Rational Numbers Basics

Questions and Answers

What is the set of rational numbers denoted by, and what is the condition for q?

The set of rational numbers is denoted by Q, and the condition for q is q ≠ 0.

What is the formula for subtracting two rational numbers, and what is the key step in this process?

The formula is (p/q) - (r/s) = (ps - qr) / qs, and the key step is to subtract the numerators and keep the common denominator.

What is the formula for multiplying two rational numbers, and what is the key step in this process?

The formula is (p/q) × (r/s) = (pr) / (qs), and the key step is to multiply the numerators and denominators separately.

What are the three properties of rational numbers, and what do they describe?

The properties are the Commutative Property, the Associative Property, and the Distributive Property, which describe the behavior of rational numbers under addition, multiplication, and combination of operations. Signup and view all the answers

What is the formula for dividing two rational numbers, and what is the key step in this process?

The formula is (p/q) ÷ (r/s) = (p/q) × (s/r) = (ps) / (qr), and the key step is to invert the second number and then multiply. Signup and view all the answers

How do you simplify the expression (2/3) - (1/4) using the formula for subtracting rational numbers?

First, find the common denominator, which is 12. Then, rewrite the expression as (8/12) - (3/12) = (8 - 3) / 12 = 5/12. Signup and view all the answers

What is the result of multiplying (2/3) and (3/4) using the formula for multiplying rational numbers?

The result is (2/3) × (3/4) = (2 × 3) / (3 × 4) = 6/12 = 1/2. Signup and view all the answers

Study Notes

Rational Numbers

Definition

A rational number is a number that can be expressed as the quotient or fraction of two integers, i.e., p/q, where p and q are integers and q ≠ 0.
The set of rational numbers is denoted by Q.
Rational numbers can be represented in decimal form, where the decimal expansion either terminates or repeats.

Subtraction

To subtract two rational numbers, subtract the numerators and keep the common denominator.
Formula: (p/q) - (r/s) = (ps - qr) / qs

Multiplication

To multiply two rational numbers, multiply the numerators and denominators separately.
Formula: (p/q) × (r/s) = (pr) / (qs)

Properties

Commutative Property: The order of the numbers does not change the result of addition and multiplication.
Associative Property: The order in which we add or multiply numbers does not change the result.
Distributive Property: a(b + c) = ab + ac, where a, b, and c are rational numbers.

Division

To divide two rational numbers, invert the second number (i.e., flip the numerator and denominator) and then multiply.
Formula: (p/q) ÷ (r/s) = (p/q) × (s/r) = (ps) / (qr)

Rational Numbers

Definition

A rational number is a number that can be expressed as the quotient or fraction of two integers, p/q, where p and q are integers and q ≠ 0.
The set of rational numbers is denoted by Q.
Rational numbers can be represented in decimal form, where the decimal expansion either terminates or repeats.

Operations

Subtraction

To subtract two rational numbers, subtract the numerators and keep the common denominator.
Formula: (p/q) - (r/s) = (ps - qr) / qs

Multiplication

To multiply two rational numbers, multiply the numerators and denominators separately.
Formula: (p/q) × (r/s) = (pr) / (qs)

Properties

Commutative Property

The order of the numbers does not change the result of addition and multiplication.

Associative Property

The order in which we add or multiply numbers does not change the result.

Distributive Property

a(b + c) = ab + ac, where a, b, and c are rational numbers.

Division

To divide two rational numbers, invert the second number (i.e., flip the numerator and denominator) and then multiply.
Formula: (p/q) ÷ (r/s) = (p/q) × (s/r) = (ps) / (qr)