A Short Note On Subtraction

Elementary mathematics is a very crucial part of all defence exams and so it is very important to grasp the basics of arithmetics. The exam pattern is designed to test your problem-solving ability in mathematics which is the reason a separate paper in mathematics is allotted. Candidates cannot afford to skip mastering the basic arithmetic functions. Subtraction has different properties and formulas than other operations, which is elaborated upon in this article. 

Subtraction is denoted by using a minus ( – ) sign. Its properties and formulas are different from the other basic operations of arithmetics. One-digit numbers are easy to subtract, but for bigger numbers, we divide the numbers into sections based on their place values.  While solving such issues, we may come across some scenarios that need borrowing and others that do not. Subtracting with regrouping is another name for subtracting with borrowing. We apply the regrouping approach whenever the minuend is less than the subtrahend. To make the minuend greater than the subtrahend, we need to borrow one number from the previous section while regrouping.

Formula 

When subtracting two numbers, we employ some of the terms included in the expression:

• Subtrahend: The number that is subtracted from the other number.

• Minuend: The number from which the subtrahend is to be deducted.

• Subtracting the subtrahend from the minuend gives the final result, which is the difference 

• Minuend – Subtrahend = Difference 

Properties

Anti-commutativity

It is anti-commutative, which suggests that reversing the words in a difference from left to right gives the opposite result. If x and y are any two numbers then,  x – y = – (y – x)

Non-associativity

When attempting to define repeated subtraction, one must understand that it is non-associative. 

x-y-z can be expressed as  (x – y) – z or x – (y – z)  but the two choices provide different results. To answer this problem, one must first construct a sequence of procedures, with alternative sequences generating different outcomes.

Thus, it does not have an associativity property.

Predecessor 

Subtraction of one has a specific meaning in terms of integers: for every integer x,  (x – 1) is the greatest integer smaller than x, often known as the antecedent of x.

 Units

When subtracting two integers with different units of measurement, such as kilos or pounds, the units must match.  The unit of the result will be the same as the units of the original numbers being subtracted. 

Subtracting natural numbers

While subtracting 2 natural numbers, unlike addition, the first one must be bigger than that of the second. If the first number is smaller than the second, the result is not a natural number. 

You can, for example, do 20 – 5 = 15 (because the first number is bigger than the second one), but not: 5 – 20 (because the second number is less than the first one).

As a result, the subtraction somehow doesn’t meet the commutative property: the order of the terms in it cannot be changed. As a result, anytime we subtract, we must begin from the left side and work our way to the right.

Also, according to the non- associativity property, we can only subtract from left to right, and cannot subtract in any order, unlike addition. 

Example, if we have: 20 – 5 – 7 -1

We need to solve in following order: 

1. 20 – 5 = 15

2. 15 – 7 = 8

3. 8 – 1 = 7

It is not possible for us to subtract 5 -7 first and then others. 

Subtracting integers

Note: Integer in bold sign has higher numerical value.

subtracting Integers in Steps

Step 1: Convert the integer subtraction problem to an integer addition problem. Here’s how to do it:

• Keep the first number, for now,

• Second, switch to addition as the procedure.

• Mark the second number with the opposite sign

• Finally, do a standard addition of integers.

Here is a quick summary of how to add integers, which will be helpful in the final step.

1.  Adding two numbers having the same sign 

Keep the common sign after adding their absolute values.

2. Adding two different-sign integers

Take the sign of the integer with the bigger absolute value after subtracting their absolute values

Subtracting Rational numbers

When the denominator is the same simply, subtract the numerators and keep the denominator the same.

When the denominator is different,

 5 / 6 – 7 / 9

• Find the (L.C.M) of rational numbers’ denominators and write it as the denominator. LCM = 18

• Divide LCM by each rational number’s denominator. 18 / 6 = 3; 18 / 9 = 2  Multiply the answer you get with the numerator of the rational number in question. 5 x 3 = 15, 7 x 2 = 14

• use a negative sign to link the products in the numerator. 15 / 18 — 14 / 18 

• Finally, subtract the numerators to get the difference. 1/ 18

Conclusion

Given the nature of the defence exam, it is imperative to be adept at solving mathematics questions. The basic operations of arithmetics play a huge role in determining your success in it. Practice the formulas and rules given above. Only through practice, mastery can be developed over the concepts.