Integers

Integers are "*directed-whole-numbers*". That is, "received an amount of 3" and "given an amount of 3" are two different numbers -- +3 (aligned to the chosen direction) and -3 (opposed to the direction) respectively.

Learn in this

• What are directed numbers?

• How to handle direction in numbers?

• Sign and Absolute values of directed numbers

• *First Principles* of Comparison, addition, subtraction, multiplication, and division : directed numbers.

• *Procedural Simplification by sign and place value*

• Numerical Expressions with Directed Numbers - Precedence order and Sequence

Directed Numbers

This page introduces integers as directed numbers. The positive numbers are aligned to the chosen direction and the negative numbers are opposed to the chosen direction.

Handling Direction in Integers

This page explains handing positive and negative numbers, specifically the meaning of negative of a positive number, and negative of a negative number.

Ordinal Property of Integers

This page explains the ordered sequence of integers with negative and positive numbers in number-line. Negative numbers are positioned to the left of 0.

...,-2, -1, 0, 1, 2, ...

Sign and Absolute Value of Integers

This page introduced finding sign of an integer and absolute value of an integer. The symbol to represent absolute value is also introduced.

Comparing Integers

In this page, comparing two integers to find one as larger, equal or smaller is explained.

The *comparison in first principles* is explained for directed numbers and it is extended to *comparison as ordered sequence*.

A *simplified procedure by sign and place-value of large numbers* is explained.

Predecessor and Successor (Integers)

It was earlier studied that the integers (both positive and negative numbers) are in an ordered sequence. In this page, the predecessor and successor of an integer is explained.

Largest or Smallest in Integers

In this page, finding largest or smallest number among three or more integers is explained.

Ascending and Descending Orders in Integers

In this page, arranging three or more integers in ascending or descending order is explained.

Integer Addition : First Principles

The first principles of addition is, combine two quantities and count or measure the combined quantity. This page explains the same for integers, which are directed whole numbers.

Integer Addition : Simplified Procedure

This page extends the addition in first principles into a simplified procedure for addition of integers, which is called *sign property of integer addition*.

Integer Subtraction : First Principles

The first principles of subtraction is, taking away a part of the quantity and counting or measuring the remaining quantity. This page explains the same for integers, which are directed whole numbers.

Integer Subtraction : Simplified Procedure

This page extends the subtraction in first principles into a simplified procedure for subtraction of integers, which is called *sign property of integer subtraction*.

The procedure is summarized as -- subtraction is addition of additive inverse.

Integer Multiplication : First Principles

The first principles of multiplication is, repeatedly combine a quantity a number of times and count or measure the combined quantity. This page explains the same for integers, which are directed whole numbers.

Integer Multiplication: Simplified Procedure

This page extends the multiplication in first principles into a simplified procedure for multiplication of integers, which is called *sign property of integer multiplication*.

Integer Division : First Principles

The first principles of division is, splitting a quantity into a number of parts and count or measure one part. This page explains the same for integers, which are directed whole numbers.

Integer Division : Simplified Procedure

This page extends the division in first principles into a simplified procedure for division of integers, which is called *sign property of integer division*.

Integers: Simplification of Expressions

In this page, handling of integers, both positive and negative numbers, in numerical expressions is explained.

The precedence order PEMDAS / BODMAS is explained.

For operations of same precedence order, the sequence of operation "simplification from left to right" is explained.

Integers : Precedence Order PEMA / BOMA

Redefining the precedence order in arithmetics with PEMA or BOMA. As part of integers, the revised version is in handling subtraction -- handle subtraction as inverse of addition.

eg: $x-y-x$ This cannot be simplified using PEMDAS / BODMAS, instead, it can be simplified as $x+(-y)+(-x)$ and applying properties of arithmetics, it is $(-y)+x+(-x)=-y$