Natural Numbers 1 2 3 4 5 Whole numbers are natural numbers and zero Whole Numbers 0 1 2 3 4 5 N is a subset of W Integers are whole numbers and opposites of naturals ID: 386427
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Slide1Slide2
Natural numbers are counting numbers.
Natural Numbers
= {1, 2, 3, 4, 5…}Slide3
Whole numbers are natural numbers and zero.
Whole Numbers
= {0, 1, 2, 3, 4, 5…}Slide4
N is a subset of W.Slide5
Integers are whole numbers and opposites of naturals.
Integers
= {...−3, −2, −1, 0, 1, 2, 3…}Slide6
N and W are subsets
of Z.Slide7
Rational numbers are integers and all fractions.
Rational NumbersSlide8
=
{
a
b
a
b & b ≠ 0
}
,
,Slide9Slide10
Irrational numbers are totally different from rational numbers. The two have nothing in common.
Irrational NumbersSlide11
Rationals
and irrationals are disjoint sets.
In other words, they have no common element.Slide12
Irrationals
2, , 5 7, 1.305276...
pSlide13
Real numbers are rational and irrational.
Real NumbersSlide14
= & irrationalsSlide15
There are an infinite number of rational numbers between each pair of integers. This is called the density of numbers.Slide16
Rational Numbers
A rational number is any number that can be written
in the form
, where
a
and
b
are integers and b ≠ 0.
abSlide17
Lowest Terms
A rational fraction
is in
lowest terms if the GCF of
a
and
b
is one.
abSlide18
Rename in lowest terms.
Example 1
1218
12 = 2 • 2 • 3
18 = 2 • 3 • 3
GCF = 2 • 3 = 6
1218
2 x 6
3 x 6
=
2
3
=Slide19
Rename in lowest terms.
2490
2490
2 x 2 x 2 x 3
2 x 3 x 3 x 5
=
4
15
=
2 x 2 x 2 x 3
2 x 3 x 3 x 5
=
Example 2Slide20
Rename in lowest terms.
3042
5
7
=
ExampleSlide21
Rename in lowest terms.
3,0004,200
5
7
=
ExampleSlide22
Rename in lowest terms.
7290
4
5
=
ExampleSlide23
A
proper fraction
is one whose numerator is less than its denominator.Slide24
If the numerator is greater than or equal to the denominator, the fraction is greater than or equal to one and is called an
improper fraction
.Slide25
A
mixed number
is actually the sum of a whole number and a fraction.Slide26
Renaming Improper Fractions as Mixed Numbers
Divide the numerator by the denominator.
Write the quotient as the whole number.
Write the remainder over the divisor as a fraction.
If possible, reduce the fraction to lowest terms.Slide27
Rename as a mixed number.
19 7
7 19
2
- 14
5
5 7
=
2
Example 3Slide28
Rename as a mixed number.
12 8
8 12
1
- 8
4
4 8
1
1 2
=
1
Example 3Slide29
Rename the improper fraction as a mixed number.
78 36
1 6
=
2
ExampleSlide30
Rename the improper fraction as a mixed number.
93 8
−
5 8
= −11
ExampleSlide31
Evaluate the expression
when
y
= 38 and
z
= 2. Write the answer as a mixed number in lowest terms.
y
3
z
3 19
6
- 18
1
1 3
=
6
38
3(2)
38
6
=
19
3
=
19 x 2
3 x 2
=
Example 4Slide32
Evaluate when
x
= 2,
y
= – 3, and
z
= 5.
3 5
=
6
3
x
–
y
z
ExampleSlide33
Evaluate when
x
= 2,
y
= – 3, and
z
= 5.
2 5
=
5
–
y
3
x
2
z
ExampleSlide34
Evaluate when
x
= 2,
y
= – 3, and
z
= 5.
4 25
= −
(3
x)
2
3
yz
2
ExampleSlide35
Renaming Mixed Numbers as Improper Fractions
Multiply the whole number by the denominator.
Add the numerator to the product.
Write the sum over the denominator.
If possible, reduce the fraction to lowest terms.Slide36
Rename as an improper
fraction in lowest terms.
16 5
=
1 5
3
1 5
3
5(3) + 1
5
=
15 + 1
5
=
Example 5Slide37
Rename as an improper
fraction in lowest terms.
31 4
=
6 8
7
6 8
7
8(7) + 6
8
=
56 + 6
8
=
62
8
=
31 x 2
4 x 2
=
Example 5Slide38
Rename the mixed number as an improper fraction.
31 11
=
9 11
2
ExampleSlide39
Rename the mixed number as an improper fraction.
4 5
− 12
64 5
= −
Example