Converse, Inverse, Contrapositive

Given an if-then statement "if p, then q", we can create three related statements:

A conditional statement consists of two parts, a hypothesis in the “if” clause and a conclusion in the “then” clause.  For instance, “If it rains, then they cancel school.” 
              "It rains" is the hypothesis.
              "They cancel school" is the conclusion.

To form the converse of the conditional statement, interchange the hypothesis and the conclusion.
            The converse of "If it rains, then they cancel school" is "If they cancel school, then it rains."

To form the inverse of the conditional statement, take the negation of both the hypothesis and the conclusion.
            The inverse of “If it rains, then they cancel school” is “If it does not rain, then they do not cancel school.”

To form the contrapositive of the conditional statement, interchange the hypothesis and the conclusion of the inverse statement. 
            The contrapositive of "If it rains, then they cancel school" is "If they do not cancel school, then it does not rain."

Statement	If p, then q.
Converse	If q, then p.
Inverse	If not p, then not q.
Contrapositive	If not q, then not p.

If the statement is true, then the contrapositive is also logically true. If the converse is true, then the inverse is also logically true.

Example 1:

Statement	If two angles are congruent, then they have the same measure.
Converse	If two angles have the same measure, then they are congruent.
Inverse	If two angles are not congruent, then they do not have the same measure.
Contrapositive	If two angles do not have the same measure, then they are not congruent.

In the above example, since the hypothesis and conclusion are equivalent, all four statements are true. But this will not always be the case!

Example 2:

Statement	If a quadrilateral is a rectangle, then it has two pairs of parallel sides.
Converse	If a quadrilateral has two pairs of parallel sides, then it is a rectangle.(FALSE!)
Inverse	If a quadrilateral is not a rectangle, then it does not have two pairs of parallel sides. (FALSE!)
Contrapositive	If a quadrilateral does not have two pairs of parallel sides, then it is not a rectangle.

Classification of Numbers

In mathematics, you'll see many references about numbers. Numbers can be classified into groups and initially it may seem somewhat perplexing but as you work with numbers throughout your education in math, they will soon become second nature to you. You'll hear a variety of terms being thrown at you and you'll soon be using those terms with great familiarity yourself. You will also soon discover that some numbers will belong to more than one group. For instance, a prime number is also an integer and a whole number. Here is a breakdown of how we classify numbers:

Natural Numbers
Natural numbers are what you use when you are counting one to one objects. You may be counting pennies or buttons or cookies. When you start using 1,2,3,4 and so on, you are using the counting numbers or to give them a proper title, you are using the natural numbers.

Whole Numbers
Whole numbers are easy to remember. They're not fractions, they're not decimals, they're simply whole numbers. The only thing that makes them different than natural numbers is that we include the zero when we are referring to whole numbers. However, some mathematicians will also include the zero in natural numbers and I'm not going to argue the point. I'll accept both if a reasonable argument is presented. Whole numbers are 1, 2, 3, 4, and so on.

Integers
Integers can be whole numbers or they can be whole numbers with a negative signs in front of them. Individuals often refer to integers as the positive and negative numbers. Integers are -4, -3, -2, -1, 0, 1, 2, 3, 4 and so on.

Rational Numbers
Rational numbers have integers AND fractions AND decimals. Now you can see that numbers can belong to more than one classification group. Rational numbers can also have repeating decimals which you will see be written like this: 0.54444444... which simply means it repeats forever, sometimes you will see a line drawn over the decimal place which means it repeats forever, instead of having a ...., the final number will have a line drawn above it.

Irrational Numbers
Irrational numbers don't include integers OR fractions. However, irrational numbers can have a decimal value that continues forever WITHOUT a pattern, unlike the example above. An example of a well known irrational number is pi which as we all know is 3.14 but if we look deeper at it, it is actually 3.14159265358979323846264338327950288419.....and this goes on for somewhere around 5 trillion digits!

Real Numbers
Here is another category where some other of the number classifications will fit. Real numbers include natural numbers, whole numbers, integers, rational numbers and irrational numbers. Real numbers also include fraction and decimal numbers.

In summary, this is a basic overview of the number classification system, as you move to advanced math, you will encounter complex numbers. I'll leave it that complex numbers are real and imaginary.

Integers

Converse, Inverse, Contrapositive

Given an if-then statement "if p, then q", we can create three related statements:

Here is your cheat sheet to help you remember what to do with positive and negative numbers (integers) with adding, subtracting, multiplying and dividing.
Time Required: 20 Minutes & Practice
Here's How:
1.         Adding Rules:
Positive + Positive = Positive: 5 + 4 = 9
Negative + Negative = Negative: (- 7) + (- 2) = - 9
Sum of a negative and a positive number: Use the sign of the larger number and subtract
(- 7) + 4 = -3
6 + (-9) = - 3
(- 3) + 7 = 4
5 + ( -3) = 2
2.         Subtracting Rules:
Negative - Positive = Negative: (- 5) - 3 = -5 + (-3) = -8
Positive - Negative = Positive + Positive = Positive: 5 - (-3) = 5 + 3 = 8
Negative - Negative = Negative + Positive = Use the sign of the larger number and subtract (Change double negatives to a positive)
(-5) - (-3) = ( -5) + 3 = -2
(-3) - ( -5) = (-3) + 5 = 2
3.         Multiplying Rules:
Positive x Positive = Positive: 3 x 2 = 6
Negative x Negative = Positive: (-2) x (-8) = 16
Negative x Positive = Negative: (-3) x 4 = -12
Positive x Negative = Negative: 3 x (-4) = -12
4.         Dividing Rules:
Positive ÷ Positive = Positive: 12 ÷ 3 = 4
Negative ÷ Negative = Positive: (-12) ÷ (-3) = 4
Negative ÷ Positive = Negative: (-12) ÷ 3 = -4
Positive ÷ Negative = Negative: 12 ÷ (-3) = -4
Tips:
1.         When working with rules for positive and negative numbers, try and think of weight loss or poker games to help solidify 'what this works'.
2.         Using a number line showing both sides of 0 is very helpful to help develop the understanding of working with positive and negative numbers/integers.
What You Need
·         Calculator
·         Free Worksheets Below
·         An Understanding of the 4 Operations
·         A Number Line

·         Pencil and Eraser

Prime Numbers

Prime numbers are the numbers that are bigger than one and cannot be divided evenly by any other number except 1 and itself. If a number can be divided evenly by any other number not counting itself and 1, it is not prime and is referred to as a composite number. Prime numbers are whole numbers that must be greater than 1. Zero and one are not considered prime numbers. Learn how to determine which numbers are prime. Remember, when talking about prime numbers, we are referring to whole numbers.

Difficulty: Average

Time Required: Depends on Your Grade or Previous Math Experience with Prime Numbers

Here's How:

1.         Factorization. Understand what a factor is before you start to work with prime numbers. Let's take the number 20. 
Let's factor 20:
2 x 2 x 5
2 and 5 are the prime factors for 20.
This means that 20 is not a prime number.
Factors are the numbers multiplied to get the product.
We can also find more factors of 20:
5 x 4 and 2 x 10. 
All the factors of 20 are: 1, 2, 4, 5, 10.

One more. Is 9 a prime number?
Let's factor 9:
3 x 3
Factors of 9 are 1, 3 and 9.
9 is not a prime number.

2.         Separate Piles:
When learning about prime numbers, one of the easiest methods to start is to work with numbers in a concrete method. Use buttons, coins, dried beans etc. Start with numbers less than 100. For each number, count out that many objects.

For instance, if you want to find out if 27 is prime, begin by counting out 27 objects.
Task 1: Can 27 objects be evenly divided into 2 piles?
No
Can 27 objects be evenly divided into 3 piles?
Yes!
There are 3 piles of 9. This means that 3 and 9 are prime factors of 27. Therefore 27 is not a prime number.
Practice with objects on a variety of numbers to help understanding.

3.         Calculator Method: After using the concrete method (buttons, coins etc.) and trying to separate the 17 or 23 coins evenly into 2 or 3 piles,then try the calculator method. After all, with any concept, concrete methods should be used first.

Take your calculator and key in the number you are trying to determine is prime. Let's take 57. Divide it by 2. Does it come out to an even number? No, you'll discover it's 27.5. Now divide 57 by 3. Is it even? You will see 19 which is an even number. Is 57 prime? Yes, 19 and 3 are its factors. When using the calculator to determine prime, begin by dividing it by 2 or 3 first.

4.         Divisibility: Know your divisibility rules.
Try 2: Any number ending in 2 is an even number and it will be divisible by 2, therefore it is not prime.
Try 3. Take the number, and add the digits up, when those digits are divisible by 3, the number is not prime. Take 2469, those digits add up to 21, and 21 is divisible by 3, therefore 2469 is not a prime number.
Try 4. Take the last 2 digits of the number, is it divisible by 4? Then the number is divisible by 4. Try 8336, the last 2 digits are 36 which are divisible by 4, therefore 8336 is not prime.
Try 5: any number ending in 0 or 5 is divisible by 5 and not prime.

5.         Prime Number Calculator I don't recommend using the prime number calculator until the concept of prime numbers is fully understood. If that's the case, then prime number calculators are a quick and easy method to determine if a number is prime or not.

6.         Prime Factorization Trees This method is similar to factorization. Take a look at the image of the factor trees. There are 3 numbers being factored: 32, 21 and 40. There is more than one way to factor numbers, however, eventually you will factor until you've reached all the common factors. For instance, if I am factoring the number 30. I could begin with 10 x 3 or 15 x 2. In each case I will continue to factor 10 (2 x 5) and I will continue to factor 15 (3 x 5) and the end resulting prime factors will be the same: 2, 3 and 5. After all, 5 x 3 x 2 = 30 as does 2 x 3 x 5.

7.         Division This method is similar to the calculator method. Take your number and try to divide it by 2, if not, then try 3, if not try 4, if not try 5. This can be time consuming and not something you would want to use with very large numbers. However, for somebody starting out with prime numbers, this is a great method to help with the understanding of what makes a prime number prime.

8.         Confusing for Learners:

When working with prime numbers it's important that students know the difference between factors and multiples. These two terms are easily confused by learners.