Algebra
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Boundless Algebra
Numbers and Operations
Introduction to Arithmetic Operations
Algebra Textbooks Boundless Algebra Numbers and Operations Introduction to Arithmetic Operations
Algebra Textbooks Boundless Algebra Numbers and Operations
Algebra Textbooks Boundless Algebra
Algebra Textbooks
Algebra
Concept Version 18
Created by Boundless

Basic Operations

The basic arithmetic operations for real numbers are addition, subtraction, multiplication, and division.

Learning Objective

  • Calculate the sum, difference, product, and quotient of positive whole numbers


Key Points

    • The basic arithmetic operations for real numbers are addition, subtraction, multiplication, and division.
    • The basic arithmetic properties are the commutative, associative, and distributive properties.

Terms

  • difference

    The result of subtracting one quantity from another.

  • sum

    The result of adding two quantities.

  • quotient

    The result of dividing one quantity by another.

  • product

    The result of multiplying two quantities.

  • commutative

    Referring to a binary operation in which changing the order of the operands does not change the result (e.g., addition and multiplication).

  • associative

    Referring to a mathematical operation that yields the same result regardless of the grouping of the elements.


Full Text

The Four Arithmetic Operations

Addition

Addition is the most basic operation of arithmetic. In its simplest form, addition combines two quantities into a single quantity, or sum. For example, say you have a group of 2 boxes and another group of 3 boxes. If you combine both groups together, you now have one group of 5 boxes. To represent this idea in mathematical terms:

$2+3=5$

Subtraction

Subtraction is the opposite of addition. Instead of adding quantities together, we are removing one quantity from another to find the difference between the two. Continuing the previous example, say you start with a group of 5 boxes. If you then remove 3 boxes from that group, you are left with 2 boxes. In mathematical terms:

$5-3=2$

Multiplication

Multiplication also combines multiple quantities into a single quantity, called the product. In fact, multiplication can be thought of as a consolidation of many additions. Specifically, the product of $x$ and $y$ is the result of $x$ added together $y$ times. For example, one way of counting four groups of two boxes is to add the groups together:

$2+2+2+2=8$

However, another way to count the boxes is to multiply the quantities:

$2 \cdot 4 = 8$

Note that both methods give you the same result—8—but in many cases, particularly when you have large quantities or many groups, multiplying can be much faster.

Division 

Division is the inverse of multiplication. Rather than multiplying quantities together to result in a larger value, you are splitting a quantity into a smaller value, called the quotient. Again, to return to the box example, splitting up a group of 8 boxes into 4 equal groups results in 4 groups of 2 boxes:

$8 \div 4 = 2$

The Basic Arithmetic Properties

Commutative Property

The commutative property describes equations in which the order of the numbers involved does not affect the result. Addition and multiplication are commutative operations:

  • $2+3=3+2=5$
  • $5 \cdot 2=2 \cdot 5=10$

Subtraction and division, however, are not commutative.

Associative Property

The associative property describes equations in which the grouping of the numbers involved does not affect the result. As with the commutative property, addition and multiplication are associative operations:

  • $(2+3)+6=2+(3+6)=11$
  • $(4 \cdot 1) \cdot 2=4 \cdot (1 \cdot 2)=8$

Once again, subtraction and division are not associative.

Distributive Property

The distributive property can be used when the sum of two quantities is then multiplied by a third quantity. 

  • $(2+4) \cdot 3 = 2 \cdot 3+4\cdot 3 = 18$
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