5. Approximate Numbers
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.
Later, on this page
Exact numbers arise from counting.
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?
This section answers those questions for you.
Significant Digits
Significant digits give an indication of the accuracy of a number. A digit which is 0 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].
Examples
(Assume all numbers are measurements)
| Number | Significant Digits | Comments |
| 12.378 | 5 | All non-zero digits |
| 12.30 | 4 | The measurement is between 12.295 and 12.305 |
| 0.0587 | 3 | The two zeros are place holders. |
| 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 the example 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
Accuracy refers to the number of significant digits in a number.
Precision refers to the decimal position of the last significant digit.
Example:
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:
The number 80.53 rounded to three significant digits is 80.5.
Rounded to two significant digits, we have 81.
NOTE: We use the symbol ≈ for "is approximately equal to".
Operations with Approximate Numbers
When adding or subtracting approximate numbers, the result should have the precision of the least precise number.
Example:
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
When multiplying or dividing approximate numbers, the result should have the accuracy of the least accurate number.
Example:
When multiplying 3.564 and 2.37, our final answer should have three significant digits.
3.564 × 2.37 = 8.44668 ≈ 8.45
When finding the square root of a number, the result has the same accuracy as the number.
Example:
√22.97 should be written correct to 4 significant digits:
√22.97 ≈ 4.793 (same accuracy).
Exercise
Two jets flew at 938 km/h and 1450 km/h respectively. How much faster was the second jet?
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
where the bar above the zero indicates it is a significant digit. This number has 5 significant digits.
Another example:
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).
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