002 has 3 decimal-places.
0.05 has 2 decimal-places.

Now the integer forms are divided.
eg: 2÷5=0.4

The number of decimal places of divisor is subtracted from dividend number of decimal places.
eg: 3-2=1.

The quotient form is modified to have the number of decimal points give by the difference above.
eg: 0.4 is modified to 0.04, that is decimal point moved to the left by 1.


Consider 1.55÷.005=310".


Difference in decimal places is 2-3=-1

So the product decimal place moves 1 place to the right.


Decimal Division -- Simplified Procedure :
The signs (+ve / -ve) are handled as in Sign-property of Integer Division
 •  +ve ÷ +ve = +ve
 •  +ve ÷ -ve = -ve
 •  -ve ÷ +ve = -ve
 •  -ve ÷ -ve = +ve

The decimal places are removed and the division is carried out as per Whole number Division by Place Value.

 •  The decimal-point is modified in the result
 •  The decimal-point is moved to the left the number of times equal to the difference number of decimals in dividend minus number of decimals in divisor. A positive difference moves the decimal point to the left, and a negative difference moves the decimal point to the right.


The outline of material to learn "decimals" is as follows.

Note: goto detailed outline of Decimals

•   Decimals - Introduction

    →   Decimals as Standard form of Fractions

    →   Expanded form of Decimals

•   Decimals - Conversion

    →   Conversion between decimals and fractions

    →   Repeating decimals

    →   Irrational Numbers

•   Decimals - Arithmetics

    →   Comparing decimals

    →   Addition & Subtraction

    →   Multiplication

    →   Division

•   Decimals - Expressions

    →   Expression Simplification

    →   PEMA / BOMA