Download Decimals

College Mathematics Notes
Section 1.2
Page 1 of 8
Chapter 1: Pre-Algebra
Section 1.2: Decimals
Big Idea for this section: A decimal gives you another way to represent a fraction. It is based on a base-10
positional notation, so many of the rules you already know for the arithmetic of whole numbers apply to
decimals.
Big Skills for this section: You should be able to do arithmetic with decimals. That means be able to add,
subtract multiply, and divide fractions, and convert back and forth between fractions and decimals.
Section 1.2.1: The Decimal Representation (for Terminating Decimals)
 Positional notation as increasing factors of 10…
 Positional notation as decreasing factors of 10…  once get past 1  10, you’re in the decimal range…
 The positional notation can be used to write a fraction equivalent:
1
1
1
 7
 5
10
100
1000
3
7
5
 

10 100 1000
300
70
5



1000 1000 1000
375

1000
3  125

8  125
3

8
0.375  3 

Or you can use the trick of writing a fraction with the digits after the decimal in the numerator and the
next larger power of 10 in the denominator:
5128
10,000
2  2  2  641

22225555
641

25555
641

1250
0.5128 
375
1000
3  125

8  125
3

8
0.375 
Practice:
1. Write 0.3125 as a fraction.
College Mathematics Notes
Section 1.2
Page 2 of 8
Section 1.2.2: General Conversion between Fractions and Decimals
 To convert from fraction to a decimal, you can try to write the fraction as an equivalent fraction with a
denominator that is a power of 10.
1

2
3

4
7

16
1

3

Or you can do long division, remembering to put a decimal at the end if quotient and dividend. Stop when
you get zero for a remainder…
3

4
7

16

Or you can punch it into your calculator.

For some fractions, you never get a remainder of zero no matter how long you do the long division. These
fractions have a decimal representation that repeats forever, and are called non-terminating or repeating
decimals. You know you have a repeating decimal when the long-division pattern keeps on repeating.
1

3
College Mathematics Notes

Section 1.2
A bar over the repeating pattern is frequently used to represent a repeating decimal.
1

3
1

11

Page 3 of 8
To convert a repeating decimal to a fraction…
Practice:
College Mathematics Notes
Section 1.2
Page 4 of 8
Section 1.2.3: Rounding Decimals
 A place to round to will be specified for you with phrases like “round to two decimal places” or “round
to the nearest hundredth”. The idea is that you will re-write the number, but only up to that place, and
with the last digit rounded according to the following rules:
o If the next digit is less than 5, leave the last digit alone
o If the next digit is 5 or more, add 1 to the last digit.

For an example, here is how to round the number 10,547.395 to different specified decimal places:
10,547.395 rounded to
2 places
1 place
the nearest unit
the nearest ten
the nearest hundred
the nearest thousand
Decimal Place of Rounding
hundredth’s place
tenth’s place
one’s place
ten’s place
hundred’s place
thousand’s place
Practice:
2. Round 1.2479 to the nearest thousandth.
Result
10,547.40
10,547.4
10,547.
10,550
10,500
11,000
College Mathematics Notes
Section 1.2
Page 5 of 8
Section 1.2.4: Adding and Subtracting Decimals


To add: Line up the decimal points and add the digits from right to left, carrying as necessary.
To subtract: Line up the decimal points and subtract the digits from right to left, borrowing as necessary.
Practice:
3. 15.156 + 0.9957 =
4. 10.295 – 7.87 =
5. 158.34 – 77.473 =
College Mathematics Notes
Section 1.2
Page 6 of 8
Section 1.2.5: Multiplying Decimals
 Multiply as if there were no decimals present
 The answer will have as many decimal places as all the decimal places of both factors added up.
Practice:
1. 2.5  1.7 =
College Mathematics Notes
Section 1.2
Page 7 of 8
Section 1.2.6: Dividing Decimals


Move the decimal point the same number of places in the divisor and dividend until the divisor is a
whole number.
Then do standard long division, putting a decimal point in the quotient right above the decimal point in
the dividend.
Practice:
2. 3.82  1.2 =
College Mathematics Notes
Section 1.2
Section 1.2.7: Applications of the Arithmetic of Decimals
Practice:
3.
Page 8 of 8