Representation of Irrational Numbers on Number Line MCQs Quiz | Class 9

This quiz covers Class IX Mathematics (Code 041), Unit I: Number Systems. It focuses on the Representation of Irrational Numbers on the Number Line, specifically testing concepts related to geometrical construction, the square root spiral, and the unique point representation of real numbers. Answer the questions below and click submit to review your score and download the PDF solution.

Understanding Representation of Irrational Numbers

In the Class 9 Mathematics curriculum under Number Systems, a key skill is learning how to locate irrational numbers on a number line. Unlike rational numbers, which are easily placed using integers or fractions, irrational numbers require geometric construction based on the Pythagorean Theorem.

1. The Concept of Unique Point Representation

The fundamental axiom of the real number system states that every real number is represented by a unique point on the number line. Conversely, every point on the number line represents a unique real number. This includes both rational and irrational numbers.

2. Locating Square Roots of Integers

To locate numbers like sqrt(2), sqrt(3), or sqrt(5), we use right-angled triangles.

  • For sqrt(2): Construct a right triangle with base 1 unit and height 1 unit. The hypotenuse will be sqrt(2).
  • For sqrt(3): Build upon the hypotenuse of the sqrt(2) triangle. Use sqrt(2) as the base and 1 unit as the height. The new hypotenuse is sqrt(3).
  • Square Root Spiral: Continuing this process creates a spiral shape, allowing us to find sqrt(n) for various integers.

3. Locating Square Roots of Decimals (Geometric Proof)

To find the value of sqrt(x) for any positive real number x (like 9.3) geometrically:

  1. Draw a line segment AB = x units.
  2. Extend AB to C such that BC = 1 unit.
  3. Find the midpoint O of AC.
  4. Draw a semi-circle with center O and radius OA.
  5. Draw a perpendicular at B intersecting the semi-circle at D.
  6. The length BD is equal to sqrt(x). Using B as center and radius BD, an arc can be drawn to cut the number line at the required point.

Extra Practice Questions

  • 1. Construct the square root spiral up to sqrt(5).
  • 2. Locate sqrt(10) on the number line using a right triangle with sides 3 and 1.
  • 3. Verify geometrically that sqrt(9.3) is approximately 3.05.
  • 4. Show that the point corresponding to pi exists on the real number line.
  • 5. Draw a line segment of length sqrt(4.5) using the semi-circle method.