Opener/Warm Up #1 (8/25/10) Complete the following…show work 6 + 3 ∙ 5 2. ½ + ¾ 3. |-5| 4. 9(-3)(2) 5. (2/3)( ¼ ) 6. 15÷(3+2)+(-2)(3)

Algebra 2 Numbers and Functions The Complex Number System and Operations with Numbers

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 number repeats.

Terminating Decimals Terminating decimals are decimals that contain a finite number of digits. Examples: 36.8 0.125 4.5

The Complex Number System Real Numbers Rational Integers Whole Numbers Natural Numbers Irrational 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

Real Numbers Real numbers consist of all the rational and irrational numbers. The real number system has many subsets: Natural Numbers Whole Numbers Integers

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)

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.

Integers Integers are the set of whole numbers and their opposites. {…,-3, -2, -1, 0, 1, 2, 3,…}

Whole Numbers Whole numbers are the set of numbers that include 0 plus the set of natural numbers. {0, 1, 2, 3, 4, 5,…}

Natural Numbers Natural numbers are the set of counting numbers. {1, 2, 3,…}

Venn Diagram of the Real Number System Rational Numbers Irrational Numbers

Example Classify all the following numbers as natural, whole, integer, rational, or irrational. List all that apply. 117 -12.64039… -½ 6.36 -3

To show how these number are classified, use the Venn diagram To show how these number are classified, use the Venn diagram. Place the number where it belongs on the Venn diagram. Rational Numbers Irrational Numbers Integers Whole Numbers 6.36 -12.64039… Natural Numbers -3 117

Solution -3 is an integer and a rational number. Now that all the numbers are placed where they belong in the Venn diagram, you can classify each number: 117 is a natural number, a whole number, an integer, and a rational number. is a rational number. 0 is a whole number, an integer, and a rational number. -12.64039… is an irrational number. -3 is an integer and a rational number. 6.36 is a rational number. is an irrational number.

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 Multiplication Commutative 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(1/a) = 1 Distributive Property a(b + c) = ab + ac

Examples of Properties Name the property displayed: -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 Parenthesis/Grouping Symbols – Perform operations within the innermost grouping symbols according to Steps 2-4 below Exponents/Powers – Perform operations indicated by exponents Multiplication and/or Division – Perform multiplication and division in the order they occur left to right Addition and/or Subtraction – Perform addition and subtraction in the order they occur 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.

Examples – Using Order of Operations Evaluate the following: 22(12 + 8) 5 2. 52 ÷ (2 + 11) 3. 7 • 12 + 30 ÷ 5