Arithmetic Operations – Definition, Properties

Basic Arithmetic Operations

1. Addition (+): Combines two numbers or values to get their sum. For example, 3+2=5.
2. Subtraction (-): Subtracts one number or value from another, yielding the difference. For instance, 7−4=3.
3. Multiplication (× or ∗): Multiplies two numbers or values to produce a product. For example, 4×3=12.
4. Division (/): Divides one number or value by another, resulting in a quotient. For example, 8/4=2.

Addition Definition

Addition is a basic arithmetic operation that combines two or more numbers or values to yield a sum. The operation is denoted by the symbol +. For instance, adding 3 and 4 gives 7, represented as 3+4=7. Addition is foundational in mathematics and everyday calculations.

1. Positive Integers: The sum of two positive integers is a positive integer. For example, 3+4=7.
2. Commutative: The order of addition does not affect the sum. For example, 3+4=4+3.
3. Associative: Grouping order does not change the sum when adding three or more numbers. For instance, (2+3)+4=2+(3+4).
4. Identity Element: Adding zero to any number does not change its value. For example, 5+0=5.
5. Negative Numbers: The sum of a positive integer and a negative integer depends on their magnitudes. For instance, 3+(−2)=1 and 2+(−5)=−3.

Subtraction Definition

Subtraction is an arithmetic operation that involves taking away one number from another to yield a difference. It is denoted by the symbol “-” (minus). For example, 7−3=4 shows how 3 is subtracted from 7 to produce

1. Positive Integers: Subtracting a smaller positive integer from a larger one yields a positive integer. For example, 5−3=2.
2. Order Matters: Subtraction is not commutative, meaning the order in which numbers are subtracted changes the result. For example, 5−3≠3−5.
3. Associative: Subtraction is not associative, meaning grouping order changes the result. For instance, (5−3)−1≠5−(3−1).
4. Negative Numbers: Subtracting a positive integer from a smaller positive integer yields a negative integer. For instance, 3−5=−2.
5. Identity Element: Subtracting zero from any number does not change its value. For example, 7−0=7.

Multiplication Definition

Multiplication is a basic arithmetic operation that combines two or more numbers or values to yield a product. It is denoted by the symbols ×× or ∗∗. For example, multiplying 4 by 3 gives 12, represented as 4×3=124×3=12.

1. Positive Integers: The product of two positive integers is a positive integer. For example, 3×4=123×4=12.
2. Commutative: The order of multiplication does not affect the product. For example, 3×4=4×33×4=4×3.
3. Associative: Grouping order does not change the product when multiplying three or more numbers. For instance, (2×3)×4=2×(3×4).
4. Identity Element: Multiplying any number by one does not change its value. For example, 5×1=5.
5. Multiplicative Zero: Multiplying any number by zero results in zero. For instance, 8×0=0.

Division Definition

Division is a basic arithmetic operation that splits a number into equal parts. It is denoted by the symbol // or a horizontal line. For example, dividing 10 by 2 yields a quotient of 5, represented as 10/2=5. Division is fundamental in mathematics and everyday calculations.

1. Non-Zero Divisor: Division by zero is undefined. For instance, 5/0 does not have a value.
2. Identity Element: Dividing any number by one does not change its value. For example, 8/1=8.
3. Zero Dividend: Dividing zero by any non-zero number results in zero. For instance, 0/5=0.
4. Division of Equals: Dividing a number by itself (except for zero) results in one. For example, 9/9=1.
5. Sign Rules:
   • Dividing two positive numbers or two negative numbers yields a positive quotient. For example, 12/4=3 and (−12)/(−4)=3.
   • Dividing a positive number by a negative number, or a negative number by a positive number, results in a negative quotient. For instance, 12/(−4)=−3 and (−12)/4=−3.

Basic Arithmetic Properties

1. Commutative Property:
   • Addition and multiplication are commutative, meaning the order of operands doesn’t affect the result.
   • For example, 3+4=4+3 and 2×5=5×2.
2. Associative Property:
   • Addition and multiplication are associative, meaning grouping order doesn’t change the result.
   • For example, (1+2)+3=1+(2+3) and (2×3)×4=2×(3×4).
3. Identity Property:
   • Addition and multiplication have identity elements.
   • The identity element for addition is zero, making 5+0=5.
   • The identity element for multiplication is one, making 7×1=7.
4. Distributive Property:
   • Multiplication distributes over addition.
   • For example, 2×(3+4)=2×3+2×4, yielding 2×7=6+8=14.
5. Inverse Property:
   • Addition and multiplication have inverses.
   • The additive inverse of 𝑎 is −𝑎, making 𝑎+(−𝑎)=0.
   • The multiplicative inverse of 𝑎 (if 𝑎≠0) is 1/𝑎1/a, making 𝑎×(1/𝑎)=1a×(1/a)=1.

If 𝐴=7 and 𝐵=3, calculate the following:

1. 𝐴+𝐵
2. 𝐴−𝐵
3. 𝐴×𝐵
4. 𝐴/𝐵

Answer:

1. Addition: 𝐴+𝐵=7+3=10
2. Subtraction: 𝐴−𝐵=7−3=4
3. Multiplication: 𝐴×𝐵=7×3=21
4. Division: 𝐴/𝐵=7/3≈2.33

Question 2:

Consider three numbers 𝐴=5, 𝐵=4, and 𝐶=3. Solve the following:

1. 𝐴+(𝐵×𝐶)
2. (𝐴+𝐵)×𝐶
3. 𝐴/𝐵+𝐶

Answer:

1. 𝐴+(𝐵×𝐶)=5+(4×3)=5+12=17
2. (𝐴+𝐵)×𝐶=(5+4)×3=9×3=27
3. 𝐴/𝐵+𝐶=5/4+3=1.25+3=4.25

Question 3:

Given two integers 𝐴=12 and 𝐵=8, find:

1. 𝐴÷𝐵
2. 𝐴%𝐵

Answer:

1. Quotient: 𝐴÷𝐵=12/8=1.5
2. Remainder: 𝐴%𝐵=12%8=4

Question 4:

Simplify the expression 𝐴⋅(𝐵+𝐶) for 𝐴=2A=2, 𝐵=3B=3, and 𝐶=5C=5.

Answer:

First, evaluate the addition inside the parentheses: 𝐵+𝐶=3+5=8

Next, multiply by 𝐴⋅(𝐵+𝐶)=2⋅8=16

FAQs

What are the four basic arithmetic operations?

The four basic arithmetic operations are:

1. Addition: Combining two or more numbers to yield a sum.
2. Subtraction: Subtracting one number from another to yield a difference.
3. Multiplication: Multiplying two numbers to yield a product.
4. Division: Dividing one number by another to yield a quotient.

What are the five arithmetic operators?

The five key arithmetic operators are:

1. Addition (+): Combines two numbers.
2. Subtraction (-): Subtracts one number from another.
3. *Multiplication (× or ): Multiplies two numbers.
4. Division (/): Divides one number by another.
5. Modulo (%): Returns the remainder after dividing one number by another.

What are the 6 mathematical operations?

The six key mathematical operations include:

1. Addition: Combining numbers.
2. Subtraction: Subtracting numbers.
3. Multiplication: Multiplying numbers.
4. Division: Dividing numbers.
5. Modulo: Finding the remainder after division.
6. Exponentiation: Raising a number to a power.

What are the basic rules of arithmetic?

The basic rules of arithmetic include:

1. Commutative: Order doesn’t affect addition or multiplication.
2. Associative: Grouping doesn’t change the result for addition or multiplication.
3. Identity Elements: 0 for addition, 1 for multiplication.
4. Distributive: Multiplication distributes over addition.

What is the golden rule for solving equations?

The golden rule for solving equations is to maintain balance on both sides of the equation. Any operation performed on one side must be mirrored on the other. This ensures the equation remains valid and allows for correct solutions.

What is an example of an arithmetic operation?

An example of an arithmetic operation is:

3+5=83+5=8

Here, two numbers are combined through addition to yield a sum. This demonstrates a simple yet fundamental operation used in arithmetic, applicable in various contexts.

What is the correct order of arithmetic?

The correct order of arithmetic operations is:

1. Parentheses: Resolve expressions within parentheses first.
2. Exponents: Evaluate powers and roots.
3. Multiplication and Division: From left to right.
4. Addition and Subtraction: From left to right.

This order is often remembered by the acronym PEMDAS.

Do you multiply or divide first?

Multiplication and division are of equal precedence in arithmetic. When evaluating an expression, you process these operations from left to right, performing whichever one appears first in the order of the expression