General Mathematics · Class 9-10 · Chapter 1 · Real numbers
Decimal places, and approximate values.
How do you write down a number whose digits never end? Somewhere you have to stop. There are two ways of stopping, and exam questions use two different words for them. The words differ, the answers often differ, and so do the marks.
1. Cutting and rounding
Taking a value to so many decimal places means keeping exactly that many digits and throwing the rest away. With √2 = 1.414213 56…, the value to three decimal places is 1.414; everything from the fourth digit on is cut off.
Taking an approximate value means keeping the same digits, but looking at the next one before throwing it away. If it is 5, 6, 7, 8 or 9, add 1 to the last digit kept; if it is 0, 1, 2, 3 or 4, leave it as it is. The book's own example shows the difference: √13 = 3.60555128…, whose value to three decimal places is 3.605 but whose approximate value is 3.606, because the next digit is 5.
In half the cases the two values agree, which is why students start treating the two words as synonyms. Pick a number below and drag the slider; the thing to watch is when the right hand box parts company with the left.
handlepick a number, drag the slider
How many places do we write?
check yourself
One question
With √13 = 3.605551…, what is its approximate value to three decimal places?
That is the value to three places, not the approximate value. The question asked for the approximation, and the next digit being 5 means 1 is added.
Right. The value to three places is 3.605 and the next digit is 5. So 1 is added to the last digit, giving 3.606. That one word makes the whole difference.
That is to two places. Three were asked for, so three digits have to stay.
That is to four places. The number of places has to be exactly what the question said.
2. Rounding a number of your own
The rule is one line long, but nearly everyone gets caught in one place: when adding 1 produces a carry. Rounding 2.996 to two decimal places means adding 1 to 99, which makes 100 and carries into the whole number part; the answer is 3.00, not 2.99 and not 2.10. And both zeros have to be written, because two decimal places were asked for.
Type a number of your own below. It shows the reason along with the answer, which is to say which digit the decision was made on.
calculatortype a number and how many places
Find the approximate value
What you type goes nowhere. The page is yours.
check yourself
Another question
What is the approximate value of 2.996 to two decimal places?
That is the value to two places, not the approximation. The next digit is 6, so 1 has to be added.
The carry has gone to the wrong place. 99 + 1 = 100, so the decimals become 00 and the 1 goes into the whole number part.
Right. The value to two places is 2.99, the next digit is 6, so 1 goes on the last digit: 99 + 1 = 100. The carry runs into the whole number part and 2 becomes 3. And both zeros have to be written, because two decimal places were asked for.
The value is right, the writing is not. Two decimal places asked for means two places shown, so 3.00.
3. Squeezing √2
The value √2 = 1.414213 56… has been used all along, but where did it come from? Nobody handed it over, and no calculator is needed; squaring finds it. 1² = 1 is less than 2 and 2² = 4 is more than 2, so √2 must lie between 1 and 2.
Now the same thing more finely: 1.4² = 1.96 is less and 1.5² = 2.25 is more, so 1.4 < √2 < 1.5. Each round settles one more decimal place. Tap the candidates below and find four places for yourself.
squeeze ittap a candidate and see its square
Pin √2 down
You can go on finding places like this as long as you like, and the job never finishes. Here is what non-terminating, non-recurring really means: the number is definite and every one of its digits is definite; it is the list that never ends. Not vague, but endless.
check yourself
Last question
Given 1.4² = 1.96 and 1.5² = 2.25, what follows about √2?
You cannot just take the midpoint. 1.45² = 2.1025, which is more than 2, so √2 is smaller than 1.45.
Right. 1.96 < 2 < 2.25, that is 1.4² < (√2)² < 1.5². For positive numbers the order of the squares follows the order of the numbers, so 1.4 < √2 < 1.5. In the same way 1.41² = 1.9881 and 1.42² = 2.0164 give the next place.
1.5² = 2.25, which is more than 2. So √2 is less than 1.5, not more.
Lying between two decimals does not make a number rational. 2 is not a perfect square, so √2 is irrational, as this chapter proves.
That is the end of the chapter. The lines from this note are cut onto the cards below.
cut from the notetap to turn over
Four cards
frontHow do value to n places and approximate value differ?
backThe first cuts; the second looks at the next digit and rounds.
frontWhat is the rounding rule?
backNext digit 5 or more: add 1 to the last digit kept. Otherwise leave it.
frontHow do you find digits of √2 without a calculator?
backSqueeze it by squaring. 1.4² = 1.96 is less, 1.5² = 2.25 is more, so 1.4 < √2 < 1.5.
frontWhat does a non-terminating, non-recurring decimal mean?
backThe number is definite and so is every digit; it is the list that never ends. Not vague, endless.