Multiplication and Division of Integers and Rational Numbers

Multiplication of Integers

When multiplying the positive integer by other integers, if the multiplier decreases by each time, the product decreases by accordingly:
From this multiplication, we observe the following rules:
  • A positive integer multiplied by another positive integer results in a positive product equal to the product of their absolute values.
  • A positive integer multiplied by a negative integer results in a negative product equal to the product of their absolute values.
Similarly, multiplying the negative integer by integers that decrease by results in products that increase by :
From this multiplication, we observe the following rules:
  • A negative integer multiplied by a positive integer results in a negative product equal to the product of their absolute values.
  • A negative integer multiplied by another negative integer results in a positive product equal to the product of their absolute values.

Multiplication of Rational Numbers

  1. The product of two numbers with the same sign is equal to the product of their absolute values with a positive sign .
  2. The product of two numbers with different signs is equal to the product of their absolute values with a negative sign .
  3. The product of any number and is always .

Commutative and Associative Properties of Multiplication

For two numbers and ,
The result remains the same regardless of the order of multiplication. This is called the commutative property of multiplication.
For three numbers , , and ,
The result remains the same regardless of which two numbers are multiplied first. This is called the associative property of multiplication.

Rules of Multiplication

For three numbers , , and :
  1. Commutative Property:
  2. Associative Property:
When multiplying three or more numbers, it is often convenient to rearrange the order of multiplication using the commutative and associative properties. For example:

Distributive Property

For numbers , , and :
This shows that multiplying a number by the sum of two numbers is equivalent to multiplying the number by each addend and then adding the results. This is called the distributive property of multiplication over addition.

Rule for Distributive Property

For three numbers , , and :

Division of Integers and Rational Numbers

Since , we can write .
This relationship between multiplication and division also holds for integers:
  • The quotient of two integers with the same sign is positive and equal to the quotient of their absolute values.
  • The quotient of two integers with different signs is negative and equal to the quotient of their absolute values.

Division of Rational Numbers

  1. The quotient of two numbers with the same sign is positive and equal to the quotient of their absolute values.
  2. The quotient of two numbers with different signs is negative and equal to the quotient of their absolute values.
  3. Dividing by a nonzero number always results in .

Reciprocal

Two numbers are reciprocals if their product is . Using reciprocals, division can be converted into multiplication. For example:

Mixed Operations

When performing calculations involving addition, subtraction, multiplication, and division:
  1. Compute powers (exponents) first.
  2. Calculate expressions within parentheses in the order of parentheses: (), {}, [].
  3. Perform multiplication and division.
  4. Perform addition and subtraction.