Download The number 8 is a rational number because it can be written as the

REAL NUMBERS
(as opposed to fake numbers?)
Two Kinds of Real Numbers
• Rational Numbers
• Irrational Numbers
Rational Numbers
• A rational number is a number that can be written as a
fraction, in which both the numerator (the number on top)
and the denominator (the number on the bottom) are whole
numbers.
• The number 8 is a rational number because it can be
written as the fraction 8/1.
• Likewise, 3/4 is a rational number because it can be
written as a fraction.
• Even a big, clunky fraction like 7,324,908/56,003,492
is rational, simply because it can be written as a
fraction.
• Every whole number is a rational number, because any whole
number can be written as a fraction. For example, 4 can be
written as 4/1, 65 can be written as 65/1, and 3,867 can be
written as 3,867/1.
Rational Numbers
• A rational number is a real
number that can be written
as a ratio of two integers.
• A rational number written in
decimal form is terminating
or repeating.
Examples of Rational
Numbers
•16
•1/2
•3.56
•-8
•1.3333…
•- 3/4
Irrational Numbers
• An irrational number is a
number that cannot be
written as a ratio of two
integers.
• Irrational numbers written as
decimals are non-terminating
and non-repeating.
Irrational Numbers
• All numbers that are not rational are considered
irrational. An irrational number can be written as
a decimal, but not as a fraction.
• An irrational number has endless non-repeating
digits to the right of the decimal point. Here are
some irrational numbers:
• π = 3.141592… = 1.414213…
• Although irrational numbers are not often used in
daily life, they do exist on the number line. In
fact, between 0 and 1 on the number line, there
are an infinite number of irrational numbers!
Examples of Irrational
Numbers
• Square roots of
non-perfect
“squares”
17
• Pi
Irrational Numbers
• Every positive real number has two
real roots – one positive (principal
root) and one negative.
• Ex: √16 = 4 and – 4 because 4 x 4 =
16 and -4 x -4 = 16
Negative real numbers have negative
roots: Ex: -√16 = -4
Your Turn
Which of the following numbers are
rational?
1
√3
−6
3½
305.83
√17
-2
3.1415926535897932384626433
Compare and Order
• Since irrational numbers never terminate,
we can compare and order irrational
numbers by locating them between two
consecutive integers.
• For example, the irrational number, √21
can be found between the perfect squares
of √16 and √25. So, we know that the
value of √21 is between 4 and 5. We can
estimate the value at 4.5. Check it out
with your calculator!
Your Turn
• Locate the following irrational
numbers between two consecutive
integers.
•
•
•
•
√232
-√14
-√75
√600
Equivalent Forms
• You can also simplify real numbers to find
equivalent forms.
• Ex: √12 is not a perfect square but it can
be simplified by finding perfect squares
within the number. For example √12 = √4 x
√3. Since √4 is a perfect square, it can be
simplified as 2. Therefore, √12 can be
expressed as 2√3.
Your Turn
• Simplify the following irrational
numbers:
• √18 = √9 x √2 =
• √40 = √4 x √10 =
• √72 = √9 x √4 x √2 =
• √120 = √12 x √10 = √4 x √3 x √10