Download Number Sets

Number Sets
Natural Numbers: N = {1, 2, 3, …}
Whole Numbers: W = {0, 1, 2, 3, …}
Integers: I = {…,–3, –2, –1, 0, 1, 2, 3, …}
a
Rational Numbers: Q = { , a and b are integers, b ≠ 0}
b
N
W
I
Q
R
Example: Find the length of the diagonal of a 1 unit by 1 unit square.
c
1 unit
1 unit
N
What kind of number is c  2 ? Can c  2 be written as a fraction.? Is c  2 in the set Q?
Reminder: The square of an even number is even number, 22  4, 42  16, 62  36 .
The square of an odd number is an odd number: 32  9, 52  25, 72  49 .
Proof:
u
u
If c  2 is in the set Q, then d can be written as a fraction c  , and
is in reduced form.
v
v
u2
We know that 2  c 2  2 , so 2v 2  u 2 .
v
2
2
2
If 2v  u then u is an even number so u is an even number.
If u is even, then u = 2k, and 2v2  u 2  (2k )2  4k 2 . But 2v 2  4k 2 can be reduced to v 2  2k 2 . This means
u
that v 2 is an even number which means v is an even number. If both u and v are even numbers, then
can be
v
reduced which contradicts the original premise.
Therefore c  2  1.41421356237... is not a fraction in the set Q so we need to expand the number set.
Definition: The number set R, the real numbers, is the set of all decimal and mixed decimal numbers, finite and
infinite. Decimal numbers can be finite, infinite repeating, and infinite non-repeating.
I. Finite Decimals
1. What about kids and calculators?
Reminder: a 0 =
an =
Activity Worksheet: “Exploring Decimals with Base 10 Blocks”
Decimals are representations of fractions using our base 10 place value system. Depending on how you
represent the unit (whole), base 10 blocks can be used to represent a variety of finite decimal numbers.
2. Example:
If Flat is the unit, how much is a Long? a Cube?
Write 4F, 3L, 2C in fraction form:
Write 4F, 3L, 2C in decimal form. How is the decimal placed?:
How is 4F, 3L, 2C read?
3. Fill in and be able to reconstruct this table:
Decimal
103
10 2
100
101
Point
1000
100
10
1
.
Thousands Hundreds
Tens
Ones
(and)
101
1
10
Tenths
102
103
1
1
100
1000
Hundredths Thousandths
4. Expanded Form:
Example:
Read the number 259.371 correctly and write in expanded form.
Read the number 9003.3009 correctly and write in expanded form.
5. Multiplying by powers of 10:
Examples: Compute:
987.65  104 =
12.345*100 =
123.45*103 =
9876  100, 000 =
When multiplying or dividing decimals by powers of 10, the effect of multiplying by 10 r is to move the decimal
point right r places; the effect of dividing by 10 r is to move the decimal point left r places.
123.45*10 3
987.65  104
6. Writing Finite Decimals as Fractions:
Examples: Write the decimal numbers as fractions:
907.04 =
0.345 =
7. Writing Fractions as Decimals:
Examples: Use long division to write the fractions as decimals:
1
1
1
=
=
=
2
3
4
1
1
1
=
=
=
7
8
9
1
=
5
1
=
10
1
=
6
1
=
11
II. Infinite Decimals
Characterizing Rational Numbers as Finite and Infinite Repeating Decimal Numbers:
1. Every rational number can be written as a finite terminating or infinite repeating decimal. For an infinite
repeating decimal, the number of digits in the repeating group is called the length of the period; the group of
decimals is called the repetend.
Use long division to write the fractions as decimals:
5
23
25 =
=
8
3
2. Terminating decimals: Let a/b be a fraction in simplest form. Then a/b has a terminating decimal
representation if and only if b has only 2’s and/or 5’s in its prime factorization.
Examples:
Which of the following is repeating? Which is terminating?
7
9
=
=
150
250
31
Express
as a decimal without using long division.
2  52
3. Every finite terminating or infinite repeating decimal can be written as a rational number.
Examples: Write these decimals as rational numbers:
907.04 =
0.12345 =
0.16 =
0.16 =
(Which of these is larger? What is the length of the period? What is the repetend?)
Which is larger 1 or 0.9 ?
5.5123 =
0.9 =
4. What are examples of infinite non-repeating decimals? Infinite non-repeating decimal numbers are said
to be irrational because they cannot be written as a ratio of two integers, in other words they cannot be written
as a fraction.
Examples:  , 34 , 1.01001000100001…
III. Computing with Decimals
1. The 5-up Rule for Rounding Decimals (page 436)
Example: Round 291.8049 to
the nearest hundred:
the nearest hundredth:
the nearest integer:
2. Adding and Subtracting Decimals
Examples:
Convert to fractional form to indicate rule:
2.71 + 37.762 =
351.42 – 417.818 =
What error are students likely to make with addition and subtraction?
3. Multiplying and Dividing Decimals
Examples:
Convert to fractional form to indicate rule (page 439):
2.71 × 37.762 =
351.42  7.8 =
IV. Ratios and Proportions
1. Definition: The ratio of a to b is written a:b and is a quotient
a
.
b
Determining Ratios:
Example:
In a bag of 120 candies 45 are red and the remainder are blue.
What is the ratio of blue candies to the total candies?
What is the ratio of blue to red candies?
Example: The ratio of boys to girls in Ms. McCormick’s classroom is 4:5? How many students are in Ms.
McCormick’s classroom?
a
a c
c
and
are two ratios and the ratios are equal, then the equality  is a proportion. A
b
b d
d
proportion is a statement that two ratios are equal.
Use the fundamental law of fractions to solve proportions.
2. Definition: If
Example: The ratio of boys to girls in Ms. McCormick’s classroom is 4:5? How many students are in Ms.
McCormick’s classroom if
a. there are 28 boys?
b. there are 81 students?
Page 460 27
IV. Percent
1. Definition: The term “percent” means “per hundred”, so r% is the ratio
Example: Express the decimal as a percent.
a. 0.75
b. 0.6
c. 1.25
d. 0.0025
r
100
e. 9
Example: Express the percent as a decimal number.
a. 15%
b. 255%
c. 0.25%
Example: Express the percent as a fraction.
a. 15%
b. 255%
c. 0.25%
Example: Express the fraction as percents, to the nearest tenth of a percent.
1
1
16
a.
b.
c.
8
11
5
2. Solving percent problems:
p
a
 can be solved for p, a, or b. To solve
100 b
percent problems successfully, one must identify the correct whole and parts. Here, p is the whole number
percent, b is the whole, a is the part.
Example
a. What percent of 72 is 12?
b. What is 12% of 72?
c. Twelve percent of what number is 72?
Page 470
4 d, 7, 13 compute, 19
Because percents are a way to express a ratio, the proportion
“How I Spend my School Daze”