CK12 FlexBooks explains Real numbers can be broken down into different types of numbers such as rational and irrational numbers. They can be visualized using number lines and operated on using set symbols and operators.
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Basics
Real NumBeRs
of
AlgebraI
Study Guides
Big Picture
Real numbers are used to measure the quantity of things in life. Almost any number you can think of is most likely to
be a real number. Real numbers can be broken down into different types of numbers such as rational and irrational
numbers. They can be visualized using number lines and operated on using set symbols and operators. General
guidelines and rules are created to work with real numbers.
Key Terms
, as long as the denominator is not equal to 0.
Rational Number: Ratio of one integer to another:
Integer: A rational number where the denominator is equal to 1. Includes natural numbers, negative natural
numbers, and 0.
Natural Numbers: Counting numbers such as 1, 2, 3.
Whole Numbers: All natural numbers and 0.
NonInteger: A rational number where the denominator is not equal to 1.
Proper Fraction: Numerator is less than denominator. Represents a number less than one.
Improper Fraction: Numerator is greater than denominator. Represents a number greater than 1.
Equivalent Fractions: Two fractions that represent the same amount.
or �.
Irrational Number: Number that cannot be expressed as a fraction, such as
Understanding Real Numbers
Symbols
Here are some common symbols used in algebra:
1. Sum or product of two rational numbers is rational.
•
Example: 2 + 3 = 5
Symbols
•
Example:
Symbol
Meaning
+
add
2. Sum of rational number and irrational number is irra
subtract

tional.
× or ·
multiply
•
Example: 2 +
= 2 +
÷ or /
divide
3. Product of nonzero rational number and irrational
square root, nth root
or
number is irrational.
 
absolute value
•
Example: 3 ·
= 3
=
equals
•
Example: 5 · � = 5 �
yourtextbookandisforclassroomorindividualuseonly.
≠
not equal
Disclaimer:thisstudyguidewasnotcreatedtoreplace
4. Difference between two whole rational numbers is not
≈
approximately equal
always a positive number.
less than, less than or
•
Example: 5  4 = 1
<, ≤
equal to
•
Example: 5  9 = 4
greater than, greater
>, ≥
5. Quotient of a whole rational divisor and a whole
than or equal to
dividend is not always a whole number.
{ }
set symbol
•
Example:
an element of a set
( ), [ ]
group symbols
•
Example:
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Basics
Real NumBeRs
of
coNt.
Algebra
Exponents and nth Roots
Exponents
nth Roots
Exponent is a shorthand notation for repeated multi
The nth root is the inverse operation of raising a number
plication.
to the nth power. So the inverse operation of xn = y is
.
•
·
2 · 2
2 = 23. We say that 2 is raised to the power
•
of 3.
= 4 because 42 = 16
•
is called the radical sign
•
·
2 · 2 · 2 · 2
2 = 25. We say that 2 is raised to the
•
When n=2, we usually write
, not
, and we call
power of 5.
it the square root.
•
For x , we say that x is raised to the power of n.
n
•
When n=3, we call it the cube root.
• x
and n are variables, symbols that are used to
represent a value.
If the nth root can’t be simplified (reduced) to a rational
•
If n=2, we can also say x squared. If x=3, we
number without the radical sign (
), the number is
irrational.
say x cubed.
•
Examples:
= 8, so it is a rational number.
cannot be reduced any further and is irrational.
We can get an approximate value for irrational square
roots by using the calculator. In decimal form, the
number will look like an unending string of numbers.
•
Example:
≈ 1.414 when rounded to three decimal
places.
Fractions and Decimals
A rational number is just a ratio of one number to another written in fraction form as
.
•
Every whole number is a rational number where the denominator equals 1.
•
A denominator equal to 1 is sometimes called the invisible denominator because it is not usually written out:
.
•
Fraction bar: the line that separates the numerator and the denominator. The denominator b ≠ 0.
•
A proper fraction represents a number less than one because a < b, while an improper fraction represents a
number greater than one because a > b.
•
A negative fraction is usually written with the negative sign to the left of the fraction
•
Example:
could equal
or
•
Improper fractions can be rewritten as an integer plus a proper fraction (mixed number).
•
Example:
•
Whenever we can write two fractions equal (=) to each other, we have equivalent fractions.
•
Example:
, so
and
are equivalent fractions.
•
The fractions
are equivalent as long as c ≠ 0.
A fraction can be converted into a decimal  just divide the numerator
by the denominator.
Figure: Equivalent fractions
•
Examples:
•
The ... means that the decimal goes on forever.
Not all decimals can be converted into fractions.
•
If the numbers after the decimal point (.) never repeats and never ends, the number is irrational.
•
Any number that can’t be written as a fraction is irrational.
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Basics
Real NumBeRs
of
coNt.
AlgebraI
Sets
Sets are used to define groups of elements. In math,
Difference: the set of elements that belong to A only
sets can be used to define different types of numbers,
•
Denoted as A \ B
such as even and odd numbers. Outside of math, sets
can also be used for other elements such as sets of keys
or sets of clothing. The different types of sets (as shown
below) are used to classify the objects in the sets.
We can list the elements (members) of a set inside the
symbols { }. If A = {1, 2, 3}, then the numbers 1, 2,
and 3 are elements of set A.
•
Numbers like 2.5, 3, and 7 are not elements of A.
•
•
If A is the group of whole numbers and B is the
We can also write that 1
A, meaning the number
group of natural numbers, A \ B is 0
1 is an element in set A.
The order here matters! B \ A means the set of
•
If there are no elements in the set, we call it a null
elements belonging to B only.
set or an empty set.
Complement Set: all elements in a set that is not A
Union: the set of all elements that belong to A or B
•
Denoted as Ac
•
Denoted as A
B
•
B \ A is equal to Ac. If A is the group of whole
•
The union of rational numbers and irrational
numbers and B is the group of natural numbers,
numbers is all real numbers.
Ac is null (there are no elements in set B that is not
Intersection: the set of elements that is true for both
also in A)
A and B
Disjoint Sets: when sets A and B have no common
•
Denoted as A
B
elements.
•
Rational and irrational numbers are disjoint sets.
Notes
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