overview

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.

recap

"Division" can be described as

dividend split into divisor number of parts.

$11\xf74=$ $2$ 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.

$\frac{3}{4}\xf7\frac{2}{3}$

$=\frac{3}{4}\times \frac{3}{2}$ (multiply by inverse)

$=\frac{9}{8}$.

splitting

Decimals are fractions with standardized place-values.

Consider $0.6\xf73=0.2$.

By first principles, the dividend $0.6=\frac{6}{10}$ is split into divisor $3$ number of equal parts : $\frac{2}{10};\frac{2}{10};\frac{2}{10}$. And one part is $\frac{2}{10}$, or equals $0.2$ in decimals.

Consider $0.6\xf70.3=2$".

By first principles, the dividend $0.6=\frac{6}{10}$ is split into divisor $0.3=\frac{3}{10}$ number of equal parts :

This is done in two steps:

first step: $\frac{6}{10}$ is split into $\frac{1}{10}$ (denominator of $\frac{3}{10}$) times is $\frac{60}{10}$.

Second step: the result from the first step $\frac{60}{10}$ is split into $3$ (numerator of $\frac{3}{10}$) times is $\frac{20}{10}$

The result is $\frac{20}{10}$, or equals $2$ in decimals.

aligned or opposed to direction

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

$0.2$*aligned in direction* is "$+0.2$".

$0.2$*opposed in direction* is "$-0.2$".

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

Consider $0.6\xf7(-0.3)=-2$

By first principles, the dividend $0.6=\frac{6}{10}$*aligned in direction* is split into the divisor $0.3=\frac{3}{10}$*opposed in direction* number of equal parts :

This is done in two steps,

first step: $\frac{6}{10}$*aligned in direction* is split into $\frac{1}{10}$*aligned in direction* (denominator of $\frac{3}{10}$) times is $\frac{60}{10}$*aligned in direction*.

Second step: the result from the first step $\frac{60}{10}$*aligned in direction* is split into $3$ (numerator of $\frac{3}{10}$)*opposed in direction* times is $\frac{20}{10}$*opposed in direction*.

The result is $\frac{20}{10}$*opposed 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.

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 $\xf7$ +ve = +ve

• +ve $\xf7$ -ve = -ve

• -ve $\xf7$ +ve = -ve

• -ve $\xf7$ -ve = +ve

This is applicable to decimals.

In Fractions, we have studied that *division is inverse of multiplication*.

For example, to divide $\frac{4}{5}\xf7\frac{3}{2}$, it is modified to $\frac{4}{5}\times \frac{2}{3}$

Decimals are divided keeping in mind the place-value representation, which is a form of fractions.

For example, to divide $1.8\xf7.06$, it is equivalently thought as $\frac{18}{10}\xf7\frac{6}{100}$ and modified to multiplication $\frac{18}{10}\times \frac{100}{6}$.

simplify

Consider division of $0.002\xf70.05$

This is equivalently $\frac{2}{1000}\xf7\frac{5}{100}$

$=\frac{2}{1000}\times \frac{100}{5}$

$=\frac{2\times 100}{1000\times 5}$

$=\frac{2}{50}$

$=0.04$

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: $2\xf75=0.4$

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 $1$.

example

Consider $1.55\xf7.005=310$".

$155\xf75=31.$

Difference in decimal places is $2-3=-1$

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

$1.55\xf70.005=310$

summary

**Decimal Division -- Simplified Procedure** :

The signs (+ve / -ve) are handled as in *Sign-property of Integer Division*

• +ve $\xf7$ +ve = +ve

• +ve $\xf7$ -ve = -ve

• -ve $\xf7$ +ve = -ve

• -ve $\xf7$ -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.

Outline

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__