Real Numbers
Irrational NumbersNumbers that cannot be written as a fraction
√2, π
Rational NumbersNumbers that can be written as a fraction
Decimals that repeat
Decimals that stop
√25, ½, 5, 0.123, 0.333333…
Real NumbersSet of all irrational and rational numbers
Real Numbers
IntegersPositive and negative counting numbers (plus 0){…-3, -2, -1, 0, 1, 2, 3…)
Whole NumbersCounting numbers starting at 0{0, 1, 2, 3…}
Natural NumbersCounting numbers starting at 1{1, 2, 3…}
Real Numbers
Infinite sets- not countableWhole numbers greater than 8
{3, 4, 5 …}
Finite sets- countableIntegers between 2 and 17
{2, 5, 7, 19, 23}
Real Numbers
Estimating the value of an irrational numberCompare perfect square values
List perfect squares close to your value
√67
√49 = 7; √64 = 8; √81 = 9
67 is between 64 and 81 so √67 is between 8 and 9
8 < √67 < 9
Real Numbers
1. Which of the following represents an infinite set of numbers?a. {1/2, 1/3, ¼, 1/5}
b. {Negative integers}
c. {-3, -1, 0, 1, 3}
d. {Natural numbers between 5 and 20}
Real Numbers1. Which of the following represents an infinite
set of numbers?a. {1/2, 1/3, ¼, 1/5}
This set has a clear start and stop, we see exactly 4 values in the set so it is countable or finite
b. {Negative integers}
integers go off to infinite so this set is not countable
c. {-3, -1, 0, 1, 3}
We can count the 5 values in this set.
d. {Natural numbers between 5 and 20}
We can list and count the values in this set. 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20
Real Numbers
2. Which of the following is an irrational number?
a. √5
b. √9 = 3 whole numbers are rational
c. 7 = 7/1 whole numbers are rational
d. 3.78 = 378/100 decimals that stop are rational
Real Numbers
3. Between which two consecutive integers is √113 ?
a. 12 and 13
b. 8 and 9
c. 10 and 11
d. 11 and 12
Real Numbers3. Between which two consecutive integers is
√113 ?
a. 12 and 13
b. 8 and 9
c. 10 and 11
d. 11 and 12
82 = 64; 92 = 81; 102 = 100; 112 = 121; 122 = 144; 132 = 169
Number Properties
Number Properties Rap
Math Properties
Number Properties
Commutative PropertyNumbers can be added or multiplied in any order.
1 + 2 = 2 + 12(3) = 3(2)
Associative PropertyWhen adding, changing the grouping doesn’t matter.
(1 + 2) + 3 = 1 + (2 + 3)
When multiplying, changing the grouping doesn’t matter.
2(3x4) = (2x3)4
Number Properties
IdentityAdding 0 doesn’t change a value
Multiplying by 1 doesn’t change the value
InverseAdding the opposite gives you 0
Multiplying by the reciprocal gives you 1
Distributive Property3(a + b) = 3a + 3b
Number Properties
ClosureWhen you add or multiple real numbers together the answer will also be a real number.
Number Properties
The multiplicative inverse is the reciprocal. We use it to make a number turn into 1.
Integers
Adding two positive integers
Just add.
Answer will be a positive
Adding a positive and a negative
Subtract
Answer will be the same as the larger of the two numbers
Adding two negatives
Just add
Answer will be negative
Absolute Value
Absolute Value is the distance a number is from zero on the number line.
|-2| = 2
|3 – 6| = |-3| = 3
Order of Operations
Order of Operations Rap
Order of ops rap 2
Order of Operations
Parenthesis 22 – 2[5 + 3(5)]
Brackets (more parenthesis) 22 – 2[5 + 15]
22 – 2[20]
Multiplication 22 – 40
Subtraction -18
Order of Operations
2(-48 / 4 x 3)
2(-12 x 3)
2(-36)
-72This one is tricky…we have to multiply and divide at the same time from left to right.
Scientific Notation
A number written as a product of a power of 10 and a decimal number greater than or equal to 1 and less than 10.
3.72 x 106
When adding and subtracting the exponents must be the same…or we have to rewrite them in standard form first.
3.72 x 106 + 1.5 x 106 = (3.72 + 1.5) x 106 = 5.22 x 106
Scientific Notation
MultiplyingMultiply the factors, add the exponents
DividingDivide the factors, subtract the exponents
Scientific Notation
Since the exponents have the same value we can add the factors 7.8 and -4.2.
(We end subtracting)
7.8 – 4.2 = 3.6
So our answer is 3.6 x 1020