Algebra Chapter 2

Division of Integers

• Division is not commutative for integers, meaning ( a ÷ b \neq b ÷ a ).
• Example: ( (–8) ÷ (–4) = 2 ) while ( (–4) ÷ (–8) \neq 2 ).
• Division of integers can yield different types of results:
  • ( a ÷ 0 ) is undefined; ( 0 ÷ b = 0 ) for any ( b \neq 0 ).

Division by One

• Dividing any integer by ( 1 ) results in the same integer.
  • Example: ( (–8) ÷ 1 = (–8) ).
• This rule holds true for negative integers as well.

Division by Negative One

• Dividing an integer by ( –1 ) changes the sign of the integer.
  • Example: ( (–1) × 5 = –5 ) implies ( 5 ÷ (–1) = (–5) ).

Inverse Relationship with Multiplication

• Division is the inverse operation of multiplication.
  • Example: For ( 3 × 5 = 15 ), it follows that ( 15 ÷ 5 = 3 ) and ( 15 ÷ 3 = 5 ).
• Each multiplication statement corresponds to two division statements.

Observations on Division Results

• When dividing a negative integer by a positive integer, the quotient is negative.
  • Example: ( (–32) ÷ 4 = (–8) ).
• Conversely, dividing a positive integer by a negative integer also results in a negative quotient.

Negative Divisions

• Dividing two negative integers results in a positive quotient.
  • Example: ( (–12) ÷ (–6) = 2 ).

General Division Rules

• For two positive integers ( a ) and ( b ):
  • ( a ÷ (–b) = (–a) ÷ b ) for ( b \neq 0 ).
  • ( (–a) ÷ (–b) = a ÷ b ) for ( b \neq 0 ).

Non-closure Property

• Integers are not closed under division, as not every division operation yields another integer.

Test your knowledge of division with negative numbers in this algebra quiz. Analyze statements and inferences to determine the outcomes of various division operations. Perfect for students looking to reinforce their understanding of integer results.