Learn on PengiThe Art of Problem Solving: Prealgebra (AMC 8)Chapter 1: Properties of Arithmetic

Lesson 5: Subtraction

In this Grade 4 AMC Math lesson from The Art of Problem Solving: Prealgebra, students learn to define subtraction as the addition of a negative (a - b = a + (-b)) and apply this definition to prove key properties such as self subtraction, subtracting zero, and subtraction of negation. The lesson also covers why subtraction is neither commutative nor associative, and how multiplication distributes over subtraction. Part of Chapter 1: Properties of Arithmetic, this lesson builds algebraic reasoning by converting subtraction problems into addition using opposites.

Section 1

Subtraction Property

Property

a−b=a+(−b)a - b = a + (-b)

Subtracting a number is the same as adding its opposite.

Examples

  • The expression 14−814 - 8 can be rewritten as adding the opposite: 14+(−8)14 + (-8), which both equal 66.

Section 2

Subtraction Properties Involving Zero

Property

For any number xx, the properties of subtraction involving zero are:

  • x−0=xx - 0 = x
  • x−x=0x - x = 0
  • 0−x=−x0 - x = -x

Examples

  • 7−0=77 - 0 = 7
  • −5−(−5)=0-5 - (-5) = 0
  • 0−9=−90 - 9 = -9

Explanation

These properties highlight the special role of zero in subtraction. Subtracting zero from any number does not change the number, which is similar to the identity property of addition. Subtracting a number from itself always results in zero, demonstrating the concept of an additive inverse. Finally, subtracting a number from zero yields its opposite or additive inverse.

Section 3

Subtraction of a Negative Number

Property

When subtracting a negative number, the result equals adding the positive version of that number:

x−(−y)=x+yx - (-y) = x + y

Examples

Lesson overview

Expand to review the lesson summary and core properties.

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Section 1

Subtraction Property

Property

a−b=a+(−b)a - b = a + (-b)

Subtracting a number is the same as adding its opposite.

Examples

  • The expression 14−814 - 8 can be rewritten as adding the opposite: 14+(−8)14 + (-8), which both equal 66.

Section 2

Subtraction Properties Involving Zero

Property

For any number xx, the properties of subtraction involving zero are:

  • x−0=xx - 0 = x
  • x−x=0x - x = 0
  • 0−x=−x0 - x = -x

Examples

  • 7−0=77 - 0 = 7
  • −5−(−5)=0-5 - (-5) = 0
  • 0−9=−90 - 9 = -9

Explanation

These properties highlight the special role of zero in subtraction. Subtracting zero from any number does not change the number, which is similar to the identity property of addition. Subtracting a number from itself always results in zero, demonstrating the concept of an additive inverse. Finally, subtracting a number from zero yields its opposite or additive inverse.

Section 3

Subtraction of a Negative Number

Property

When subtracting a negative number, the result equals adding the positive version of that number:

x−(−y)=x+yx - (-y) = x + y

Examples