Mathematics · Classes 9-10 · Chapter 1

Infinite Decimal Fractions and Approximate Values.

Try writing out the decimal for √2 and ask when you can stop. You never can. Yet there is a clear rule for how many places to keep and what the last digit becomes, and exam questions lean on exactly that rule.

1. A decimal that never stops and never repeats

Some decimal fractions have infinitely many digits after the decimal point, and no digit or group of digits repeats in a cycle. These are called infinite decimal fractions. For example, 5.134248513942301... A repeated decimal never ends either, but its digits come round again and again. Here they do not.

Find √2 by division. Each time, double the root found so far, put a new digit on its right, and bring down zeros two at a time. You get 1, then 24 × 4 = 96, 281 × 1 = 281, 2824 × 4 = 11296, and on it goes. The procedure can run for ever without ending, so √2 = 1.4142135... is an infinite decimal fraction. Tap through it one step at a time.

√5 = 2.2360679... is neither finite nor repeating. It is an infinite non-repeating decimal, so it is irrational. Likewise √2 × √5 = √10 is an infinite decimal. But do not assume a √ sign always means infinite. Simplify first, then see what kind of decimal you get.

The value of an irrational number can be worked out to as many decimal places as you like. The question tells you how many it wants.

2. Value and approximate value are not the same

The value up to some number of decimal places is just those digits written exactly as they are. For the approximate value you must look at the next place. If that digit is 5, 6, 7, 8 or 9, add 1 to the last digit you kept. If it is 0, 1, 2, 3 or 4, the digit stays as it was, and then the value and the approximate value are the same. So always work out one place more than the approximation asks for.

Take 5.4325893... Up to 4 decimal places the value is 5.4325. The next digit is 8, so the approximate value is 5.4326. Up to 2 places the next digit is 2, so both are 5.43.

Example 21 uses 4.4623845... Drag the handle from 1 to 5 places and compare the value with the approximate value. The circled digit is the one that decides whether the last digit changes.

3. Approximate values of square roots

The division method runs like a machine. Pick a number; each press brings one more digit, with the value and the approximate value beside it.

Example 20: by division, √13 = 3.605551... We want the approximate value to 3 decimal places, so we look at 4 places: 3.6055. The fourth digit is 5, so the last 5 of 3.605 gets 1 added, and the approximate value is 3.606. Tap through the steps, and watch the digit that comes out as 0 along the way.

Now √29 = 5.3851... Up to 2 decimal places the value is 5.38. The next digit is 5, so the approximate value is 5.39.

Exercise questions ask for a square root to 4 decimal places and then its approximate value to 3 places. The rule is the same: work out 4 places, then look at the fourth digit to round at the third. For the root of a repeated decimal, first write its digits out in full, for example 5.13̇02̇ = 5.1302302...

The answers: √12 = 3.4641, approximate value 3.464. √0.2̇5̇ = 0.5025, approximate value 0.503. √1.3̇4̇ = 1.1590, approximate value 1.159: the answer page prints 1.160, but the next digit is 0, so by the rule it is 1.159. √5.13̇02̇ = 2.2650, approximate value 2.265.

4. Check yourself

Start with the first question.

In the second, watch the next digit.

The third is the last one.

The whole note on seven cards.

Source: Mathematics, Classes 9-10 (NCTB), Chapter 1 “Real Numbers”, the “Infinite Decimal Fractions and Approximate Values” part.