Extended Number

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Sample objectives

What this unit covers

• Sets: Comprehend and apply set terminology, notation, and Venn diagrams involving two or three sets. Use Venn diagrams to illustrate sets and express relationships between them. The following set notation will be utilized: • n(A) indicates the number of elements in set A • ‘… is an element of …’ • ‘… is not an element of …’ • A' denotes the complement of set A • ∅ represents the empty set • Universal set is denoted as U • A ⊆ B indicates that A is a subset of B • A ⊈ B indicates that A is not a subset of B • A ∪ B signifies the union of sets A and B • A ∩ B signifies the intersection of sets A and B. Example definitions of sets include: A = {x: x is a natural number}, B = {(x, y): y = mx + c}, C = {x: a ≤ x ≤ b}, D = {a, b, c, …}.
• Fractions, decimals and percentages: Candidates should be able to express fractions in their simplest form, including proper fractions, improper fractions, mixed numbers, decimals, and percentages. They must also recognize equivalence and convert between these forms, including converting recurring decimals to fractions and vice versa, such as expressing 0.17o as a fraction.
• Indices I: Understand and apply indices, including positive, zero, negative, and fractional values. For example, calculate values such as 2^6 = 64 and 4^-2 = 1/16. Additionally, utilize the rules of indices to solve expressions like 2^-3 × 2^4, (2^3)^2, and 2^3 ÷ 2^4.
• Standard form: Utilize the standard form A × 10, where n is any integer (positive or negative) and 1 ≤ A < 10. Convert numbers to and from standard form, and perform calculations using values expressed in standard form.
• Estimation: Round numbers to a specified level of precision, including both decimal places and significant figures. For example, express 5764 rounded to the nearest thousand. Make estimations for calculations that involve numbers, quantities, and measurements, such as estimating the value of 41.3 by rounding each number to 1 significant figure. Round results to an appropriate level of accuracy based on the context of the problem, like rounding 9.79 and 0.765.
• Limits of accuracy: Determine the upper and lower bounds for measurements that have been rounded, such as identifying the upper bound of a length measured to the nearest metre. Additionally, calculate the upper and lower bounds for results derived from calculations that involve rounded data, including finding the upper bound of the perimeter or area of a rectangle with dimensions rounded to the nearest centimetre, and determining the lower bound of speed based on rounded distance and time values.
• Percentages: 1 Determine a specified percentage of a given amount. 2 Represent one amount as a percentage of another. 3 Compute percentage increases or decreases. 4 Perform calculations involving simple and compound interest, which may include scenarios with repeated percentage changes. Note that formulas will not be provided. 5 Execute calculations using reverse percentages, such as finding the original price when given the selling price and the percentage profit. Percentage calculations may involve: deposits, discounts, profit and loss (expressed as either a value or a percentage), earnings, and percentages exceeding 100%.
• Exponential growth and decay: Understand and apply the concepts of exponential growth and decay, such as in scenarios involving depreciation or changes in population. Knowledge of the constant e is not necessary.
• Surds: Comprehend and apply the concept of surds, which involves simplifying expressions such as √20 = 2√5 and √(200 - 32) = 6√2. Additionally, rationalize denominators in expressions like √10 = 2√5/5 and 1/(1 + √3) = (1 - √3)/(2 - 1).