Relationships Among Mathematical Operations

Instructor: Yuanxin (Amy) Yang Alcocer

Amy has a master's degree in secondary education and has taught math at a public charter high school.

In this lesson, we'll learn how the four mathematical operations relate to each other, including addition, subtraction, multiplication, and division. We'll also look at and solve some examples.

The Four Mathematical Operations

No matter how simple or complex a math problem, it can be solved using one or more of the four mathematical operations - addition, subtraction, multiplication, and division. When teaching math to others, we must have a solid understanding of each of these operations. We must also understand how these operations relate to each other.

Let's briefly go over the four mathematical operations:

  • Addition uses the plus symbol (+) to show that two or more numbers are being combined or totaled.
  • Subtraction uses a dash or the subtraction symbol (-) to show that one number is being reduced by another number.
  • Multiplication uses an ''x'', an asterisk or a simple dot to indicate repeated addition.
  • Division uses a dash with a dot on both the top and bottom of the dash; it can also be denoted via a horizontal line or a slash. Division refers to the splitting of a number of items evenly into groups.

Addition and Subtraction

In terms of how these operations are related, let's start by talking about addition and subtraction. Because addition involves combining or putting together numbers and subtraction involves the reducing or taking away of numbers, addition and subtraction are known as opposite operations.

To reverse an addition operation, we subtract. For example, let's say we have $1 and we add $3. To reverse this situation, we subtract $3 dollars, which leaves us with what we started with: $1:

  • $1 + $3 = $4
  • $4 - $3 = $1

Another opposite pair of operations is that of multiplication and division. Multiplication involves the combining or putting together of groups, while division involves splitting a number up into groups. So, just as we can reverse an addition operation using subtraction and vice versa, we can reverse multiplication using division, and vice versa. For example:

  • 12 x 3 = 36
  • 36 / 3 = 12

Addition and Multiplication

In addition to relating to each other as opposite pairs of operations, the operations are related to each other in other ways.

Addition, for example, is related to multiplication in that multiplication is the repeated addition of one number. So, when we multiply 4 by 3, we're adding 4 three times, like this:

  • 4 x 3 = 4 + 4 + 4

Unfortunately, division isn't the same as repeated subtraction.

Multiplication and Exponentiation

Another operation related to multiplication is that of exponentiation, or when we repeat a multiplication operation. For example, an exponent of 4 means we multiply a given number four times. So if we're given 3 to the fourth power (3^4), we're repeatedly multiplying 3 four times (3 x 3 x 3 x 3). Exponents are denoted by either a caret or as a superscript to the right of the number being multiplied.

Examples of Operations

Becoming familiar with relationships among operations allows us to not only switch between different operations but also to spot different forms of the same mathematical expression. Let's take a look at a couple of examples:

Fractional Operation

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