Natural Numbers/Counting Numbers (N)




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§ 1.2 Symbols and Sets Numbers
Objectives

  • Sets of Numbers

  • Number Line

  • Translation

  • Absolute Value


Sets are collections of objects. In mathematics we deal with many sets of numbers. One method of showing a set, called roster form, lists the set within braces, in the following way
{object1, object2, object3, …}
The ellipsis (…) indicate that a set continues on in the manner indicated infinitely.
Another method of listing sets is called set builder notation. In this method a set is described using braces, a variable, a straight line “ | ” (meaning such that) and then a description of the set. This method will be demonstrated shortly.
Some sets of numbers that we deal with in mathematics:

Natural Numbers/Counting Numbers (N)

{ 1 , 2 , 3 , 4 , 5 , … } The counting numbers excluding zero

Note: These can also be called the positive numbers and can be

written { + 1 , + 2 , . . . } but the positive sign is assumed

and need not be written unless it is unclear what we mean.
Whole Numbers (W)

{ 0 , 1 , 2 , 3 , 4 , . . . } The counting numbers with zero

Example: {x | x  N or 0} is a way to describe the Whole Numbers

that uses the Natural Number and

demonstrates set builder notation.

Integers (I)

{ . . . , - 3 , - 2 , - 1 , 0 , 1 , 2 , 3 , . . . } All positive whole numbers, their

opposites and zero
Rational Numbers (R)

{x | x  p/q , p,q  I, q  0} All numbers that can be represented as the

quotient of integers




Example: 2/3, 5, 0, - 1/2, 0.33333
Irrational Numbers (Q)

All numbers that can't be represented as the quotient of

integers, i.e. numbers that when represented in decimal form are non-terminating,

non-repeating decimals
Example: 2 (read the square root of 2)
Example:  (this is the symbol that represents pi,

approximately 3.14)

Real Numbers ()

All numbers that can be represented on a number line, i.e. all

rational and irrational numbers.
A number line is a continuing line that represents the real numbers. At the center is zero, moving to the left from zero are the negative numbers (getting smaller and smaller) and moving to the right from zero are the positive numbers (getting larger and larger). This is known as the order property of the real numbers.






We may wish to make a statement about the relationship of two or more numbers pictured on a number line. We call such a statement a mathematical statement. Mathematical statements may or may not be true, but we will always use the following symbols to make a mathematical statement.
Inequality Symbols

 Does Not Equal

< Strictly Less Than

> Strictly Greater Than

 Less Than or Equal To

 Greater Than or Equal To
Some helpful hints in reading the greater than and less than symbols:

The arrow points to the smaller number.

Put teeth in the mouth and the smaller eats the larger (you may have learned this as a

child, and therefore I pose this here to jog a memory!)
We have both strictly less than and less than or equal to, and both can be used to refer to the same case, but the less than or equal to can also indicate equality. The less than or equal to refers to a case in which either the strict inequality or the equality holds true. For a case where we would compare 7 and 15, either the < or  will work. We see the usefulness of the  and  later when we come to solution sets of linear inequalities.
Example: Compare the following using <, > or =




a) -10 -100 b) 18/3 24/3



c) 5.2 5.1 d) -3.25 -3.2




e) 2/5 4/7 f) -2/5 -4/7

Another property of the number line is that there is always one unit between each whole number that is marked on the number line. This gives us some insight into the idea of absolute value. The absolute value of a number is defined as the number of units a number is away from zero on the number line. Absolute value does not take into account whether a number is positive or negative – it is strictly the number without its sign! We indicate absolute value in the following way

| number | the absolute value of number
Example: What is the absolute value 7 ?

Example: | - 15 |


Example: Write the absolute value of - 7 in symbols.
Now we will combine the last concept (comparing numbers) with the absolute value.
Example: Compare the following using <, > or =
a) | -1 | | 1 | b) | -8 | | 0 |



c) | -5 | -4

Supplemental Information to Section 1.2
A subset is a set which is contained by another set. This means that all the members of one set are contained within the other set. A set is a subset of itself as well. The following symbol is used to indicate that one set is a subset of another

subset of
Example: Write using symbols: The integers are a subset of the real numbers

Example: Is the following statement true or false?

The whole numbers are a subset of the natural numbers

The following symbol is used to tell us that a set contains no elements .

 or { } the null set or empty set
Example: Write the answer to the following using symbols.

The set of negative numbers contained within the

natural numbers.


§1.3 Fractions

First, we need to review some vocabulary for fractions. Recall that
2 Numerator

3  Denominator
Remember also that a fraction can represent a division problem!
Example: What is the numerator of 5/8 ?

Example: What is the denominator of 19/97 ?

Fractions represent a part of something. The numerator represents how many pieces of the whole are represented. The denominator tells us how many pieces that the whole has been divided into.
Example: For the picture drawn on the board:

a) Represent the shaded area as a fraction

b) Represent the unshaded area as a fraction


We like to represent fractions in what we refer to as lowest terms, which means that the numerator and denominator have no factors in common except one. There are two technical ways of putting a fraction into lowest terms. The first way uses greatest common factors and the other uses prime numbers.
The greatest common factor method goes as follows:

Step 1: Find the GCF of numerator and denominator

  1. List all factors of numerator and denominator

  2. Find the largest (greatest) that both have in common

Step 2: Factor the numerator and denominator using GCF

Step 3: Cancel the GCF from the denominator and numerator

Step 4: Rewrite the fraction
Example: Reduce 12/24 to its lowest terms.
Step 1: 12 – 1, 2, 3, 4, 6, 12

24 – 1, 2, 3, 4, 6, 8, 12, 24

GCF = 12

Step 2: 12 = 12 1

24 12  2

Step 3: Cancel the 12's

Step 4: Rewrite 1 .

2
Example: Reduce 42/45 to its lowest terms.


Before discussing the second method, we need to discuss the two types of numbers. All numbers are either composite or primes. Composite numbers have more factors other than one and themselves. Said another way, each composite number contains 1 and itself as factors as well as at least one other number. A prime number has only 1 and itself as factors.
In order to find all the prime factors of a composite number, we will use a method called prime factorization. The method goes like this: 1) What is the smallest prime number that our number is divisible by? 2) What times that prime gives our number? 3) Once we have these two factors we circle the prime number and focus on the one that isn’t prime. 4) If there is one that isn’t prime, we ask the same two questions again, until we have found all the prime numbers that our number is divisible by. 5) Then we rewrite our composite number as a product of all the circled primes. 6) Finally, we can use exponential notation to write them in a simplified manner. When multiplied together all the primes must yield the composite number or there is an error.
12

/ \

2 6

/ \

2 3 12 = 223 = 22  3


Whether you use a factor tree as I have here, or use one of the other methods is up to you, but I find that the very visual factor tree works nicely.
Example: Find the prime factorization of 15 and 24

The prime factorization method goes like this:

Step 1: Factor numerator and denominator into prime factors

Step 2: Cancel all factors in common in both numerator and denominator.

Step 3: Rewrite the fraction.
Example: Reduce 12/24 to its lowest terms.
Step 1: 12 = 2 2 3 .

24 2  2  2  3

Step 2: Cancel the 2 of the 2's and the 3's

Step 3: Rewrite 1/2 .
Example: Reduce 42/45 to its lowest terms.

In order to add and subtract fractions, you must also know how to build a higher term. To build a higher term you must know the Fundamental Theorem(Principle) of Fractions. Essentially this principle says that as long as you do the same thing (multiply or divide by the same number) to both the numerator and denominator you will get an equivalent fraction. Here it is in symbols:
Fundamental Theorem (Principle) of Fractions

a c = a or a c = a

b  c b b  c b

This is used to build an equivalent fraction. An equivalent fraction is a fraction which represents the same quantity. For example:

1/4 and 2/8

are equivalent fractions.
To create an equivalent fraction we use the fundamental principle of fractions. Here is a process:

Step 1: Decide or know what the new denominator is to be.

Step 2: Use division to decide what the "c" will be as in the fundamental

principle of fractions.

Step 3: Multiply both the numerator and denominator by the "c"

Step 4: Rewrite the fraction.
Example: Write an equivalent fraction to

a) 2/3 with a denominator of 9.

b) 8/15 with a denominator of 30.

Now, let's review how to multiply fractions. Multiplying fractions is very easy, but should never be confused with adding fractions.

Step 1: Cancel if possible

Step 2: Multiply numerators

Step 3: Multiply denominators

Step 4: Reduce/Change to mixed number if necessary
Example: Multiply.

a) 2/35/7 b) 3/8 x 2/5

What if we wish to multiply mixed numbers? If we wish to multiply mixed numbers we must first convert them to improper fractions. Let's recall how:

Step 1: Multiply the whole number and the denominator

Step 2: Add the numerator to the product

Step 3: Put the sum over the denominator
Example: Multiply. (11/2)( 1/2 )


Sometimes as a result of multiplying two mixed numbers we may get an improper fraction and we may need to convert it to a mixed number. It is always easiest to convert to a mixed number when the improper fraction is in its lowest terms. These are the steps to converting an improper fraction to a mixed number:

Step 1: Reduce the improper fraction to its lowest terms

Step 2: Divide the denominator into the numerator (numerator ÷ denominator)

Step 3: Write the whole number and put the remainder over the denominator.
Example: Multiply 3 2/3 ( 1/2 )
Before we discuss dividing fractions we must define a reciprocal. A reciprocal can be defined as flipping the fraction over, which means making the denominator the numerator and the numerator the denominator. Another way that I frequently speak of taking a reciprocal is saying to invert it. The actual definition of a reciprocal is the number that when multiplied by the number at hand, will yield the identity element of multiplication (one).

To divide fractions, we must use the following steps:

Step 1: Invert the divisor (that is the second number; the one that you're dividing by)

Step 2: Multiply the inverted divisor by the dividend (the first number; the number

that you are dividing into pieces)

Step 3: Reduce the answer if necessary.
Example: Divide

a) 5/82/3 b) 5/83/4




c) 5/83/5 d) 4 3/7 ÷ 31/7

e) 7/8 ÷ 3 1/4

Now, let's discuss addition and subtraction of simple fractions. To add and subtract fractions with common denominators all that must be done is to add or subtract the numerators and carry along the common denominator.
Example: Add

a) 4/51/5 b) 23/105 + 4/105
In order to add or subtract fractions with unlike denominators we must first find a common denominator. The best way to do this is to find the least common denominator (LCD) which is the least common multiple (LCM). The LCM is the lowest number which both denominators go into or said another way is the lowest multiple that all numbers have in common.
Let’s outline and practice the best method for finding an LCM/LCD.

Step 1: Find the prime factorization of all denominators, writing in exponential

notation

Step 2: Write all the unique prime numbers in the prime factorizations

Step 3: Write the primes to their highest exponent

Step 4: Multiply

Example: Find the LCD of 22 and 33.

Here are the steps that you use in order to add two fractions that do not have common denominators:

Step 1: Find the LCM/LCD

Step 2: Build equivalent fractions using LCM/LCD

Step 3: Add or subtract the new fractions

Step 4: Simplify by reducing to lowest terms and/or changing to a mixed #
Example: Add/Subtract.

a) 5/225/33 b) 1/3 + 2/5

c) 1/4 + 2/3 d) 5/8 + 3/4

What if we need to add or subtract mixed numbers or fractions from whole numbers? Then we have two methods of accomplishing our task. The first method is changing a mixed number into an improper fraction. We already discussed how to change a mixed number into an improper fraction in our discussion of multiplication.
Example: 11/5 + 23/5

The second method is to add or subtract the whole numbers, and then to add or subtract the numerators of the fractions (provided that they are common denominators – if they aren’t then they need to be made into equivalent fractions with the LCD). There are two problems that are likely to arise in using this method. The first is that the fractions when added will be more than one whole, in which case we will need to recall that a mixed number such as 11/4 means 1 + 1/4 and therefore we can convert the improper fraction to a mixed number and add it to the whole number.
Example: Add and notice what happens with the fractional portion:

13/4 + 51/4
The other case is if the fraction we are subtracting from is smaller than the fraction being subtracted. In this case we must borrow.

Example: Subtract and notice what happens in trying to subtract the fractions

(Remember subtraction is not commutative so you can’t subtract 3 from 2 and get 1 and we don’t

want to get a negative one either!)

52/5  23/5

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