Decimal numbers are used when you need more precision than whole numbers provide. Decimals are based on units of ten (tenths) and multiples of tenths. The value of a digit in a decimal depends on the place of the digit.
1.234567 – the 1 is in the whole number position
1.234567 – the 2 is in the “tenth” position
1.234567 – the 3 is in the “hundredth” position
1.234567 – the 4 is in the “thousandth” position
1.234567 – the 5 is in the “ten thousandth” position
1.234567 – the 6 is in the “hundred thousandths” position
1.234567 – the 7 is in the “millionths” position
Because measured quantities are often used in calculations, the precision of the calculation is limited by the precision of the measurements on which it is based. For example, if you calculate the area based on measurements to the nearest foot, you would give the answer in the nearest foot, not the nearest tenth or hundredth of a foot. This is using significant figures to help determine how to express answers. A calculated number cannot be more accurate than the measurements from which it is based.
Significant figure rules to determine how many significant figures in a number:
1. All non-zero numbers are significant
2. Zeros within a number are significant (3076 and 60.02 contain four significant figures
3. Zeros that do nothing but set the decimal point are not significant (630,000 has only two significant numbers)
4. Trailing zeros that aren’t needed to hold the decimal point are significant (If a problem states – “the height is 200 inches” – assume the height is known to one significant number.)
When adding, subtracting, multiplying and dividing numbers:
1. The measurements are added or subtracted, the answer can contain no more decimal places than the least accurate measurement.
22.25 ft + 7.125 ft + 11 ft = 40 feet
When added together, you get 40.375 ft, but you should give the sum as “40” feet due to rounding and giving the significant number.
2. When measurements are multiplied or divided, the answer can contain no more significant figures than the lease accurate measurement.
(29.5) / (7) = 4, not 4.214
3. When the answer in a calculation contains too many significant numbers, it must be rounded off.
Losing Significant Figures
You may sometimes lose significant figures when performing calculations. If you subtract 21.75 – 21.50 = 0.25. The answer, 0.25 has two significant numbers even though the original values contained four significant figures. When that happens, don’t worry about it. Just give the answer in the number of significant figures it has.