7-2 Operations on Decimals
Multiplying Decimals
Scientific Notation
Dividing Decimals
Mental Computation
Rounding Decimals
Estimating Decimal Computations Using Rounding
Round-off Errors
Adding Decimals
Add 2.16 and 1.73.
We can change the problem to one we already know how to solve, that is, to a sum involving
fractions.
Multiplying Decimals
If there are n digits to the right of the decimal point in one number and m digits to the right of
the decimal point in the second number, multiply the two numbers ignoring the decimals, and
then place the decimal point so that there are m + n digits to the right of the decimal point in
the product.
Scientific Notation
Scientists use scientific notation to handle either very large or very small numbers.
For example, the distance light travels in one year is 5,872,000,000,000 miles, called a light
year, is expressed as 5.872 · 1012.
The mass of an electron, 0.00054875 atomic mass units, is expressed as 5.4875 · 10−4.
Scientific Notation
In scientific notation, a positive number is written as the product of a number greater than or
equal to 1 and less than 10 and an integer power of 10.
Example 1 Write each of the following in scientific notation:
a. 413,682,000
b. 0.0000231
c. 83.7
d.
−
10,000,000
Example 2 Convert each of the following to standard numerals:
a. 6.84 · 10−5
b. 3.12 · 107
c.
−
(4.08 · 104)
Scientific Notation
Calculators with an
key can be used to represent numbers in scientific notation.
For example, to find (5. 2 · 1016) (9.37 · 104), press
Dividing Decimals
Divide 128.6 by 4.
Dividing Decimals
When the divisor is a whole number, the division can be handled as with
whole numbers. The decimal point can be placed directly over the
decimal point in the dividend.
When the divisor is not a whole number, as in 1.2032 ÷ 0.32, we can obtain a whole-number
whole
divisor by expressing the quotient as a fraction, and then multiplying the numerator and
denominator of the fraction by 100. This corresponds to rewriting the prob
problem in form (a)
as an equivalent problem in form (b), as follows:
In elementary school texts, this process is
usually described as “moving” the decimal
point two places to the right in both the
dividend and the divisor.
Mental Computation
Some tools for doing mental computations with whole numbers can be used for decimal
numbers:
Rounding Decimals
Example 3 Round each of the following numbers:
a. 7.456 to the nearest hundredth
b. 7.456 to the nearest tenth
c. 7.456 to the nearest unit
d. 7456 to the nearest thousand
e. 745 to the nearest ten
f. 74.56 to the nearest ten
Estimating Decimal Computations Using Rounding
Rounded numbers can be useful for estimating answers to computations.
When computations are performed with rounded numbers, the results may be significantly
different from the actual answer.
Other estimation strategies, such as front
front-end,
end, clustering, and grouping to nice numbers, that
were investigated with whole numbers also work with decimals
Round-off Errors
Round-off errors are typically compounded when computations are involved.
When computations are done with approximate numbers, the final result should not be
reported using more decimal places than the number used with the fewest decimal places.
That is, an answer can be no more accurate than the least accurate number used to find it.
Non-zero digits are always significant.
Zeroes before other digits are non-significant.
Zeroes between other non-zero digits are significant.
Zeroes to the right of a decimal point are significant.
To avoid uncertainty, zeroes at the end of a number are significant only if to the right of
a decimal point.
7-2 Homework # A-1, 4, 5, 6, 7, 13, 14, 15, 16, 17, 18, 19, 20, 21, 26, 28, B-4, 7, 14, 17, 23