4.2: Rounding

We have been using decimal numbers throughout our lives, but let’s take some time to formalize the language so we can learn how to round decimals appropriately.

Reading decimal numbers:

Fraction	Decimal	Name
\(\dfrac{1}{10}\)	0.1	One-tenth
\(\dfrac{1}{100}\)	0.01	One-hundredth
\(\dfrac{1}{1000}\)	0.001	One-thousandth
\(\dfrac{1}{10000}\)	0.0001	One-ten-thousandth
\(\dfrac{1}{100000}\)	0.00001	One-hundred-thousandth

The place values to the right of the decimal are given the following names:

Tenths, hundredths, thousandths, ten-thousandths, hundred-thousandths, millionths, etc.

As a result, we read 5.643 as “five and six-hundredth-forty-three- thousandths since the last digit after the decimal ends in the thousandths place.

Examples:

1. 45.6 is read as “forty-five and six tenths”
2. In the number 543.7892, the 8 is in which place value? Since the 8 is two digits after the decimal point, the place value is hundredths.
3. 0.0143 is read as “one-hundred-forty-three ten-thousandths

For rounding decimals, to round to the nearest decimal place,

1. Consider the number to the right of the desired rounding place value
   1. If the digit is less than 5 (4 or less), we truncate at the desired rounding place value, that is, we drop all the digits after the desired rounding place value
   2. If the digit is 5 or more, we round up by adding 1 to the desired rounding place value before dropping all the remaining digits to the right

Examples:

1. Round 32.784 to the nearest hundredth.Since 8 is in the hundredths place, we consider the number immediately to the right (in the thousandths place), since 4 is less than 5, we drop all the digits after the hundredths place to obtain 32.78.
2. Round 32.786 to the nearest hundredth.Since 8 is in the hundredths place, we consider the number immediately to the right (in the thousandths place), since 6 is more than 5, we add 1 to 8 (in the hundredths place, this is called rounding up) and we drop all the digits after the hundredths place to obtain 32.79.
3. Round 6.48327 to the nearest thousandth.We start by consider the value of the digit immediately after the thousandths place, which is 2, since 2 is less than 5, we drop all the digits after the thousandths place to obtain 6.483 as the rounded number to the nearest thousandth.
4. Round 6.4592 to the nearest
   1. Tenth \(\Rightarrow\) Since 5 immediately follows the tenths place, we round the 4 in the tenths place up one number to 5 (by adding 1 to 4) to obtain 6.5 as the rounded number to the tenths place.
   2. Hundredth \(\Rightarrow\) Since 9 immediately follows the hundredths place, we round the 5 in the hundredths place up to 6 to obtain 6.46 as the rounded number to the hundredths place.
   3. Thousandth \(\Rightarrow\) Since 2 immediately follows the thousandths place, we round the 5 in the hundredths place up to 6 to obtain 6.46 as the rounded number to the hundredths place.
5. Round 6.4597 to the nearest thousandth (to three decimal places).Since the fourth digit is 5 or higher, we must round the third digit after the decimal up by adding 1, but if we add 1 to 9, we would get 10, so we think of 59 (the second and third places after the decimal and round it up by 1 to obtain 60, hence 6.4597 rounded to three decimal places or the nearest thousandth becomes 6.460. We must write the zero in the thousandth place in this case to show the rounding to three decimal places, without the zero on the end, the rounding would be incorrect.

Significant Figures

Significant figures are used to assure the accuracy of a calculation based on the numbers used within the calculation. When a problem doesn’t specify how to round an answer or it is not implied in its context (such as money), we use significant figures.

The following digits are considered significant:

• Digits between 1-9. Example: 875 would consist of three significant figures.
• Any zeros between non-zero digits. Example: 20.07 and 5608 would both consist of four significant figures.
• The trailing zeros in a decimal number are significant

The following digits represent ambiguous cases, so the best method to determine the significant figures is to convert the number to scientific notation:

• Any zeros in numbers that do not contain a decimal point. Example: 530,000 depending on the situation, this could represent two significant figures or six significant figures, so let’s convert it to scientific notation to obtain: \(5.3\ X\ {10}^4\) which would now represent two significant figures. In general, if the zeros are used to locate the decimal point, they are not significant figures, whereas if the zeros are being used for accuracy, they are included as significant figures.

The following digits are not significant:

• If a number is less than 1, the zeros that occur after the decimal point but before a non-zero digit. Example: 0.0075 would consist of two significant figures

When performing calculations, the number of significant figures of the final answer is equivalent to the fewest number of significant figures provided in the calculation.

Examples:

1. State the number of significant figures in each of the following numbers
   1. 3.402 \(\Rightarrow\) 4 significant figures, the non-zero digits are counted since they are between 1-9 and the zero is counted since it is between two non-zero digits
   2. 0.000472 \(\Rightarrow\) 3 significant figures; this number is less than 1, so the zeros after the decimal point but before the non-zero digits do not count, so we only count the digits between 1-9
   3. \(8.0\ X\ {10}^4\) \(\Rightarrow\) 2 significant figures; we consider the decimal value (not the power of 10), the trailing zero counts since it is used to show accuracy of the number
   4. 452,000 \(\Rightarrow\) ambiguous case, 3 to 6 significant figures depending on the context of the number
2. Round the following numbers to three significant figures
   1. 89.0146 \(\Rightarrow\) Since the zero after the decimal point is significant since it is between two non-zero digits, we round to the nearest tenth in this cased to obtain 89.0
   2. 0.005324 \(\Rightarrow\) Since this number is less than 1, the zeros immediately to the right of the decimal point are not significant, so to round to three significant figures, we round to the nearest hundred-thousandth to obtain 0.00532
   3. 872.58 \(\Rightarrow\) Since all the digits are significant, to round to three significant figures, we round to the nearest whole number (three digits from the left) to obtain 873.
   4. 8723.158 \(\Rightarrow\) Since all the digits are significant, to round to three significant figures, we round to the nearest ten (three digits from the left) to obtain 8720.
   5. 65,207 \(\Rightarrow\) Since all the digits are significant, to round to three significant figures, we round to the nearest hundred (three digits from the left) or 65,200.
3. Based on the given calculations, state the number of significant figures of the answer, then perform the calculation and round it based on the significant figures
   1. \((4.5)(2.33)(6.232)\) \(\Rightarrow\) Two significant figures since 4.5 has the least number of significant figures.\[\left(4.5\right)\left(2.33\right)\left(6.232\right)=65.34252\,\,\, \text{which rounds to 65} \nonumber\]
   2. \(\frac{(2.543)(3.516)}{0.01}\) \(\Rightarrow\) One significant figure since 0.01 has the least number of significant figures. \[\frac{(2.543)(3.516)}{0.01}=\frac{8.941188}{0.01}=894.1188 \,\,\, \text{which rounds to 900}\nonumber \]