A Short Note On Addition

Defence exams have a separate question paper for elementary mathematics. Given its importance, it is crucial to be able to solve arithmetic questions with ease. And to do that, we must have a stronghold on the basics. Addition is a very important basic operation that needs our attention if we are to get a hold of solving maths problems in the exam successfully. 

Addition is denoted by a plus + symbol. It is used not only used to add numbers but also, integers, real and complex numbers. Apart from arithmetics it is also used in algebra, to add vectors, subgroups, matrices and subspaces.

Adding numbers is a very simple task. As the difficulty level increases in mathematics, various other components also gets involved. For example, first, you will learn to add natural numbers. Then, we’ll discuss the complexity of adding positive and negative integers and finally, we’ll learn to add fractions or rational numbers. 

Properties of addition include: 

• Commutativity

An example of commutativity: 4 + 7 = 7 + 4

The commutative property essentially means that the same result is derived even after changing the order of terms. 

If x and y are numbers, then by the commutative property, x + y = y + x

• Associativity 

An example of associativity: 6 + (2 + 3) = (6 + 2) + 3

The result remains the same even after the order of operations is changed when three or more numbers are involved.

However, when addition is used with other arithmetic operations, the order matters. 

• Identity element 

An example of identity element: 7 + 0 = 0 + 7 = 7

0 is the additive identity because adding it to any number does not change the number. 

The law was first given by Brahamgupta in 628 AD

• Successor 

Successor example: 3 +1 = 4, 6 + 1 = 7, 5 + 1 = 6

For an integer x,  ( x + 1) is called the successor of x. 

• Units

If we are to add physical quantities having units, they must have common units to be added. For example, 300 millilitres can be added to 80 millilitres to form 380 millilitres. But, 5 feet, 3 inches cannot be added to 160 inches. The unit has to be the same for the addition to occur. 

In arithmetics, addition progresses from adding natural numbers to integers, rational numbers and real numbers. 

Adding natural numbers

To find 1 + 2 + 3 + 4 +…. up to n terms, the following formula is used.  This is set up in an arithmetic order

Sum of Natural Numbers: Sum = [n(n+1)]/2 

(here n is a natural number)

Adding integers

Adding two positive integers

2 + 5 = 7

You can get to positive integer seven by starting at positive integer two on a number line and moving five digits to the right. Because both numbers are positive, you can just maintain the positive sign and combine their absolute values to achieve the result, positive integer seven.

Addition of two negative integers.

 -3 + (-7 )=?

You can get to negative integer ten by starting at negative integer three on a number line and moving seven units to the left. Because both numbers are negative integers you can just maintain the minus sign and sum their absolute values to achieve the result, negative integer ten.

Adding  a positive to negative integer

-4 + 7 = ?

On the number line, if one starts at negative integer four and goes seven units to the right, you’ll arrive at the positive integer three.

Retain the sign of the integer with the highest absolute value. 

In other words, Subtract four from seven while keeping the positive sign, which gives positive integer three. 

Adding negative to a positive integer, follow these steps: 

4 + (-8)

On a number line, if one starts at positive four and goes eight points towards the left, they will arrive at negative four. Also, because the signs of these numbers differ, retain the sign of the one with the highest absolute value. In other words, subtract four from eight and retain the negative sign.

Adding rational numbers

It falls under two categories 

When the denominators of the numbers are same, the following formula applies: 

 (x/y + n/y) = (x+n) / y

When the denominators are not the same,

• The first step is to take the LCM of the denominators
• Multiply both the numerator and the denominator in such a way that the denominators remain the same, which is the calculated LCM
• Add the numerators 

For example, if we want to add 5 / 6 and 7 / 9, for which we can take the LCM of 6 and 9, which is 18.

Then, we multiply 6 with 3 to get the denominator 18. The same number is multiplied by 5 to give 15. Hence, 15 / 18.

We multiply 9 with 2 to give 18. Then we multiply 2 with 7 to give 14, hence: 14 / 18.

Now since denominators are the same, the first formula applies: (15 + 14) / 18 = 29 / 18.

Conclusion

Practice the addition of natural and rational numbers and integers using the formulas given. To excel in mathematics, whether it is arithmetics or algebra, one must master the basics of operations. With enough practice, you will be able to solve all problems about addition, successfully and it will ease studies of advanced arithmetics.