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Number - Number Sets (Natural, Integer, Rational, Irrational, and Real Numbers)

Grade 8IB

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

🔑Concepts

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Natural Numbers (N\mathbb{N}): These are counting numbers including zero, represented as N={0,1,2,3,...}\mathbb{N} = \{0, 1, 2, 3, ...\}. Some contexts define it starting from 11.

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Integers (Z\mathbb{Z}): These include all whole numbers, their negatives, and zero: Z={...,−3,−2,−1,0,1,2,3,...}\mathbb{Z} = \{..., -3, -2, -1, 0, 1, 2, 3, ...\}.

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Rational Numbers (Q\mathbb{Q}): Any number that can be expressed in the form ab\frac{a}{b}, where a,b∈Za, b \in \mathbb{Z} and b≠0b \neq 0. This includes terminating decimals (e.g., 0.250.25) and recurring decimals (e.g., 0.3˙0.\dot{3}).

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Irrational Numbers (I\mathbb{I} or Q′\mathbb{Q}'): Numbers that cannot be written as a simple fraction. Their decimal expansions are non-terminating and non-recurring. Examples include π\pi, ee, and n\sqrt{n} where nn is not a perfect square.

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Real Numbers (R\mathbb{R}): The set of all rational and irrational numbers combined. Any point on a continuous number line represents a real number.

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Subsets: The relationship between these sets is N⊂Z⊂Q⊂R\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R}. All natural numbers are integers, all integers are rational numbers, and all rational numbers are real numbers.

📐Formulae

Q={ab:a,b∈Z,b≠0}\mathbb{Q} = \{ \frac{a}{b} : a, b \in \mathbb{Z}, b \neq 0 \}

R=Q∪Q′\mathbb{R} = \mathbb{Q} \cup \mathbb{Q}'

If x=n and n is not a perfect square, then x∈Q′\text{If } x = \sqrt{n} \text{ and } n \text{ is not a perfect square, then } x \in \mathbb{Q}'

💡Examples

Problem 1:

Classify the following numbers into the smallest possible set among N,Z,Q,Q′:\mathbb{N}, \mathbb{Z}, \mathbb{Q}, \mathbb{Q}': a) −5-5, b) 2\sqrt{2}, c) 34\frac{3}{4}, d) 49\sqrt{49}

Solution:

a) −5∈Z-5 \in \mathbb{Z}, b) 2∈Q′\sqrt{2} \in \mathbb{Q}', c) 34∈Q\frac{3}{4} \in \mathbb{Q}, d) 49=7∈N\sqrt{49} = 7 \in \mathbb{N}

Explanation:

−5-5 is a negative whole number (Integer). 2\sqrt{2} is non-terminating/non-recurring (Irrational). 34\frac{3}{4} is a fraction (Rational). 49\sqrt{49} simplifies to 77, which is a counting number (Natural).

Problem 2:

Show that the recurring decimal 0.7˙0.\dot{7} is a rational number by expressing it as a fraction.

Solution:

Let x=0.777...x = 0.777... Then 10x=7.777...10x = 7.777... Subtracting the first from the second: 10x−x=7.777...−0.777...10x - x = 7.777... - 0.777... 9x=79x = 7 x=79x = \frac{7}{9}

Explanation:

Since 0.7˙0.\dot{7} can be written as the fraction 79\frac{7}{9} where both 77 and 99 are integers, it satisfies the definition of a rational number Q\mathbb{Q}.

Problem 3:

Identify if the sum of 3+23 + \sqrt{2} is rational or irrational.

Solution:

3∈Q3 \in \mathbb{Q} and 2∈Q′\sqrt{2} \in \mathbb{Q}'. The sum of a rational and an irrational number is always irrational. Therefore, 3+2∈Q′3 + \sqrt{2} \in \mathbb{Q}'.

Explanation:

Adding a terminating value to a non-terminating, non-recurring value results in a non-terminating, non-recurring decimal.