This Integer Rules Cheat Sheet is a comprehensive guide designed to demystify the fundamentals of integers, their operations, and properties for learners of all levels. Starting with "The Basics," it covers integer definitions, the concept of absolute values, the significance of zero, and techniques for comparing and ordering integers. The sheet progresses to "Integer Operations," explaining the rules for adding, subtracting, multiplying, and dividing integers, including handling division exceptions. Additionally, it delves into "Integer Properties" such as the commutative, associative, and distributive properties, rounding off with "Advanced Topics" that introduce prime and composite numbers, as well as even and odd integers, and simple factorization. This cheat sheet serves as a quick reference to understand and apply integer rules in mathematical problems efficiently.
The Basics
Integers
Integer definition: A set of positive and negative whole numbers, including zero. They can be represented on a number line.
THE NUMBER LINE
| Negative Numbers | 0 | Positive Numbers |
|---|---|---|
| -5 -4 -3 -2 -1 | 0 | 1 2 3 4 5 |
Absolute Value
The distance a number is from zero on the number line. An absolute value is never negative. Examples:
- | -5 | = 5
- | 5 | = 5
Negative of a Number
- Definition: The negative of a number is its value on the opposite side of zero on the number line.
- Notation: Denoted by a minus (-) sign in front of the number.
- Rule:
- The negative of a positive number is negative.
- The negative of a negative number is positive.
- Examples:
- Negative of +5 is -5.
- Negative of -3 is +3.
- Zero: The negative of 0 is 0 itself.
The Sign of Zero
- Neutral: Zero is neither positive nor negative.
- Identity Element: In addition, zero acts as an identity element; any number plus zero is the number itself.
- Multiplicative Effect: Any number multiplied by zero equals zero.
Comparing and Ordering Integers
- Direction: Numbers to the right are greater, to the left are smaller.
- Zero: The dividing point; positive numbers are greater than zero, negative numbers are less.
- Comparison:
- Use "<" (less than) or ">" (greater than) to compare.
- Example: -3 < 2 (read as "-3 is less than 2").
- Ordering:
- Arrange from smallest to largest or vice versa.
- Example: -5, -2, 0, 1, 3
Integer Operations
ADDING INTEGERS
SAME SIGN
Add and Keep the Sign!
- Add the absolute value of the numbers and keep the same sign.
- (positive) + (positive) = Positive
- (+4) + (+5) = +9
- (negative) + (negative) = Negative
- (-4) + (-5) = -9
DIFFERENT SIGNS
Subtract and Keep the Sign of the Bigger Number!
- Subtract the absolute value of the numbers and keep the sign of the bigger number.
- (-4) + (+5) = +1
- (+4) + (-5) = -1
MULTIPLYING INTEGERS
SAME SIGNS
- POSITIVE
- Multiply the numbers. Answer will be positive.
- (-5) × (-5) = +25
DIFFERENT SIGNS
- NEGATIVE
- Multiply the numbers. Answer will be negative
- (+5) × (-5) = -25
SUBTRACTING INTEGERS
Do not subtract integers. You must change the signs: "Add the Opposite"
- KEEP- Keep the sign of the first number
- CHANGE- Change the subtraction sign to addition
- CHANGE- Change the sign of the second number to the opposite sign. If it is positive- change to negative. If it is negative- change to positive.
- (+4) – (-4) becomes (+4) + (+4) = 8
DIVIDING INTEGERS
SAME SIGNS
- POSITIVE
- Divide the numbers. Answer will be positive.
- (-5) ÷ (-5) = +1
DIFFERENT SIGNS
- NEGATIVE
- Divide the numbers. Answer will be negative
- (+5) ÷ (-5) = -1
Integer Division Exceptions
- Fraction Results: When dividing two integers does not result in a whole number, the result is a fraction or a decimal. Example: (7 \div 2 = 3.5).
- Dividing by Zero: Division by zero is undefined. It's a rule in mathematics that you cannot divide a number by zero.
Integer Properties
Commutative Property
- Addition: a + b = b + a
- Example: 3 + (-5) = (-5) + 3
- Multiplication: a b = b a
- Example: (-4) 2 = 2 (-4)
Associative Property
- Addition: (a + b) + c = a + (b + c)
- Example: (3 + 4) + (-2) = 3 + (4 + (-2))
- Multiplication: (a b) c = a (b c)
- Example: (-3 2) 4 = -3 (2 4)
Distributive Property
- The sum of two numbers times a third number is equal to the sum of each addend times the third number.
- a (b + c) = a b + a * c
- Example: 2 (3 + (-4)) = 2 3 + 2 * (-4)
Identity Property
- Addition: Adding zero to any integer does not change its value.
- a + 0 = a
- Example: (-5) + 0 = -5
- Multiplication: Multiplying any integer by one does not change its value.
- a * 1 = a
- Example: 7 * 1 = 7
Inverse Property
- Addition: For every integer a, there exists an integer -a such that a + (-a) = 0.
- Example: 5 + (-5) = 0
- Multiplication: For every nonzero integer a, there exists a reciprocal 1/a such that a * 1/a = 1. (Note: This is more relevant for rational numbers rather than just integers, as the reciprocal of an integer may not be an integer unless a = ±1.)
Advanced topics
Prime and Composite Numbers
- Prime Numbers: Integers greater than 1 that have only two divisors: 1 and themselves.
- Examples: 2, 3, 5, 7.
- Composite Numbers: Integers that have more than two divisors.
- Examples: 4, 6, 8, 9.
- Special Case: 1 is neither prime nor composite.
Even and Odd Integers
- Even Integers: Integers divisible by 2. They end in 0, 2, 4, 6, or 8.
- Examples: -4, 0, 2, 6.
- Odd Integers: Integers not divisible by 2. They end in 1, 3, 5, 7, or 9.
- Examples: -3, 1, 5, 7.
Simple Factorization
- Definition: Breaking down an integer into a product of its factors.
- Purpose: To identify all integers that divide evenly into the original number.
- Example: The factors of 6 are 1, 2, 3, and 6.
- 6 can be represented as 1 6 or 2 3.