Exercise 1.3 — Successive Magnification

Representing decimal numbers on number line using successive magnification.

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What Exercise 1.3 Is About

Exercise 1.3 from Chapter 1 — Real Numbers focuses on two closely related skills: visualising decimal numbers with great precision on the number line using successive magnification, and constructing square roots geometrically using the square root spiral. Both techniques build a deeper, more visual understanding of real numbers — rational and irrational alike. This exercise is part of the Class 9 Mathematics syllabus for CBSE, Telangana, and Andhra Pradesh boards.

Successive Magnification — The Core Idea

Every decimal number corresponds to a unique point on the number line. Successive magnification is the method of zooming in, step by step, to locate that point with increasing accuracy. Each zoom divides the previous interval into ten equal parts. The number of zoom steps needed equals the number of decimal places in the given number.

Each zoom step: divide the current interval into 10 equal parts

Worked Example — Locating 2.874 (Question 1)

Since there are three decimal places, three rounds of magnification are needed. The diagram below shows all four levels — the whole number line, then three successive zooms closing in on 2.874.

Successive magnification to locate 2.874 on the number line Four number lines showing three levels of zoom: integers, tenths, hundredths, and thousandths, pinpointing 2.874 Zoom 1 — integers −1 0 1 2 3 4 Zoom 2 — tenths (2.0 – 2.9) 2.0 2.2 2.4 2.6 2.7 2.8 2.9 3.0 Zoom 3 — hundredths (2.80 – 2.89) 2.80 2.82 2.84 2.86 2.87 2.88 2.89 2.90 Zoom 4 — thousandths (2.870 – 2.879) 2.870 2.871 2.872 2.873 2.874 2.875 2.876 2.877 2.878 2.879 2.874
  • Zoom 1: 2.874 lies between 2 and 3 — zoom into [2, 3].
  • Zoom 2: The tenths digit is 8 — zoom into [2.8, 2.9].
  • Zoom 3: The hundredths digit is 7 — zoom into [2.87, 2.88].
  • Zoom 4: The thousandths digit is 4 — the point 2.874 is the 4th mark in [2.870, 2.880].
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Visualising a Recurring Decimal — Locating 5.2̄8̄ (Question 2)

Question 2 asks for 5.2̄8̄ = 5.282828… on the number line. Since the repeating block is "28", the first few decimal places are 5.282, which is where we zoom to.

Successive magnification to locate 5.282828… on the number line Three number lines zooming in to locate the recurring decimal 5.28-bar at 5.282 Zoom 1 — integers (3 to 7) 3 4 5 6 7 Zoom 2 — tenths (5.0 – 5.9) 5.0 5.1 5.2 5.4 5.6 5.8 6.0 Zoom 3 — hundredths (5.28 – 5.29) 5.280 5.281 5.282 5.283 5.284 5.285 5.286 5.287 5.288 5.289 5.290 5.2̄8̄ 5.2̄8̄ = 5.282828… ≈ 5.282 to 3 decimal places

The Square Root Spiral

The square root spiral builds √2, √3, √4, √5, and beyond geometrically — each new value is the hypotenuse of a right triangle with one leg of length 1. This proves that all these lengths are real and can be physically constructed, even the irrational ones.

Square root spiral (Wheel of Theodorus) showing √2 through √7 A geometric spiral where each successive right triangle with unit perpendicular leg produces the next square root value as its hypotenuse √2 √3 √4=2 √5 √6 1 1 O A Each new triangle adds a unit perpendicular leg — hypotenuse = next square root Triangle 1 → √2 (irrational) Triangle 2 → √3 (irrational) Triangle 3 → √4 = 2 (rational) Triangle 4 → √5 (irrational) Triangle 5 → √6 (irrational)

Notice that √4 = 2 is rational (4 is a perfect square) — the spiral passes through integer lengths exactly at each perfect square, confirming our rule from Exercise 1.2: √n is rational only when n is a perfect square.

Common Mistakes to Avoid

  • Zooming into the wrong interval — always check which tenth the digit falls in before moving to the next zoom level.
  • For a recurring decimal, find at least 3–4 decimal places before beginning magnification — you cannot zoom accurately from just the fraction form.
  • In the square root spiral, each new perpendicular leg must be exactly 1 unit and exactly perpendicular to the previous hypotenuse, not to the number line.
  • Mistaking √4 = 2 (rational) for √3 or √5 — only perfect square roots are rational.

What This Exercise Prepares You For

The visualisation skills built here strengthen your understanding of the entire real number line, which underlies Coordinate Geometry. The square root spiral connects back to Exercise 1.2 on Irrational Numbers and forward to surds and rationalisation later in Chapter 1. The number line work also supports all future graph-based topics in Class 9 and 10.

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