the number system the complex number system and operations with numbers
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Repeating Decimals
• Repeating decimals are decimals that contain a infinite number of digits.
• Examples: 0.333… 7.689689…
FYI…The line above the decimals indicate that numberrepeats.
9.1
Terminating Decimals
• Terminating decimals are decimals that contain a finite number of digits.
• Examples:36.80.1254.5
The Complex Number System
• All numbers in the world• Represented by ℂ
Complex Number System
Real Numbers
Imaginary Numbers
Imaginary Numbers
Imaginary numbers are all the numbers that deal with the square root of a negative number and contain the letter i in it.
Example:
You will learn more about these numbers in Algebra 2
i525
Real Numbers
• Real numbers consist of all numbers that can be represented on a number line.
• Represented by ℝ
Irrational Numbers
• Irrational numbers are any numbers that cannot be
expressed as .
• They are expressed as non-terminating, non-repeating
decimals; decimals that go on forever without
repeating a pattern.
• Examples of irrational numbers:
– 0.34334333433334…– 45.86745893…– (pi)–
b
a
2
Rational Numbers
• Rational numbers are any numbers that can be expressed in the form of , where a and b are integers, and b ≠ 0.
• They can always be expressed by using terminating decimals or repeating decimals.
• Represented by ℚ• Examples:
b
a
3.0,125.0,5
2
Complex Number System
Real Numbers
Rational
Integers
Whole Numbers
Natural Numbers
Irrational
Imaginary Numbers
Integers
• Integers are the set of whole numbers and their opposites.
{…,-3, -2, -1, 0, 1, 2, 3,…}
• Represented by ℤ
Whole Numbers
• Whole numbers are the set of numbers that include 0 plus the positive numbers.
{0, 1, 2, 3, 4, 5,…}
• Represented by 𝕎
Natural Numbers
• Natural numbers are the set of counting numbers.
{1, 2, 3,…}
• Represented by ℕ or ℙ
Venn Diagram of the Complex Numbers
Irrational Numbers
Rational Numbers
Complex Numbers
Imaginary NumbersReal Numbers
Integers
Whole Numbers
NaturalNumbers
Example
• Classify all the following numbers as natural, whole, integer, rational, or irrational. List all that apply.
a. 117
b. 0
c. -12.64039…
d. -½
e. 6.36
f. -3
FYI…For Your Information
• When taking the square root of any number that is not a perfect square, the resulting decimal will be non-terminating and non-repeating. Therefore, those numbers are always irrational.
Properties of Real Numbers
Property Addition MultiplicationCommutative a+b = b+a ab = ba
Associative (a+b)+c = a+(b+c) (ab)c = a(bc)
Identity a + 0 = a a•1 = a
Inverse a + (-a) = 0 a = 1Opposite Reciprocal
Distributive Property
a(b + c) = ab + ac
a
1
Examples of PropertiesName the property displayed:
1. -2 + (x – 5) = (-2 + x) – 5
2. (-2) ( -½ ) = 1
3. 2(4 – 5) = (4 – 5)2
4. x(y – w) = xy – xw
Order of Operations
1. Parenthesis/Grouping Symbols
2. Exponents3. Multiplication and Division
– left to right4. Addition and/or
Subtraction – left to right
Grouping Symbols
Grouping symbols include parenthesis, braces, brackets, numerators and denominators of fractions and underneath a radical or inside absolute value symbols.