In Algebra we care about different sets of
In Algebra we care about different sets of numbers and which numbers are part of different sets.
Natural Numbers
Natural Numbers • 1, 2, 3, 4, 5, …
Natural Numbers • • 1, 2, 3, 4, 5, … Numbers we count with
Natural Numbers • • • 1, 2, 3, 4, 5, … Numbers we count with Positive whole numbers
Natural Numbers • • 1, 2, 3, 4, 5, … Numbers we count with Positive whole numbers Symbol = N
Whole Numbers
Whole Numbers • 0, 1, 2, 3, 4, 5, …
Whole Numbers • • 0, 1, 2, 3, 4, 5, … Natural numbers & 0
Whole Numbers • • • 0, 1, 2, 3, 4, 5, … Natural numbers & 0 Symbol = W
Integers
Integers • … , -3, -2, -1, 0, 1, 2, 3, …
Integers • • … , -3, -2, -1, 0, 1, 2, 3, … Whole numbers and their opposites
Integers • • • … , -3, -2, -1, 0, 1, 2, 3, … Whole numbers and their opposites Symbol = Z
Rational Numbers
Rational Numbers • Symbol = Q
Rational Numbers • • Symbol = Q Numbers that can be written as the quotient of two integers
Rational Numbers • • • Symbol = Q Numbers that can be written as the quotient of two integers “Normal” fractions
Rational Numbers • For example … ¾ 5/ 3
Rational Numbers • For example … ¾ -½ 5/ 3 4 3/ 7
Rational Numbers • For example … ¾ -½ 2. 25 5/ 3 4 3/ 7 -. 66666…
Rational Numbers • For example … ¾ -½ 2. 25 42 5/ 3 4 3/ 7 -. 66666… -11
Rational Numbers • • • Symbol = Q Numbers that can be written as the quotient of two integers “Normal” fractions
Irrational Numbers • • Symbol = I or Ir Not rational Can’t be written as a quotient of integers “Weird” numbers
Examples of Irrational Numbers • Special numbers e
Examples of Irrational Numbers • Most trig function values sin(52) tan(107)
Examples of Irrational Numbers • Decimals that don’t end and don’t repeat. 27227722277722227777…
Real Numbers • Symbol = R
Real Numbers • • Symbol = R Rational and irrational numbers together
Real Numbers • • • Symbol = R Rational and irrational numbers together Every number on the number line
Real Numbers • • Symbol = R Rational and irrational numbers together Every number on the number line Every number you know
Properties of Real Numbers • a. k. a. “Field Properties”
A field is just any set that has the same properties as the real numbers. • The properties of numbers are essentially the postulates of algebra.
Properties of Addition and Multiplication
Commutative Property
3+5=5+3 2(-9) = -9 2 Order doesn’t matter when you add of multiply.
Commutative Property
Associative Property
-17 + (17 + 39) = (-17 + 17) + 39 (7 4) 9 = 7(4 9) You can group together what you want to when you add of multiply.
Associative Property
Identity Property
7+0=4 0+2=2 5 1=5 1(-4) = -4 When you add 0 or multiply by 1, you get back what you started with.
Identity Property
Inverse Property
Inverse Property
Closure Property
When you add or multiply two real numbers, the answer is a unique real number.
When you add or multiply two real numbers, the answer is a unique real number. There is only one answer when you add or multiply.
When you add or multiply two real numbers, the answer is a unique real number. You get back the same kind of thing you started with.
Closure Property
Distributive Property
3(2 x + 3 y – 5) = 6 x + 9 y – 15 When you multiply a number times parentheses, take the number times each term in parentheses one at a time.
Distributive Property
Properties of Equality
Reflexive Property of =
2=2
2=2 Everything is equal to itself.
2=2 Everything is equal to itself. Numbers never change.
Symmetric Property of =
If 2 + 3 = 5, then 5 = 2 + 3
If 2 + 3 = 5, then 5 = 2 + 3 You can flip an equation around without changing its meaning.
Example … 129 = 3 x – 15
Example … 129 = 3 x – 15 same as 3 x – 15 = 129
Example … 129 = 3 x – 15 same as 3 x – 15 = 129 Either way x = 48
Symmetric Property of =
Transitive Property of =
If a list of things are equal, then the first equals the last.
Addition Property of =
You can add or subtract the same thing from both sides of an equation without changing anything. x + 14 = 44 x = 30
Multiplication Property of =
You can multiply or divide both sides of an equation by the same thing without changing anything. 5 x = 30 x=6
Defined Operations
Definition of Subtraction
Subtraction means adding the opposite 4 – (-7) = 4 + 7
Definition of Division
Sets of numbers • Natural • Whole • Integers • Rational • Irrational • Real
Properties Commutative Associative Identity Inverse Closure Distributive
Properties Reflexive Symmetric Transitive + Prop = X Prop = Def. of – Def. of
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