Suppose we want 3,, to 4 significant figures. We simply round the entire number to the nearest thousand, giving us 3,, What if a number is in scientific notation? In such cases the same rules apply. To enter scientific notation into the sig fig calculator, use E notation , which replaces x 10 with either a lower or upper case letter 'e'. For example, the number 5. For a very small number such as 6. When dealing with estimation , the number of significant digits should be no more than the log base 10 of the sample size and rounding to the nearest integer.
For example, if the sample size is , the log of is approximately 2. There are additional rules regarding the operations - addition, subtraction, multiplication, and division. For addition and subtraction operations, the result should have no more decimal places than the number in the operation with the least precision.
For example, when performing the operation Hence, the result must have one decimal place as well: The position of the last significant number is indicated by underlining it. For multiplication and division operations, the result should have no more significant figures than the number in the operation with the least number of significant figures. For example, when performing the operation 4.
So the result must also be given to three significant figures: 4. If performing addition and subtraction only, it is sufficient to do all calculations at once and apply the significant figures rules to the final result. If performing multiplication and division only, it is sufficient to do all calculations at once and apply the significant figures rules to the final result. For example, for the calculation Now, note that the result of the multiplication operation is accurate to 2 significant figures, and more importantly, one decimal place.
You shouldn't round the intermediate result and only apply the significant digit rules to the final result. So for this example, the final steps of the calculation are Exact values, including defined numbers such as conversion factors and 'pure' numbers, don't affect the accuracy of the calculation.
They can be treated as if they had an infinite number of significant figures. To use an exact value in the calculator, give the value to the greatest number of significant figures in the calculation. So for this example, you would enter Trailing zeros in a whole number with no decimal shown are NOT significant. Writing just "" indicates that the zero is NOT significant, and there are only TWO significant figures in this value.
This rule applies to numbers that are definitions. So now back to the example posed in the Rounding Tutorial : Round Writing just "" would give us only one significant figure. Rule 8 provides the opportunity to change the number of significant figures in a value by manipulating its form.
By rule 6, has TWO significant figures; its two trailing zeros are not significant. If we add a decimal to the end, we have Integers: When you count, the result is exact assuming that you do not loose count. On the other hand the year computer problem Y2K that received so much press is a number with four significant figures. The count of years is exact.
Rational fractions: Any fraction made from integers is exact. The student must be careful with these fractions. The quantity described must be inherently an integer to apply this rule. When we convert rational numbers to decimal fractions they always produce a set of repeating digits.
Computed results: Any math operation with numbers having significant figures will result in a number having significant figures. However the number of significant figures in the result depends on all the inputs to the problem.
These procedures are described in the section Rounding After Math Operations below. When computing results on a calculator we often end up with many digits displayed.
Because computation itself cannot increase our measurement accuracy we must decide how many of these figures are significant and round the result back to the appropriate number of figures.
Once you have decided how many figures you will keep look at the first digit you will reject. When the first rejected digit is less than 5 you will round down simply delete the rejected digits. When the first rejected digit is greater than 5 round up this means that you will still delete the rejected digits but now you add one to the last digit you keep. If the first digit rejected is equal to 5 round up or down so that the last digit retained is even. If we always round 5 up we will distort the results for very large data sets.
For example: Rounding After Math Operations: The rule for choosing the number of digits to retain depends on the mathematical operation you perform. The simplest case is for multiplication or division. Here the number of significant figures in the result is equal to the number of significant figures in the least accurate value used in the computation. In the following examples the least accurate number is in bold face type.
This last example using 3.
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