Unit 1: Operations with Real Numbers
integer
An integer is a number that can be written without a fractional component. For example: 21, 4, 0, and −2048 are integers, while 9.75, 5 ¹⁄₂, and √2 are not.
Subtracting positive and negative numbers
Convert the subtracted number to its opposite: 11 - (-7) = 11 + 7 = 18 -9 - (-3) = -9 + 3 = -6
Dividing by zero
In ordinary arithmetic, the expression has no meaning, as there is no number which, multiplied by 0, gives a (assuming a≠0), and so division by zero is undefined.
Identity property of addition
The identity property for addition tells us that zero added to any number is the number itself. Zero is called the "additive identity." Example: 1 + 0 = 1
Identity property of multiplication
The identity property for multiplication tells us that the number 1 multiplied times any number gives the number itself. The number 1 is called the "multiplicative identity." Example: a x 1 = a b x 1 = b
Zero divided by a number
Zero divided by any number is always 0.
positive + Bigger negative =
negative
positive - Bigger positive =
negative
positive × negative =
negative
positive ÷ negative =
negative
negative - Bigger negative =
positive
negative ÷ negative =
positive
operation
A process in which a number, quantity, expression, etc., is altered or manipulated according to formal rules, such as those of addition, multiplication, and differentiation.
Simplify
Make (something) simpler or easier to do or understand.
Integer Rules for: Adding positive and negative numbers
Same signs: add and keep the sign 15 + 35 = 50 -5 + -23 = -28 Different signs: subtract and take sign of number with larger absolute value -13 + 35 = 22
distributive property
The Distributive Property is an algebra property which is used to multiply a single term and two or more terms inside a set of parentheses. Example: a(b+c) = ab + ac
Inverse property of addition
The Inverse Property of Addition says that any number added to its opposite is equal to zero. a + (-a) = 0.
Inverse property of multiplication
The Inverse Property of Multiplication says that any number multiplied by its reciprocal is equal to 1. Example: a x 1/a = 1/a x a = 1
order of operations
"PEMDAS", which is turned into the phrase "Please Excuse My Dear Aunt Sally". It stands for "Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction".
commutative property of multiplication
Factors can by multiplied in any order and the answer will be the same. Example: 2 x 3 = 3 x 2
associative property of multiplication
The way in which three numbers are grouped when they are multiplied doesn't change the product. Example: 2 (4 x 5) = 4 (5 x 2)
commutative property of addition
The word "commutative" comes from "commute" or "move around", so the Commutative Property is the one that refers to moving stuff around. For addition, the rule is "a + b = b + a"; in numbers, this means 2 + 3 = 3 + 2.
associative property of addition
This states that you can add or multiply regardless of how the numbers are grouped. By 'grouped' we mean 'how you use parenthesis'. In other words, if you are adding or multiplying it does not matter where you put the parenthesis.
Dividing positive and negative numbers
When the dividend and the divisor have the same sign, the quotient is positive. 45 / 5 = 9 -120 / -6 = 20 When the dividend and the divisor have different signs, the quotient is negative. 35 / -5 = -7 -25 / 5 = -5
Multiplying positive and negative numbers
When the factors have the same sign, the product is positive. 5 x 6 = 30 -10 x -5 = 50 When the factors have different signs, the product is negative. -6 x 8 = -48 9 x -11 = -99
Bigger negative + positive =
negative
Rules for Adding Integers: negative + negative =
negative
Rules for Subtracting Integers: Bigger negative - negative =
negative
negative - positive =
negative
negative × positive =
negative
negative ÷ positive =
negative
Bigger positive + negative =
positive
Bigger positive - positive =
positive
Rules for Multiplication/Division of Integers: negative × negative =
positive
negative + Bigger positive =
positive
positive + positive =
positive
positive - negative =
positive
positive × positive =
positive
positive ÷ positive =
positive