Number Systems Ex 1.2: Irrational Numbers on the Number Line

What Are Irrational Numbers?

An irrational number is a number that cannot be expressed as a simple fraction p/q, where p and q are integers and q is not zero. When written as a decimal, an irrational number is non-terminating and non-repeating. This means the digits after the decimal point go on forever without any repeating pattern.

Think about the numbers we know:

• Rational Numbers: These are numbers like 5 (which is 5/1), -3/4, or 0.333... (which is 1/3). Their decimal forms either terminate (like 0.5) or repeat a pattern (like 0.333...).
• Irrational Numbers: Famous examples include √2 (approx. 1.4142135...), √3 (approx. 1.7320508...), and π (approx. 3.1415926...). Notice how there is no repeating block of digits in their decimal expansions.

Together, the collection of all rational numbers and all irrational numbers makes up the set of Real Numbers (ℝ). This means every single point on the number line corresponds to a unique real number, which can be either rational or irrational.

How to Represent √5 on the Number Line

1. Step 1: Use the Pythagorean Theorem — To find a length of √5, we use the theorem a² + b² = c². We need to find two numbers whose squares add up to 5. Let's choose 2 and 1, because 2² + 1² = 4 + 1 = 5. So, we will construct a right-angled triangle with base = 2 units and perpendicular = 1 unit. The hypotenuse will be √5.
2. Step 2: Draw the Base on the Number Line — Draw a standard number line. Mark point 'O' at 0. Mark a point 'A' at the coordinate 2. The length of the segment OA is 2 units.
3. Step 3: Draw the Perpendicular — At point A (at coordinate 2), use a protractor or set square to draw a line segment AB of length 1 unit, perpendicular to the number line.
4. Step 4: Form the Hypotenuse — Join the points O and B. The line segment OB is the hypotenuse of the right-angled triangle OAB. According to Pythagoras' theorem, the length of OB = √(OA² + AB²) = √(2² + 1²) = √5.
5. Step 5: Transfer the Length to the Number Line — Now, use a compass. Place the metal tip at O (the origin) and the pencil tip at B. The radius of your compass is now exactly √5 units. Without changing the compass width, draw an arc from B downwards until it cuts the number line. Let's call this intersection point 'P'. The point P on the number line represents the number √5.

Exam Tip: Mastering True/False Questions

Exercise 1.2 includes 'True or False' questions that test your understanding of number classifications. Always justify your answer with a clear reason or a counter-example.

Here’s how to think through them:

• "Every irrational number is a real number." This is TRUE. Why? Because the definition of Real Numbers is the set of all rational and irrational numbers combined.
• "Every point on the number line is of the form √m, where m is a natural number." This is FALSE. Why? Think about negative numbers. -2 is a point on the number line, but you cannot find a natural number 'm' such that √m = -2. The square root of a natural number is always positive.
• "Every real number is an irrational number." This is FALSE. Why? Provide a counter-example. The number 5 is a real number, but it is rational (5/1), not irrational. This statement incorrectly excludes all rational numbers from the set of real numbers.

Practice Questions with Solutions

• Q: State whether the following statement is true or false. Justify your answer: "Every integer is a real number." A: Step 1: Recall the definitions. Integers are the set {..., -3, -2, -1, 0, 1, 2, 3, ...}. Real numbers are the set of all rational and irrational numbers. Step 2: Check if integers fit into the definition of real numbers. Every integer 'z' can be written as a rational number z/1. Step 3: Since all rational numbers are real numbers, it follows that all integers are also real numbers. Final answer: True. Every integer is a real number because it can be expressed as a rational number (e.g., -5 = -5/1), and all rational numbers are part of the set of real numbers.
• Q: Are the square roots of all positive integers irrational? If not, give an example of the square root of a number that is a rational number. A: Step 1: Consider the question. It asks if √x is always irrational for any positive integer x. Step 2: Test some positive integers. Let's test x = 1, 2, 3, 4, 5... √1 = 1 (Rational) √2 = 1.414... (Irrational) √3 = 1.732... (Irrational) √4 = 2 (Rational) √9 = 3 (Rational) Step 3: We have found examples (√1, √4, √9) where the square root of a positive integer is a rational number. These are the square roots of perfect squares. Final answer: No, the square roots of all positive integers are not irrational. For example, √4 = 2, which is a rational number.
• Q: Show how √3 can be represented on the number line. A: Step 1: To get √3, we first need to construct √2. We use Pythagoras' theorem with sides 1 and 1. On a number line, draw a base OA of 1 unit from O (origin). Draw a perpendicular AB of 1 unit at A. The hypotenuse OB will have length √(1² + 1²) = √2. Step 2: Now, use the length √2 as the new base. The segment OB on the number line has length √2. At point B, draw a perpendicular segment BC of length 1 unit. Step 3: Join O and C. Triangle OBC is a right-angled triangle. By Pythagoras' theorem, the hypotenuse OC = √(OB² + BC²) = √((√2)² + 1²) = √(2 + 1) = √3. Step 4: Place the compass point at O and the pencil point at C. Draw an arc that intersects the number line at a point P. This point P represents √3. Final answer: By constructing a right-angled triangle with sides √2 and 1, the resulting hypotenuse is √3, which can be transferred to the number line with a compass.
• Q: Represent √10 on the number line. A: Step 1: Use Pythagoras' theorem to find two numbers whose squares add up to 10. We can use 3 and 1, since 3² + 1² = 9 + 1 = 10. Step 2: Draw a number line. Mark the origin O at 0 and a point A at 3. The length OA is 3 units. Step 3: At point A, draw a perpendicular line segment AB of length 1 unit. Step 4: Join points O and B. The length of the hypotenuse OB is √(OA² + AB²) = √(3² + 1²) = √10. Step 5: Place the compass point at the origin O and the pencil at B. Draw an arc to intersect the positive number line at a point P. The point P represents the number √10. Final answer: By constructing a right-angled triangle with a base of 3 units and a height of 1 unit, the hypotenuse is √10. This length can be marked on the number line using a compass.

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