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Number and Algebra - Number sets and properties (Natural, Integers, Rational, Real)

Grade 10IB

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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Natural Numbers (N\mathbb{N}): This set includes counting numbers starting from 0, represented as N={0,1,2,3,...}\mathbb{N} = \{0, 1, 2, 3, ...\}. Visually, these can be seen as discrete, equally spaced points on a number line beginning at zero and extending infinitely to the right.

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Integers (Z\mathbb{Z}): This set consists of all whole numbers, including positive numbers, negative numbers, and zero: Z={...,−3,−2,−1,0,1,2,3,...}\mathbb{Z} = \{..., -3, -2, -1, 0, 1, 2, 3, ...\}. On a number line, these are the distinct 'tick marks' extending infinitely in both directions from the origin.

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Rational Numbers (Q\mathbb{Q}): Any number that can be expressed as a fraction pq\frac{p}{q} where p,q∈Zp, q \in \mathbb{Z} and q≠0q \neq 0. This includes terminating decimals (like 0.750.75) and recurring decimals (like 0.3ˉ0.\bar{3}). Visually, these numbers fill many gaps between integers on the number line.

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Irrational Numbers: Numbers that cannot be expressed as a simple fraction and have non-terminating, non-recurring decimal expansions. Common examples include π\pi, ee, and 2\sqrt{2}. On a number line, these occupy the 'gaps' left by rational numbers that cannot be measured by ratios.

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Real Numbers (R\mathbb{R}): The union of the sets of Rational and Irrational numbers. The set R\mathbb{R} represents every possible point on a continuous, unbroken number line, often called the 'Real Line'.

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Set Hierarchy and Subsets: The number sets are nested within each other. This is visually represented by a Venn Diagram where the smallest circle N\mathbb{N} is inside Z\mathbb{Z}, which is inside Q\mathbb{Q}, and all are contained within the largest set R\mathbb{R}. Symbolically: N⊂Z⊂Q⊂R\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R}.

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Absolute Value: The distance of a number from zero on the number line, denoted by ∣x∣|x|. Since distance is always non-negative, ∣−5∣=5|-5| = 5 and ∣5∣=5|5| = 5. Visually, it measures the magnitude regardless of the direction from the origin.

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Scientific Notation: A way of expressing very large or very small numbers in the form a×10ka \times 10^k, where 1≤∣a∣<101 \le |a| < 10 and kk is an integer. Visually, a positive kk moves the decimal to the right (large numbers), while a negative kk moves it to the left (small numbers).

📐Formulae

Q={pq:p,q∈Z,q≠0}\mathbb{Q} = \{ \frac{p}{q} : p, q \in \mathbb{Z}, q \neq 0 \}

a×10k where 1≤∣a∣<10,k∈Za \times 10^k \text{ where } 1 \le |a| < 10, k \in \mathbb{Z}

∣x∣=x2|x| = \sqrt{x^2}

Percentage Error=∣va−veve∣×100%\text{Percentage Error} = |\frac{v_a - v_e}{v_e}| \times 100\%

Interval notation: [a,b]⇒a≤x≤b\text{Interval notation: } [a, b] \Rightarrow a \le x \le b

💡Examples

Problem 1:

Convert the recurring decimal x=0.727272...x = 0.727272... into a fraction in its simplest form.

Solution:

Step 1: Let x=0.72‾x = 0.\overline{72}. Step 2: Multiply by 100100 (since two digits repeat): 100x=72.7272...100x = 72.7272.... Step 3: Subtract the original equation from this new equation: 100x−x=72.7272...−0.7272...100x - x = 72.7272... - 0.7272... 99x=7299x = 72 Step 4: Solve for xx: x=7299x = \frac{72}{99}. Step 5: Simplify the fraction by dividing both numerator and denominator by 9: x=811x = \frac{8}{11}.

Explanation:

To convert a repeating decimal to a fraction, we multiply the decimal by a power of 10 that aligns the repeating parts, subtract to cancel the infinite decimal, and then solve for the variable.

Problem 2:

Classify the following numbers into the most specific set (N,Z,Q,or R\mathbb{N}, \mathbb{Z}, \mathbb{Q}, \text{or } \mathbb{R}): −5-5, 7\sqrt{7}, 34\frac{3}{4}, and 1212.

Solution:

  1. −5-5: It is a whole number but negative, so it belongs to Integers (Z\mathbb{Z}).
  2. 7\sqrt{7}: This is a non-terminating, non-recurring decimal, so it is an Irrational number, which is part of the Real Numbers (R\mathbb{R}).
  3. 34\frac{3}{4}: This is a ratio of two integers, so it is a Rational Number (Q\mathbb{Q}).
  4. 1212: This is a counting number, so it is a Natural Number (N\mathbb{N}).

Explanation:

The goal is to find the 'smallest' or most specific set that contains the number. Note that a Natural number is also an Integer and a Real number, but N\mathbb{N} is the most specific classification.