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# 5. Approximate Numbers

## Approximation

Approximate numbers arise from measurement or calculation. We can never perform a completely accurate measurement with a ruler, tape measure or thermometer. There is always some inaccuracy involved, since we can always get a more accurate answer if we use a ruler (or other measuring device) with smaller units.

By comparison, exact numbers arise from counting. For example, the number of pens we can have is either 0 pens, 1 pen, 2 pens, 3 pens and so on. Such quantities are exact.

Why does it matter? Our calculators often give us long answers containing many decimals. How many decimal places should we use in our answer? How many significant digits? What do we do when we multiply or add numbers with different significant digits?

## Significant Digits

All digits greater than 0 in a number are significant. For example, say we measure a pipe diameter and get 26.832 cm. This number has 5 significant digits.

Rounding: We can round off 26.832 to 2 decimal places and get 26.83. (This means our measurement is closer to 26.83 cm than it is to 26.84 cm. Another way of thinking about this is that 26.83 is between 26.825 and 26.835.) Our rounded number 26.83 has only 4 significant digits.

What about 0? When is it significant?

Let's consider the number 26.830. This suggests greater accuracy than our rounded number 26.83. The zero in 26.830 is significant.

A zero digit is significant if it is not a place holder. [Another way of thinking about this is that the number of significant digits is the number of digits we write when we write the number in scientific notation].

Let's now round our earlier measurement 26.832 cm to the nearest 10. This is 30 cm. The zero in this number serves as a place holder - it is not a significant digit.

### Example 1 - Significant digits

(Assume all numbers are measurements)

 Number Significant Digits Comments a) 12.378 5 All non-zero digits b) 12.30 4 The measurement is between 12.295 and 12.305 c) 0.0587 3 The two zeros are place holders. d) 3600 2 The measurement is between 3550 and 3650

NOTE: We are assuming that for numbers greater than 1, the last non-zero number is significant.

So in example d) above, 3600, we assume it is a number correct to the nearest 100, since the 6 is the last non-zero integer. The two zeroes in 3600 are place-holders.

## Accuracy and Precision

Say we get several students to measure the weight of an object.

The accuracy of a measurement refers to how close it is to the actual true weight of our object. You are accurate if your measurement is very close to the true weight.

On the other hand, the precision of measurements refers to how close the measurements are to each other. On most scales there is a "zero" adjustment. We need to "zero" the scales when no object is on them. If this is not done, it is quite possible for the measurements to be precise (all close together) but quite inaccurate (a long way from the true measurement).

If the scales were zeroed properly, and the students were good at measuring, it is possible for the measurements to be accurate (close to the true measure) and precise (close to each other).

Let's now talk about accuracy and precision of numbers.

Significant digits give us an indication of the accuracy of a number arising from a measurement. The more significant digits in the number, the more accurate it indicates the measurement to be.

For example, we conduct our weight measurement activity again, but this time with 2 different scales. One set of scales indicates whole-numbered kilograms only, whereas the second one is in grams.

The students who use the first set of scales can only give whole-number answers like 7 kg. The other students get an answer of 6,748 grams (that is, 6.748 kg). The first answer is not very close to the true weight (it only has one significant digit), whereas the second one is much closer (it has 4 significant digits).

Here's another example. The measurement 26.832 cm from above is more accurate than the rounded figure 26.83 cm. This means 26.832 cm is closer to the actual diameter of the pipe than 26.83 cm is (assuming the person measuring is doing a good job).

Significant digits can also give us an indication of the precision of a number. The precision of a number refers to the decimal position of the last significant digit.

### Example 2 - Accuracy and precision

Comparing the two numbers 0.041 and 7.673, we see that 7.673 is more accurate because it has four significant digits, where 0.041 only has two.

The numbers have the same precision, as the last significant digit is in the thousandths position for both.

## Rounding Off Decimals

### Example 3 - Rounding

The number 80.53 rounded to three significant digits is 80.5.

Rounded to two significant digits, we have 81.

### "Approximately equal to" symbol

Notation: We use the symbol ≈ for "is approximately equal to".

Hint: There is no magic about rounding. You just consider which is closer. Is 80.53 closer to 80.5 or 80.6? When we round some more (to whole numbers), we ask is 80.53 closer to 80 or 81?

## Operations with Approximate Numbers

Precision of the answer: When adding or subtracting approximate numbers, the result should have the precision of the least precise number.

### Example 4

When adding 2.3, 5.704 and 12.67, our final answer should be correct to one decimal place.

2.3 + 5.704 + 12.67 = 20.674 ~~ 20.7

### Accuracy when multiplying or dividing

When multiplying or dividing approximate numbers, the result should have the accuracy of the least accurate number.

### Example 5

When multiplying 3.564 and 2.37, our final answer should have three significant digits.

3.564 xx 2.37 = 8.44668 ~~ 8.45

### Accuracy when finding square root

When finding the square root of a number, the result has the same accuracy as the number.

### Example 6

sqrt(22.97) should be written correct to 4 significant digits:

sqrt(22.97) ~~ 4.793

Both numbers have the same accuracy.

## Exercise

Two jets flew at 938 km/h and 1450 km/h respectively. How much faster was the second jet?

When comparing 1450 and 938, the least precise of the 2 numbers is 1450 (it is correct to the nearest 10) so we need to write the answer correct to the nearest 10. We are using the result from above:

When adding or subtracting approximate numbers, the result should have the precision of the least precise number.

So we have

1450 − 938 = 512

Now 512 correct to the nearest 10 is 510. So the required answer is 510 km/h.

Be careful, though, since sometimes the result may look a bit silly.

Following this thinking, we would write

10 − 7 = 3 ≈ 0,

since "10" is written correct to the nearest 10 and the answer, 3, to the nearest 10 is 0.

## Another Possibility

In the jets example, what if the 1450 is really correct to the nearest whole number? How would we know, since it does not say so in words? Remember, we are assuming it is correct to the nearest 10 since the last non-zero digit is the 5.

## Bar Notation

Let’s take another number arising from a measurement in an experiment:

360\ 000

We would assume that this is a number correct to the nearest 10 000, because the 6 is the last non-zero digit. But what if the experimenter knew it was correct to the nearest 10 and wanted to indicate this without using words?

The experimenter could write it as

360\ 0bar(0)0

where the bar above the zero indicates it is a significant digit. This number has 5 significant digits.

### Another example

1\ 40bar(0)\ 000

This indicates the number is correct to the nearest 1000. This number has 4 significant digits (the 1, 4, 0 and 0 at the front).