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Rational and Irrational Numbers - Properties of rational and irrational numbers

Grade 9ICSE

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

🔑Concepts

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Rational Numbers: A rational number is any number that can be expressed in the form pq\frac{p}{q}, where pp and qq are integers and q≠0q \neq 0. On a number line, these are points that can be represented by terminating or recurring decimals, such as 12\frac{1}{2} (0.5) or 13\frac{1}{3} (0.3ˉ0.\bar{3}).

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Irrational Numbers: These are numbers that cannot be written in the form pq\frac{p}{q}. Their decimal expansions are non-terminating and non-recurring (e.g., 2=1.414213...\sqrt{2} = 1.414213...). On the number line, they occupy the spaces between rational numbers that cannot be captured by fractions.

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Real Numbers: The set of all rational and irrational numbers together forms the set of Real Numbers. Every point on a continuous number line represents a unique real number, creating a complete and unbroken visual representation of all possible values.

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Surds: A surd is an irrational root of a positive rational number. For example, 3\sqrt{3} is a surd because its value cannot be expressed as a exact fraction, whereas 4\sqrt{4} is not a surd because it equals 2. Visually, surds represent lengths of sides of shapes (like the diagonal of a unit square) that cannot be measured exactly with a rational scale.

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Properties of Operations: The sum, difference, product, or quotient of two rational numbers is always rational. However, the sum or product of a rational and an irrational number is always irrational (e.g., 2+32 + \sqrt{3} or 232\sqrt{3}). The sum or product of two irrational numbers can be either rational or irrational.

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Density Property: Between any two distinct rational numbers, there are infinitely many rational and irrational numbers. No matter how much you zoom into a section of the number line, you will always find more numbers within any given interval.

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Rationalizing the Denominator: This is the process of removing an irrational number (like a square root) from the denominator of a fraction. This is done by multiplying both the numerator and denominator by a suitable factor (the conjugate), such as multiplying 1a\frac{1}{\sqrt{a}} by aa\frac{\sqrt{a}}{\sqrt{a}} to get aa\frac{\sqrt{a}}{a}.

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Conjugates: For an expression like a+ba + \sqrt{b}, the conjugate is a−ba - \sqrt{b}. When these two are multiplied, the result is always a rational number (a2−ba^2 - b), which is the fundamental principle used to simplify complex irrational denominators.

📐Formulae

Rational Form: pq,q≠0\text{Rational Form: } \frac{p}{q}, q \neq 0

Difference of Squares: (a+b)(a−b)=a2−b\text{Difference of Squares: } (a + \sqrt{b})(a - \sqrt{b}) = a^2 - b

Product Rule: a×b=ab\text{Product Rule: } \sqrt{a} \times \sqrt{b} = \sqrt{ab}

Quotient Rule: ab=ab\text{Quotient Rule: } \frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}

Rational number between aextandb=a+b2\text{Rational number between } a ext{ and } b = \frac{a+b}{2}

Irrational number between aextandb=ab (if abextisnotaperfectsquare)\text{Irrational number between } a ext{ and } b = \sqrt{ab} \text{ (if } ab ext{ is not a perfect square)}

💡Examples

Problem 1:

Rationalize the denominator of 35−2\frac{3}{\sqrt{5} - \sqrt{2}}.

Solution:

Step 1: Identify the conjugate of the denominator 5−2\sqrt{5} - \sqrt{2}, which is 5+2\sqrt{5} + \sqrt{2}. Step 2: Multiply the numerator and denominator by this conjugate: 35−2×5+25+2\frac{3}{\sqrt{5} - \sqrt{2}} \times \frac{\sqrt{5} + \sqrt{2}}{\sqrt{5} + \sqrt{2}} Step 3: Apply the formula (a−b)(a+b)=a2−b2(a-b)(a+b) = a^2 - b^2 in the denominator: 3(5+2)(5)2−(2)2\frac{3(\sqrt{5} + \sqrt{2})}{(\sqrt{5})^2 - (\sqrt{2})^2} Step 4: Simplify the denominator: 3(5+2)5−2=3(5+2)3\frac{3(\sqrt{5} + \sqrt{2})}{5 - 2} = \frac{3(\sqrt{5} + \sqrt{2})}{3} Step 5: Cancel the common factor 3: 5+2\sqrt{5} + \sqrt{2}

Explanation:

To rationalize, we use the conjugate to create a difference of squares in the denominator, which effectively removes the radical signs.

Problem 2:

Find one rational and one irrational number between 2 and 3.

Solution:

Step 1: To find a rational number, calculate the arithmetic mean: Rational Number=2+32=2.5\text{Rational Number} = \frac{2 + 3}{2} = 2.5 Step 2: To find an irrational number, calculate the square root of their product: Irrational Number=2×3=6\text{Irrational Number} = \sqrt{2 \times 3} = \sqrt{6} Step 3: Check if 6\sqrt{6} is irrational. Since 6 is not a perfect square, 6≈2.449...\sqrt{6} \approx 2.449... is irrational.

Explanation:

The mean of two numbers is always between them and is rational if the inputs are rational. The square root of the product is a standard way to find an irrational number, provided the product is not a perfect square.