overview

The division in first principles was explained as splitting of integer dividend into integer divisor number of parts. In that, one part is the quotient and the part of dividend that could not be split is the remainder.
A simplified procedure "Sign-property of Integer Division" is explained to find the sign of the quotient and remainder, whereas the absolute values of the quotient and remainder are calculated by whole number division of absolute values of dividend and divisor.

watch the sign

The division $14\xf73$is understood as

$\text{received:}14$ is split into $3$ equal parts and one part is put-in (positive divisor).

The result of the division is
$14\xf73=$ quotient $4$ and remainder $2$

The division $(-14)\xf73$is understood as

$\text{given:}14$ is split into $3$ equal parts and one part is put-in (positive divisor).

The result of the division is
$(-14)\xf73=$ quotient $-4$ and remainder $-2$

The division $14\xf7(-3)$is understood as

$\text{received:}14$ is split into $3$ equal parts and one part is taken-away (negative divisor).

The result of the division is
$14\xf7(-3)=$ quotient $-4$ and remainder $2$

The division $(-14)\xf7(-3)$ is understood as

$\text{given:}14$ is split into $3$ equal parts and one part is taken-away (negative divisor).

The result of the division is
$(-14)\xf7(-3)=$ quotient $4$ and remainder $-2$

Summary of integer division illustrative examples:

• $14\xf73=4\text{(Q})\&2\text{(R})$ : $\text{received:}14$ is split into $3$ parts is quotient $\text{received:}4$ and remainder $\text{received:}2$.

• $(-14)\xf73=-4\text{(Q})\&-2\text{(R})$ : $\text{given:}14$ is split into $3$ parts is quotient $\text{given:}4$ and remainder $\text{given:}2$.

• $14\xf7(-3)=-4\text{(Q})\&2\text{(R})$ : $\text{received:}14$ is split into $-3$ parts is quotient $\text{given:}4$ and remainder $\text{received:}2$.

• $(-14)\xf7(-3)=4\text{(Q})\&-2\text{(R})$ : $\text{given:}14$ is split into $-3$ parts is quotient $\text{received:}4$ and remainder $\text{given:}2$.

Based on this, the division is simplified as

• +ve $\xf7$ +ve $=$ +ve with +ve remainder

• +ve $\xf7$ -ve $=$ -ve with +ve remainder

• -ve $\xf7$ +ve $=$ -ve with -ve remainder

• -ve $\xf7$ -ve $=$ +ve with -ve remainder

examples

Find the result of the division $22\xf7(-1)$

The answer is "$-22$"

Find the result of the division $0\xf7(-32)$

The answer is "both the above"

Find the result of the division $-180\xf7(-88)$

The answer is "quotient $2$ and remainder $-4$".

summary

**Integer Division -- Simplified Procedure** : The sign of the quotient and remainder are decided by the signs of dividend and divisor as:

**Sign-property of Integer Division**

• positive $\xf7$ positive $=$ positive with positive remainder

• positive $\xf7$ negative $=$ negative with positive remainder

• negative $\xf7$ positive $=$ negative with negative remainder

• negative $\xf7$ negative $=$ positive with negative remainder

Sign of the remainder is that of the dividend.

The absolute values of the quotient and remainder are calculated by whole number division of absolute values of dividend and divisor.

Outline

The outline of material to learn integers is as follows.

Note: * click here for detailed outline of Integers (directed numbers) *

→ __Introduction to Directed Numbers__

→ __Handling Direction__

→ __Ordinal Property__

→ __Sign and Absolute Value__

→ __Comparing Integers__

→ __Predecessor & Successor__

→ __Largest & Smallest__

→ __Ascending & Descending__

→ __Addition: First Principles__

→ __Addition: Simplified Procedure__

→ __Subtraction: First Principles__

→ __Subtraction: Simplified Procedure__

→ __Multiplication: First Principles__

→ __Multiplication: Simplified Procedure__

→ __Division: First Principles__

→ __Division: Simplfied Procedure__

→ __Numerical Expressions with Integers__

→ __PEMA / BOMA__