Rational Numbers and Algebra

Podcast

Play an AI-generated podcast conversation about this lesson

Questions and Answers

Match the following rational numbers with their equivalent forms:

= 10/10 0 = 10/10 -2 = -20/10 -5/6 = -20/24

Match the following statements with their explanations:

Rational numbers can be found between any two given rational numbers = Countless rational numbers can be found between two rational numbers -2 can be written as -20/10 = Conversion of rational numbers to equivalent forms 0 can be written as 0/10 = Conversion of rational numbers to equivalent forms Countless rational numbers can be found between -2 and 0 = Rational numbers can be found between any two given rational numbers

Match the following with their corresponding numerators:

= -10 0 = 0 -5/6 = -20 -2 = -20

Match the following rational numbers with their denominators:

<ul> <li>= 10 0 = 10 -5/6 = 24 -2 = 10</li> </ul> Signup and view all the answers

Match the following rational numbers with their equivalent forms:

-5/6 = -20/24 -2 = -20/10 0 = 0/10 <ul> <li>= -10/10</li> </ul> Signup and view all the answers

Match the following with their corresponding explanations:

Countless rational numbers can be found between -5/6 and 5/8 = Rational numbers can be found between any two given rational numbers -5/6 can be written as -20/24 = Conversion of rational numbers to equivalent forms 5/8 can be written as 15/24 = Conversion of rational numbers to equivalent forms -20/24 and 15/24 are equivalent forms = Conversion of rational numbers to equivalent forms Signup and view all the answers

Match the following with their corresponding properties:

Rational numbers = Can be found between any two given rational numbers -5/6 = Can be written in equivalent forms 0 = Can be written in equivalent forms <ul> <li>= Can be written in equivalent forms</li> </ul> Signup and view all the answers

Match the following with their corresponding characteristics:

Rational numbers = Can be written in the form a/b where a and b are integers -5/6 = Has a denominator of 6 0 = Has a numerator of 0 <ul> <li>= Has a numerator of -10</li> </ul> Signup and view all the answers

Match the following expressions with their simplified forms:

2 3 5 3 1 = 1/14 − 3 2 3 1 2 = 2/35 Signup and view all the answers

Match the following expressions with the properties used to simplify them:

2 3 5 3 1 = Commutativity − 3 2 3 1 2 = Distributivity Signup and view all the answers

Match the following numbers with their additive inverses:

8 = -8 -5 = 5 2 = -2 9 = -9 Signup and view all the answers

Match the following values of x with the verification of the equation -(-x) = x:

11 = True -13 = True 15 = False -17 = True Signup and view all the answers

Match the following numbers with their multiplicative inverses:

8 = 1/8 -5 = -1/5 2 = 1/2 19 = 1/19 Signup and view all the answers

Match the following expressions with their equivalent forms:

2 3 5 3 1 = 2 3/5 1 3/5 <ul> <li>3 2 3 1 2 = -3 2/5 1 3/5</li> </ul> Signup and view all the answers

Match the following equations with their solutions:

<ul> <li>3 2 3 1 2 = 2/35 2 3 5 3 1 = 1/14</li> </ul> Signup and view all the answers

Match the following expressions with their correct orders of operations:

2 3 5 3 1 = Multiplication first, then division <ul> <li>3 2 3 1 2 = Division first, then multiplication</li> </ul> Signup and view all the answers

Match the property with the correct operation for rational numbers:

Commutative = Multiplication Not commutative = Subtraction Signup and view all the answers

Match the operation with the correct statement for rational numbers:

Addition = a + b = b + a Subtraction = a - b ≠ b - a Multiplication = a × b = b × a Division = a ÷ b ≠ b ÷ a Signup and view all the answers

Match the expression with the correct calculation for rational numbers:

$\frac{-3}{8} + \frac{1}{7}$ = $\frac{1}{7} + \frac{-3}{8}$ $\frac{2}{5} - \frac{3}{4}$ = $\frac{3}{4} - \frac{2}{5}$ $\frac{-7}{6} × \frac{-4}{5}$ = $\frac{-4}{5} × \frac{-7}{6}$ $\frac{-5}{3} ÷ \frac{4}{7}$ = $\frac{4}{7} ÷ \frac{-5}{3}$ Signup and view all the answers

Match the operation with the correct characteristic for rational numbers:

Addition = Always commutative Subtraction = Never commutative Multiplication = Always commutative Division = Never commutative Signup and view all the answers

Match the statement with the correct operation for rational numbers:

a + b = b + a = Addition a - b ≠ b - a = Subtraction a × b = b × a = Multiplication a ÷ b ≠ b ÷ a = Division Signup and view all the answers

Match the operation with the correct example for rational numbers:

Addition = $\frac{-3}{8} + \frac{1}{7}$ Subtraction = $\frac{2}{5} - \frac{3}{4}$ Multiplication = $\frac{-7}{6} × \frac{-4}{5}$ Division = $\frac{-5}{3} ÷ \frac{4}{7}$ Signup and view all the answers

Match the operation with the correct reason for rational numbers:

Addition = Commutative property holds Subtraction = Commutative property does not hold Multiplication = Commutative property holds Division = Commutative property does not hold Signup and view all the answers

Match the operation with the correct result for rational numbers:

Addition = Equal result Subtraction = Not equal result Multiplication = Equal result Division = Not equal result Signup and view all the answers

Study Notes

Commutative Properties of Rational Numbers

Addition is commutative for rational numbers, meaning that the order of numbers does not change the result: a + b = b + a.
Subtraction is not commutative for rational numbers, meaning that the order of numbers changes the result: a - b ≠ b - a.
Multiplication is commutative for rational numbers, meaning that the order of numbers does not change the result: a × b = b × a.
Division is not commutative for rational numbers, meaning that the order of numbers changes the result: a ÷ b ≠ b ÷ a.

Rational Numbers Between Two Given Numbers

There are countless rational numbers between any two given rational numbers.
Example: Find three rational numbers between -2 and 0. Solution: Convert -2 and 0 to rational numbers with the same denominators, and then find rational numbers between them.
Example: Find ten rational numbers between -5/6 and 5/8. Solution: Convert -5/6 and 5/8 to rational numbers with the same denominators, and then find rational numbers between them.

Multiplication of Rational Numbers

The commutative property of multiplication can be used to simplify multiplication of rational numbers.
Example: Find the value of (2 -3 × (-1 × 3 - 3)) × (- 3/5).
Solution: Use the commutative property of multiplication to simplify the expression.

Exercise 1.1

Problems involve using properties of rational numbers to simplify expressions.
Examples: Find the value of expressions such as (-2 × 3/5 + 3/2 × -5/6) and (-3/7 × 2/5 - 1/2 × 3/7).
Problems also involve finding additive inverses and multiplicative inverses of rational numbers.