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Number Systems - Representing Real Numbers on the Number Line

Grade 9CBSE

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

🔑Concepts

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Every real number is represented by a unique point on the number line, and every point on the number line represents a unique real number. This 1:1 correspondence is why the number line is referred to as the Real Number Line.

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The process of 'Successive Magnification' is used to visualize the position of a real number with a terminating or non-terminating recurring decimal expansion. This involves looking through a conceptual magnifying glass to divide an interval of length 11 into 1010 equal parts, and then further dividing the relevant sub-interval into 1010 smaller parts repeatedly until the desired decimal place is reached.

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Irrational numbers like 2\sqrt{2}, 3\sqrt{3}, or n\sqrt{n} (where nn is a positive integer) can be located using the Pythagoras Theorem. Visually, this is done by constructing a right-angled triangle on the number line where the hypotenuse length equals n\sqrt{n}. For example, to find 2\sqrt{2}, construct a square of side 11 unit starting from the origin; the diagonal of this square is 2\sqrt{2}.

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To represent x\sqrt{x} for any positive real number xx geometrically, we follow a specific construction: Mark a point AA and a point BB such that AB=xAB = x units. Extend ABAB to CC such that BC=1BC = 1 unit. Find the midpoint OO of ACAC and draw a semicircle with radius OCOC. Draw a perpendicular line to ACAC passing through BB and intersecting the semicircle at DD. The distance BD=xBD = \sqrt{x}.

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The square root of a positive real number xx exists for all x>0x > 0. On the number line, this represents the distance of the point x\sqrt{x} from the origin 00. If we treat BB as the origin in the geometric construction, the point DD can be swung down to the number line using a compass to mark the position of x\sqrt{x}.

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Rational numbers have decimal expansions that are either terminating (like 0.250.25) or non-terminating recurring (like 0.333...0.333...). These can be located precisely or approximated to several decimal places using successive magnification steps on the number line.

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Irrational numbers have decimal expansions that are non-terminating and non-recurring (like π\pi or 2\sqrt{2}). While we can approximate their position using magnification, geometric construction using the Pythagoras Theorem provides their exact location on the real number line.

📐Formulae

Pythagoras Theorem: c2=a2+b2c^2 = a^2 + b^2 or c=a2+b2c = \sqrt{a^2 + b^2}

Locating n\sqrt{n}: n=(n−1)2+12\sqrt{n} = \sqrt{(\sqrt{n-1})^2 + 1^2}

Radius for geometric x\sqrt{x} construction: r=x+12r = \frac{x + 1}{2}

Distance of point BB from center OO in x\sqrt{x} construction: OB=x−x+12=x−12OB = x - \frac{x + 1}{2} = \frac{x - 1}{2}

💡Examples

Problem 1:

Represent 2\sqrt{2} on the number line.

Solution:

  1. Take a number line and mark point OO as 00 and point AA as 11. Thus, OA=1OA = 1 unit.
  2. Draw a perpendicular ABAB at AA such that AB=1AB = 1 unit.
  3. Join OBOB. In right-angled ΔOAB\Delta OAB, by Pythagoras Theorem: OB=OA2+AB2=12+12=2OB = \sqrt{OA^2 + AB^2} = \sqrt{1^2 + 1^2} = \sqrt{2}
  4. Using a compass with center OO and radius OBOB, draw an arc that intersects the number line at point PP.
  5. The point PP on the number line represents 2\sqrt{2}.

Explanation:

This approach uses the Pythagoras theorem to create a line segment of length 2\sqrt{2} and then transfers that length onto the horizontal number line using a compass.

Problem 2:

Visualize the representation of 3.7653.765 on the number line using successive magnification.

Solution:

  1. 3.7653.765 lies between 33 and 44. Divide the segment between 33 and 44 into 1010 equal parts.
  2. 3.7653.765 lies between 3.73.7 and 3.83.8. Focus on this interval and divide it into 1010 equal parts.
  3. Now, 3.7653.765 lies between 3.763.76 and 3.773.77. Focus on this interval and divide it into 1010 equal parts.
  4. The fifth division point in this sub-interval represents 3.7653.765.

Explanation:

Successive magnification zooms into the number line step-by-step. Each step increases the precision by one decimal place until the specific point 3.7653.765 is located.