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IB MYP 4-5 Maths- Irrational numbers – Study Notes

IB MYP 4-5 Maths- Irrational numbers - Study Notes - New Syllabus

IB MYP 4-5 Maths- Irrational numbers – Study Notes

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  • Irrational numbers

    IB MYP 4-5 Maths- Irrational numbers – Study Notes – All topics

    Rational & Irrational Numbers

    Real Numbers:

    The set of real numbers includes all numbers that can be found on the number line. They are divided into two major groups: rational and irrational numbers.

    1. Rational Numbers: Numbers that can be written as fractions or ratios of two integers. This includes:

    • Terminating decimals: \( 4.5,\ 0.3 \)
    • Repeating decimals: \( 0.\overline{3},\ \frac{10}{11} \)
    • Integers: \( -3,\ -2,\ 0,\ 1,\ 2 \)
    • Fractions: \( \frac{27}{4},\ \frac{5}{9} \)

    Subsets within Rational Numbers:

    • Integers: Whole numbers and their negatives (e.g. \( -3,\ -1,\ 0,\ 2 \))
    • Whole Numbers: Non-negative integers (e.g. \( 0,\ 1,\ 2,\ 3 \))
    • Natural Numbers: Counting numbers (e.g. \( 1,\ 2,\ 3 \))

    2. Irrational Numbers: Numbers that cannot be written as fractions and have non-terminating, non-repeating decimal expansions:

    • Square roots of non-perfect squares: \( \sqrt{2},\ \sqrt{17},\ -\sqrt{11} \)
    • Famous constants: \( \pi,\ e \)

     Number Line Representation

    Rational Numbers 

    A rational number is any number that can be expressed as a fraction or ratio of two integers, where the denominator is not zero.

    \( \frac{a}{b} \), where \( a, b \in \mathbb{Z} \) and \( b \neq 0 \)

    Examples of Rational Numbers:

    • \( \frac{1}{2}, -\frac{4}{7}, 3, -10, 0 \)
    • \( 0.75 = \frac{3}{4},\quad -2.5 = -\frac{5}{2} \)
    • \( 0.333\ldots = \frac{1}{3} \) (recurring decimal)

    Important Notes:

    • All integers are rational numbers because \( a = \frac{a}{1} \).
    • Terminating and recurring decimals are rational.
    • Irrational numbers cannot be written as fractions (e.g. \( \pi, \sqrt{2} \)).

    Operations on Rational Numbers:

    • You can add, subtract, multiply, and divide rational numbers (except divide by zero).
    • Always convert mixed numbers to improper fractions before operations.

    Example:

    Determine whether the following numbers are rational:

    • \( \frac{7}{8} \)
    • \( 0.121212\ldots \)
    • \( \sqrt{9} \)
    • \( \pi \)
    • \( -4 \)
    ▶️ Answer/Explanation

     Check if each number can be written as a fraction

    \( \frac{7}{8} \): already a fraction → Rational 

    \( 0.121212\ldots = \frac{1212}{9999} \): recurring → Rational 

    \( \sqrt{9} = 3 \): integer → Rational 

    \( \pi \): non-repeating, non-terminating decimal → Irrational 

    \( -4 = \frac{-4}{1} \): integer → Rational 

    Example:

    Write the decimal \( 0.6\overline{3} \) as a fraction.

    ▶️ Answer/Explanation

     Let \( x = 0.6\overline{3} \)

    Then \( 10x = 6.3\overline{3} \)

    And \( 100x = 63.3\overline{3} \)

     Subtract

    \( 100x – 10x = 63.3\overline{3} – 6.3\overline{3} = 57 \)

    \( 90x = 57 \Rightarrow x = \frac{57}{90} = \frac{19}{30} \)

    Final Answer:

    \(\boxed{\frac{19}{30}}\)

    Irrational Numbers

    An irrational number is a number that cannot be written as a fraction \( \frac{a}{b} \), where \( a, b \in \mathbb{Z} \) and \( b \neq 0 \).

    Its decimal representation is non-terminating and non-repeating.

    Examples of Irrational Numbers:

    • \( \pi \approx 3.1415926\ldots \)
    • \( \sqrt{2} \approx 1.4142\ldots \)
    • \( \sqrt{3},\ \sqrt{5},\ \sqrt{7},\ \sqrt{11}, \ldots \)
    • \( \text{e} \approx 2.71828\ldots \) (Euler’s number)

    Important Notes:

    • If a square root cannot be simplified to a rational number, it is irrational.
    • Recurring or terminating decimals are not irrational — they are rational.
    • Irrational numbers fill the gaps on the number line between rational numbers.

    Visual Understanding:

    \( \sqrt{2} \) lies between 1 and 2. On a number line, it is not exactly representable as a fraction.

    Example:

    Determine whether each of the following is rational or irrational:

    • \( \sqrt{16} \)
    • \( \pi \)
    • \( 0.252525\ldots \)
    • \( \sqrt{7} \)
    ▶️ Answer/Explanation

    Evaluate or identify each number

    \( \sqrt{16} = 4 \Rightarrow \) Rational

    \( \pi \) is non-terminating, non-repeating → Irrational

    \( 0.252525\ldots = \frac{25}{99} \Rightarrow \) Rational

    \( \sqrt{7} \approx 2.645751\ldots \Rightarrow \) Irrational

    Example:

    Arrange the following numbers in ascending order:

    \( \sqrt{2},\ \frac{7}{5},\ \pi,\ 1.5 \)

    ▶️ Answer/Explanation

    Convert all numbers to decimal approximations

    \( \sqrt{2} \approx 1.414 \)

    \( \frac{7}{5} = 1.4 \)

    \( \pi \approx 3.1416 \)

    \( 1.5 = 1.5 \)

     Arrange from smallest to largest

    \( \frac{7}{5} (= 1.4),\ \sqrt{2} (\approx 1.414),\ 1.5,\ \pi (\approx 3.1416) \)

    Example:

    Use proof by contradiction to show that \( \sqrt{2} \) is an irrational number.

    ▶️ Answer/Explanation

     Assume the opposite

    Assume \( \sqrt{2} \) is rational. Then \( \sqrt{2} = \frac{a}{b} \), where \( a \) and \( b \) are integers with no common factors, and \( b \neq 0 \).

    Square both sides

    \( 2 = \frac{a^2}{b^2} \Rightarrow a^2 = 2b^2 \)

    Analyze parity

    So \( a^2 \) is even → \( a \) must be even. Let \( a = 2k \)

    Then \( a^2 = 4k^2 \), so:

    \( 4k^2 = 2b^2 \Rightarrow 2k^2 = b^2 \)

    This means \( b^2 \) is also even → \( b \) is even.

    Contradiction

    If both \( a \) and \( b \) are even, they share a common factor of 2 – contradicting the assumption that \( \frac{a}{b} \) is in lowest terms.

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