Addition and Subtraction of Integers

Stories, games, and talks with real things such as toys, pebbles, or counters, as well as virtual manipulatives such as interactive whiteboard objects and items that can be moved around on a computer screen, are used to establish the understanding of arithmetic in the early years of school. With practice, pupils will be able to mentally imagine and manipulate things to aid with calculations. 

The addition and subtraction methods we use have been developed from those employed on the abacus for centuries. Techniques for doing them without using an abacus were most likely devised by Hindu experts and brought to Europe by the Arabs.

Addition (usually denoted by the plus symbol +) and subtraction (usually denoted by the plus symbol -) are the basic arithmetic operations among four arithmetic operations. 

The term “addition” refers to the process of putting things together and the term “subtraction” refers to removal of things from a collection.

When you add and subtract two numbers together, you’re counting them as one larger number. In actual life, addition and subtraction occurs frequently.

History of addition:-

The earliest and most fundamental arithmetic operations are addition and subtraction. Mathematicians have known about it for about 6000 years. ‘Counting’ was sometimes thought to be an early form of addition and subtraction as marking on the barrels.

In the year 2000 B.C., Egyptians and Babylonians used addition for the first time. The symbol for subtraction was first used as a marking on barrels by German mathematicians. In the 1500s, it was adopted as an operational symbol. The addition and subtraction symbols were established in the 16th century, but before that, equations were expressed in words, which made solving problems extremely time-consuming.

Components of addition and subtraction:-

The addend, the equal sign, and the sum are the three parts of addition.

• The Addend

Furthermore, addends or summands are numbers or phrases that are added  together. The addends of the equation 10 + 6 = 16  are, for example, 10 and 6.

• The Sign of Equality

The equal sign denotes that the equation’s two parts are equivalent. The equal sign is represented by two short horizontal strokes in the addition statement 10 + 6 = 16.

• The Sum

The totals of the addends are the sum in the addition statement. For example, the sum of 10 + 7 = 17 ;   here 17 is the sum

The components of subtraction are minuend, subtrahend and the difference

In a subtraction sentence, the first number is called the minuend and the second number is called subtrahend. The difference is calculated by subtracting the subtrahend from the minuend.

Notation:-

Usually addition is written in infix notation, with the plus sign “+” between the terms and the subtraction is written with the minus sign “-” between the terms.An equals symbol is used to denote the result.

Few of the examples are as follows;

1 + 1 = 2

3 + 4 = 7

11 – 9 = 20

5 + 5 – 4 = 6

6.8 – 7.3 = -0.5

Basic properties of addition and subtraction:-

• Commutative Property of addition

Basically we can say that addition is commutative or changing the order of addends has no effect on the sum.

Example: 4 + 3 = 3 + 4 ; Here the sum of both left hand side and right hand side is the same i.e., 7 although the order is reversed.

The commutative property is not followed under subtraction because a-b is not equal to b-a for all a and b.

• Associative Property:

Just similar to commutative, addition is also associative i.e.,  changing the order in which the addends are grouped has no effect on the sum.

Example: (3 + 5) + 7 = 3 + (5 + 7) ;

              LHS =  (3 + 5) + 7 

                      = 8 + 7

                      = 15

              RHS = 3 + (5 + 7)

                      = 3 + 12

                      = 15

 Here the sum of both sides is the same i.e., equals to 15 although the groupings are different.

(N.B.- The parentheses indicate which part to solve first.)

Associative property is not followed under subtraction because (a-b)-c is not equal to

 a-(b-c) for all a,b and c.

• Identity Property:

According to the identity property of addition, the sum of 0 and any number equals that number. Here’s an illustration:

0 + 8 = 8

This is accurate since the definition of 0 is “no quantity,” so adding 0 to 8 has no effect on the quantity of 8

It should be in mind that, it doesn’t matter if the 0 appears before or after the number, according to the commutative property of addition. With the 0 after the number, here’s an example of the identity feature of addition:

8 + 0 = 8

• Additive Inverse:

A number’s opposite is its additive inverse. When you add a number to its additive inverse, the result is zero. The basic rule is to turn a positive number into a negative number and vice versa.

For example, we know that 

                            6 + (-6) = 0

That means we can say that -6 is the additive inverse of 6 and also 6 is the additive inverse of -6.

Note: Identity and inverse property for subtraction doesn’t exist.

Conclusion: –

The process of adding two or more integers to obtain a final result is known as an addition. Commutative, distributive, associative, and additive identity are the four main features of addition.