# Fundamental Operations in Mathematics | Four Basic Mathematical Operations for Simplifying Algebraic Terms

Wanna start your preparation from basics? Then here is the most basic and important concept. Learn all the Fundamental Operations and their usage in day to day life. Solve all the important problems involved in fundamental algebra operations. Know the definitions, rules, strategies, tips, tricks, and problems. Refer to the upcoming sections to get more information regarding mathematical expressions and operations.

## Fundamental Operations – Introduction

Fundamental Operations are the basic concept to solve any type of problem. We perform the operations at one time and generally, it starts from the left towards the right. If the expression contains more than 1 fundamental operation, then you can’t perform them in the order they appear in the question. There are certain rules to follow to perform various operations when more than 1 fundamental value is available. The precedence order has to be followed to solve the Fundamental Operation. The precedence order will be shown in the below sections.

### Order of Expression – DMAS Rule

The fundamental operations are expressed in an order. Generally, the order will be in the format of “DMAS” where “D” stands for “Division”, “M” stands for “Multiplication”, “A” stands for “Addition”, “S” stands for “Subtraction”. This order is considered and sequentially performed from left to right.

As there are fundamental operations in arithmetic, there are in a wat that three pairs of arithmetic operations, and for each pair of operations is the reversal of co-operation within the pairs. As addition and subtraction are reverse operations of each other. It is similar for other operations too. In regard to Algebra, everyone must understand that it is a set of rules based on fundamental operations which helps for faster calculations and an easy approach for variable problem-solving.

### Basic Fundamental Operations to Simplify Mathematical Expressions

The addition is defined by the operator (+). It is the fundamental operation of the arithmetic operators. It implies the combination of distinct quantities or sets. It involves counting numbers from one to one by incrementing the values. The result value after adding all the numbers is called the “sum”. The numbers and the initial number are called addends.

The addition of natural numbers involves 2 important properties i.e., associativity and commutativity.

Addition Operation on Positive and Negative Integers

Positive (+) + Positive (+) = Positive (+)
Negative (-) + Positive (+) = Negative (-)
Positive (+) + Negative (-) = Sign of the largest Number
Negative (-) + Positive (+) = Sign of the largest number

Examples:
5+4 = 9
(-5) + (-4) = -9
(-5) + 4 = -1
4 + (-5) = -1
(-4) + 5 = -1
5 + (-4) = 1

Example Problem:

An empty water tank of 8 feet high is available. A monkey is sitting at the bottom of that tank. The monkey tried jumping to the top of the tank. While jumping it jumps 3 ft up and slides down to 2 ft. How many jumps will the monkey take to reach the top of the water tank?

Solution:

As per the given question,

Monkey’s first jump = 3 ft up and 2 ft down

= (+3)-2=1

2nd Jump of the Monkey = 3 ft up and 2 ft down = (1+3) = 4

= (+4)-2=2

Therefore, the monkey covers 2 ft after the 2nd jump

3rd Jump of the Monkey = 3 ft up and 2 ft down = (2+3) = 5

=(+5)-2 = 3

Therefore, the monkey covers 3 ft after the 3rd jump

4th Jump of the Monkey = 3 ft up and 2 ft down = (3+3) = 6

=(+6)-2 = 4

Hence, the monkey covers 4 ft after the 4th jump

5th Jump of the Monkey = 3 ft up and 2 ft down = (4+3)=7

=(+7)-2 = 5

Therefore, the monkey covers 5 ft after the 5th jump.

6th Jump of the Monkey = 3 ft up = (5+3) = 8

As the water tank is 8 feet high, the monkey will reach the top of the water tank in the 6th jump.

#### Subtraction

The subtraction is defined by the operator (-). Subtraction is the inverse property of addition. It means that removing one quantity from another quantity. It involves incrementing down from the given number. The result value after subtracting all the numbers is called the “difference”. The number from which other number is subtracted is called subtrahend. The subtracted number is called minuend.

Subtraction Operation on Positive and Negative Integers

Negative (-) – Positive (+) = Negative (-)
Positive (+) – Negative (-) = Positive (+)
Negative (-) – Negative (-) = Sign of the largest Number
Negative (-) + Positive (+) = Sign of the largest number

Examples:

(-5)-4 = (-5) + (-4) =- 9
5 – (-4) = 5 +4 = 9
(-5) – (-4) = (-5) + 4 = -1
(-4) – (-5) = (-4) + 5 = 1

Example Problem:

At a height of 3000 feet above sea level, a plane is flying. At some time, there is a submarine that is exactly above and floating 700 feet below sea level. Calculate the vertical distance using the concept of subtraction of integers?

Solution:

Height of the plane that is flying = 3000 feet

Depth of the submarine = -700 feet ( It is negative, as it is below the sea level).

To calculate the vertical distance, we use the subtraction of integers operation.

3000-(-700) = 3000 + 700 = 3700 feet

Therefore, the vertical distance is 3700 feet.

#### Multiplication

The multiplication is defined by the operator (“x” o “*”). Multiplication is the repeated property of addition. It means that a particular number is being repeatedly added to itself several times. It involves adding the number with the same number. The result value after multiplying all the numbers is called the “product”. The number to which another number is multiplied is called multiplicand. The multiplied number is called a multiplier.

Multiplication Operation on Positive and Negative Integers

Negative (-) x Positive (+) = Negative (-)
Positive (+) x Negative (-) = Negative (-)
Negative (-) x Negative (-) = Negative (-)
Positive (+) + Positive (+) = Positive (+)

Examples:

5×4 = 12

(-3) x (-4) = 12

(-3) x 4 = – 12

3 x (-4) = -12

Example Problems:

The submarine descends 40 feet per minute from sea level. Find the relation of the submarine with the sea level 5 minutes after it starts descending?

Solution:

The submarine below sea level =40 feet

As it is below sea level, it is negative. Therefore, it is -40.

After 5 minutes, the submarine is at = -40 x 5 = -200 feet

Therefore, the final solution is -200 feet.

#### Division

The division is defined by the operator (÷). The division is the inverse property of multiplication. It means that a particular number or quantity is split into equal or the same parts. It involves the splitting of numbers in the same proportions. The result value after dividing the numbers is called the “quotient”. The original number is called “dividend”. The number which is used for dividing into groups is called “divisor”. The final result is “quotient”.

Division Operation on Positive and Negative Integers

Negative (-) ÷ Positive (+) = Negative (-)

Positive (+) ÷ Negative (-) = Negative (-)

Negative (-) ÷ Negative (-) = Positive (+)

Positive (+) ÷ Positive (+) = Positive (+)

Examples:

20÷4 = 5

(-20)÷(-4) = 5

(-20) ÷ 4 = -5

20÷(-4) = -5

Example Problem:

In a living room, there are 120 books in total and they are placed on 6 shelves. Consider that each shelf has an equal number of books, Calculate the number of books on each shelf?

Solution:

As per the given question,

The total no of books = 120

No of seats placed on shelves = 6

To determine the number of books on each shelf = 120/6 =20

Therefore, the number of books on each shelf = 20 books.

Thus the final solution is 20 books.