Q1. Given the following set of numbers, identify all the numbers that are: (a) natural numbers, (b) whole numbers, (c) integers, (d) rational numbers, (e) irrational numbers, (f) real numbers. { –13.3, –7, –2, –8/5, 0, 3, 7/22, π, 541 }

Background

Topic: Classification of Real Numbers

This question tests your understanding of the different subsets of real numbers and how to classify numbers based on their properties.

Key Terms:

• Natural numbers: Counting numbers starting from 1 (1, 2, 3, ...)

• Whole numbers: Natural numbers plus 0 (0, 1, 2, ...)

• Integers: Whole numbers and their negatives (..., -2, -1, 0, 1, 2, ...)

• Rational numbers: Numbers that can be written as a fraction of two integers (including terminating and repeating decimals)

• Irrational numbers: Numbers that cannot be written as a fraction (non-repeating, non-terminating decimals, e.g., π)

• Real numbers: All rational and irrational numbers

Step-by-Step Guidance

1. Examine each number in the set and determine if it fits the definition of each category above.

2. For natural numbers, look for positive integers greater than zero.

3. For whole numbers, include zero and all positive integers.

4. For integers, include all whole numbers and their negatives (no fractions or decimals).

5. For rational numbers, check if the number can be written as a fraction (including negative fractions and decimals that terminate or repeat).

6. For irrational numbers, identify numbers like π or any non-repeating, non-terminating decimals.

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Q2. Simplify the following rational expressions:

• a)

• b)

• c) Subtract from

• d) of

Background

Topic: Operations with Rational Numbers

This question tests your ability to add, subtract, multiply, and divide fractions.

Key Terms and Formulas:

• To add/subtract fractions: Find a common denominator.

• To multiply fractions: Multiply numerators and denominators.

• To divide fractions: Multiply by the reciprocal of the divisor.

Step-by-Step Guidance

1. For part (a), find the least common denominator (LCD) for 10 and 7, then rewrite each fraction with the LCD and subtract.

2. For part (b), rewrite the division as multiplication by the reciprocal: .

3. For part (c), rewrite both fractions with a common denominator, then subtract from .

4. For part (d), multiply by multiplying numerators and denominators.

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Q3. Use >, <, or = to make a true statement:

• a) ____

• b) ____

• c) ____

Background

Topic: Comparing Numbers and Absolute Value

This question tests your understanding of absolute value and comparing fractions.

Key Terms:

• Absolute value: The distance from zero on the number line, always positive.

• Comparing fractions: Find a common denominator or convert to decimals.

Step-by-Step Guidance

1. For part (a), calculate and and compare their values.

2. For part (b), calculate and and compare.

3. For part (c), compare and by finding a common denominator or converting to decimals.

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Q4. Evaluate:

• a)

• b) when and

• c) when

• d) when , ,

Background

Topic: Order of Operations and Substitution

This question tests your ability to use the order of operations (PEMDAS) and substitute values into expressions.

Key Terms:

• Order of operations: Parentheses, Exponents, Multiplication/Division, Addition/Subtraction

• Substitution: Replacing variables with given values

Step-by-Step Guidance

1. For part (a), follow the order of operations: division, multiplication, subtraction, addition.

2. For part (b), substitute and into the expression, then evaluate using order of operations.

3. For part (c), substitute and evaluate the expression.

4. For part (d), substitute , , into the expression, simplify each term, and add.

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