Adding Integers

• Integers are whole numbers, which can be positive, negative, or zero
• Integers do not include fractions, decimals, or mixed numbers

Addition of Integers

• Adding integers combines their values to find a total sum
• When adding integers with the same sign, add their absolute values and keep the original sign
• Example: (+3) + (+5) = +8 which is the sum of 3 and 5, keep positive sign
• Example: (-4) + (-2) = -6 which is the sum of 4 and 2, keep negative sign
• When adding integers with different signs, subtract the smaller absolute value from the larger absolute value
• Keep the sign of the integer with the larger absolute value
• Example: (+7) + (-3) = +4 because 7 - 3 = 4, keep positive sign because 7 has larger absolute value
• Example: (-9) + (+2) = -7 because 9 - 2 = 7, keep negative sign because 9 has larger absolute value
• The additive inverse property states that for any integer a, there exists an integer -a such that a + -a = 0
• Adding an integer to its opposite always results in zero
• Example: (+5) + (-5) = 0

Subtraction of Integers

• Subtracting integers is the same as adding the opposite (additive inverse) of the subtracted integer
• To subtract b from a, change the sign of b and add it to a
• This can be expressed as: a - b = a + (-b)
• Example: (+8) - (+3) = (+8) + (-3) = +5
• Example: (+5) - (-2) = (+5) + (+2) = +7
• Example: (-4) - (+1) = (-4) + (-1) = -5
• Example: (-6) - (-3) = (-6) + (+3) = -3
• Subtraction is neither commutative nor associative

Multiplication of Integers

• Multiplying integers involves finding the product of their values
• When multiplying two integers with the same sign, the result is always positive
• Example: (+4) × (+3) = +12
• Example: (-5) × (-2) = +10
• When multiplying two integers with different signs, the result is always negative
• Example: (+6) × (-1) = -6
• Example: (-7) × (+3) = -21
• The product of any integer and zero is always zero: a × 0 = 0
• Multiplication is commutative (a × b = b × a) and associative (a × (b × c) = (a × b) × c)
• The multiplicative identity is 1, meaning that any integer multiplied by 1 remains unchanged: a × 1 = a

Division of Integers

• Dividing integers involves finding the quotient when one integer is divided by another
• When dividing two integers with the same sign, the result is positive
• Example: (+10) ÷ (+2) = +5
• Example: (-15) ÷ (-3) = +5
• When dividing two integers with different signs, the result is negative
• Example: (+8) ÷ (-4) = -2
• Example: (-20) ÷ (+5) = -4
• Division by zero is undefined
• Division is neither commutative nor associative
• The result of dividing integers may not always be an integer; it can be a fraction or a decimal

Order of Operations with Integers

• When performing multiple operations with integers, follow the order of operations (PEMDAS/BODMAS)
• This means Parentheses/Brackets first
• Then Exponents/Orders
• Then Multiplication and Division from left to right
• Finally Addition and Subtraction from left to right
• Example: 2 + (3 × -4) = 2 + (-12) = -10
• Example: (10 - -2) ÷ -3 = (10 + 2) ÷ -3 = 12 ÷ -3 = -4

Properties of Integer Operations

• Closure Property: Integers are closed under addition, subtraction, and multiplication; performing these operations on integers always results in another integer, but this is not true for division
• Commutative Property: Addition and multiplication of integers are commutative, so the order of the integers has no impact on the result (a + b = b + a and a × b = b × a)
• Associative Property: Addition and multiplication of integers are associative which means the grouping of integers does not affect the result (a + (b + c) = (a + b) + c and a × (b × c) = (a × b) × c)
• Distributive Property: Multiplication of integers is distributive over addition, so a × (b + c) = (a × b) + (a × c)
• Identity Property:
  • Additive Identity: Adding zero to any integer does not change the integer (a + 0 = a)
  • Multiplicative Identity: Multiplying any integer by one does not change the integer (a × 1 = a)
• Inverse Property:
  • Additive Inverse: For any integer a, there exists an integer -a such that a + -a = 0
  • Multiplicative Inverse: Not all integers have a multiplicative inverse that is also an integer; the only integers with integer multiplicative inverses are 1 and -1

Absolute Value

• Absolute value of an integer is its distance from zero on the number line
• Denoted by |a|, where a is an integer
• Absolute value is always non-negative
• Examples: |-5| = 5, |3| = 3, |0| = 0
• When evaluating expressions containing absolute values, perform operations inside the absolute value symbols first, then take the absolute value of the result

Learn how to add integers with the same or different signs including using the additive inverse property. Understand how to apply these rules with positive and negative numbers. Test your knowledge of integer addition.