A Mathematically Derived Number of Resamplings for Noisy Optimization (GECCO2014)
Thinking Mathematically Number Theory and the Real Number System 5.4 The Irrational Numbers.
12
Thinking Mathematical ly Number Theory and the Real Number System 5.4 The Irrational Numbers
-
Upload
beatrix-caldwell -
Category
Documents
-
view
223 -
download
0
Transcript of Thinking Mathematically Number Theory and the Real Number System 5.4 The Irrational Numbers.
Thinking Mathematically
Number Theory and the Real Number System
5.4 The Irrational Numbers
The Irrational Numbers
The set of irrational numbers is the set of numbers whose decimal representations are neither terminating nor repeating.
An irrational number cannot be written as the ratio of two integers.One of the most well known irrational numbers is the ratio between the circumference and diameter of a circle known as “pi” and written π.
3.14.159
The Golden Ratio
The golden ratio has the value 215
Fibonnaci numbers 1, 1, 2, 3, 5, 8, 13, 21, 34
which is approximately 1.618 033 988
Square Roots
The principal “square root” of a positive number, n, is the positive number that when multiplied times itself produces n. This number is written n . The square root of zero is zero.
Some square roots are rational numbers. For example since 6 x 6 = 36, 36 = 6.
But, the square root operation provides us with many examples of irrational numbers. E.g. 2
Example: Square Roots
Exercise Set 5.4 #3, #11
25 = ?
173 = ?
Simplify by looking for perfect square factors
Exercise Set 5.4 #19
Simplify 80
Simplifying Square Roots
bababa 22
The Product Rule for Square Roots
If a and b represent nonnegative numbers, then √(ab) = a• b and a• b = (ab).
Exercise Set 5.4 #29
3 x 6 = ?
The Quotient Rule for Square Roots
If a and b represent nonnegative real numbers and b ≠ 0, then
b
a
b
aand
b
a
b
a
The quotient of two square roots is the square rootof the quotient.
Example: Quotients of Square Roots
Exercise Set 5.4 #35
?2
90
Adding and Subtracting Square Roots
ac + bc = (a + b)c ac - bc = (a - b)c
Important: baba
Exercise Set 5.4 #49
50 - 18 = ?
Rationalizing the Denominator
The process of rewriting a radical expression to remove the square root from the denominator without changing the value of the expression is called rationalizing the denominator.
Exercise Set 5.4 #59
Rationalize the denominator of
7
21
Thinking Mathematically
Number Theory and the Real Number System
5.4 The Irrational Numbers
