Chapter 1.4 — Exercise 1.3 — Successive Magnification

Locating 2.874 One Digit at a Time

Question 1 asks for 2.874 to be visualised this way. Since there are three digits after the decimal point, three zoom levels are needed on top of the initial whole-number line.

• 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 — 2.874 is the 4th mark inside [2.870, 2.880].

Each zoom only ever needs the next single digit of the decimal, read off in order. There's no need to know all three decimal digits of 2.874 before starting — the first zoom only cares that the whole-number part is 2, the second only cares that the tenths digit is 8, and so on. That's what makes the method work for numbers with far more decimal places than three: it never asks for more information than the current zoom level actually needs.

The Same Method on a Recurring Decimal

Question 2 raises the difficulty slightly by asking for 5.2̄8̄ = 5.282828…, a recurring rather than terminating decimal. Because the repeating block is "28", the first three decimal places are 5.282 — which is precisely where the zooming needs to converge. A recurring decimal never actually finishes, so in practice the magnification stops once enough digits have been resolved to answer the question — three or four zoom levels is normally sufficient, even though in principle the true value keeps demanding one more.

Building Square Roots With the Spiral of Theodorus

The second half of Exercise 1.3 turns to a geometric construction called the square root spiral. Starting from a right angle with both legs of length 1, the hypotenuse measures √2. Attach a new unit-length perpendicular leg to that hypotenuse, and the new hypotenuse measures √3. Repeating the process keeps producing the next square root, each one physically constructible with nothing more than a ruler and a set square. This construction is also known as the Wheel of Theodorus, after the Greek mathematician credited with it, and it's a direct visual proof that irrational lengths are not some abstract idea invented to complete the number system — they can be drawn, measured, and physically laid alongside rational lengths on the same page.

Notice that the spiral passes through an exact integer length only at √4 = 2, where 4 happens to be a perfect square — every other triangle in the sequence produces an irrational hypotenuse. That's the same rule from Exercise 1.2 showing up geometrically instead of as a decimal check.

It's also worth noticing what the spiral demonstrates about the Pythagorean theorem itself: every single triangle in the drawing is right-angled, and every hypotenuse is computed the same way — square the two legs, add them, take the square root. The spiral doesn't need a different rule for √5 than it used for √2; it's the identical theorem, applied five times in a row with a fresh triangle each time.

Precision Habits Worth Keeping

• Check which tenth (or hundredth, or thousandth) a digit falls into before zooming further — zooming into the wrong sub-interval throws off every later step.
• For a recurring decimal, expand at least 3–4 decimal places before magnifying — the fraction form alone doesn't tell you where to zoom.
• In the spiral, each new leg must be exactly length 1 and exactly perpendicular to the previous hypotenuse — perpendicular to the number line is a different (wrong) triangle.
• Don't mistake √4 = 2 for a nearby irrational value like √3 or √5 — only perfect-square roots break the pattern.
• When labelling a zoomed-in number line, keep the tick spacing visually even — an unevenly spaced diagram makes it easy to misjudge which mark a digit actually points to, even when the arithmetic underneath is correct.
• Successive magnification and the square root spiral answer two different kinds of question — where a known decimal sits, versus how to construct a length whose decimal isn't known yet — so reach for whichever one actually matches what the problem is asking.

Both techniques here — magnification and the spiral — are really the same idea in two forms: locating a real number exactly rather than approximately. Magnification does it digit by digit, algebraically, for a number whose decimal form is already known; the spiral does it construction by construction, geometrically, for a number whose decimal form would otherwise need long division to even estimate. Between the two, every real number met so far in this chapter — rational or irrational, terminating or endlessly non-repeating — has a concrete way to be pinned down on the line rather than left as an abstract symbol. Exercise 1.4 picks up right after, asking what happens when these precisely-located numbers are added, multiplied, or divided together. The same precision habit carries into Co-ordinate Geometry, where every plotted point depends on reading a number line correctly in two directions at once.