1.1 – The Real Numbers – Real Numbers

Video 1 of 4: Real Numbers
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Video 2 of 4: Properties of Fractions

Video 3 of 4: Set Notation, Unions and Intersection

Video 4 of 4: Repeating Decimals to Fractions

Real Numbers

The Real Numbers are a collection of several different types of numbers. Before we define the Real Numbers, will define a few different sets of numbers. These numbers are:

Natural Numbers: Natural numbers are the “ordering” numbers:

1, 2, 3, 4, …

These numbers go on forever to the right, which is indicated by a ellipsis (…) The natural numbers are often collectively known by their symbol $\mathbb{N}$ . A stylized N that stands for “Natural”

Whole Numbers: The whole numbers are simply the Natural Numbers, along with zero:

0, 1, 2, 3, …

These numbers are often called “counting numbers” as they can depict how many of something there is.

Integers: The integers are the positive and negative whole numbers (there is no negative zero, or rather, negative zero is still zero):

…, -3, -2, -1, 0, 1, 2, 3, …

These numbers continue on forever in both directions. Often, the set of integers is referred to by its symbol $\mathbb{Z}$ . The letter Z comes from the German word for number: “Zahlen”

Rational Numbers: are the set that includes all numbers that can be expressed as a fraction (ratio) of two integers. That is, all numbers that can be expressed as $\frac{a}{b}$ where $a$ and $b$ are integers. The set of rational numbers is often denoted by its symbol $\mathbb{Q}$ . It is a stylized Q which stands for “Quotient”. Examples of rational numbers are:

$\dfrac{1}{2}, \ \ \ \dfrac{2}{7}, \ \ \ \dfrac{723}{102458}$

Rational numbers also include all integers and decimals that repeat or terminate, but do not include decimals that continue forever without repeating.

Irrational Numbers:are numbers that cannot be written as the ratio of two integers. Examples of irrational numbers are $\pi\approx3.1415927$ . This is a decimal approximation of $pi$ . The exact value of $\pi$ is an infinitely long string of numbers that never repeats itself. Because of this $\pi$ can not be expressed exactly as the ratio of two integers, and is thus, an irrational number. Another example of an irrational number is $\sqrt{2}\approx 1.41421356$ . Again, this is a decimal approximation of $\sqrt{2}$ , a value which again is an infinitely long string of non repeating numbers. A number is said to be irrational if its decimal value never terminates and never repeats.

Real Numbers: Finally, we have the real numbers. The real numbers is the collection of rational and irrational numbers. This is all the numbers along the number line. This set is often referred to by it’s symbol $\mathbb{R}$ a stylized capital R. This stands for “Real”.

Properties of Real Numbers

Commutative Properties

The order in which real numbers are added or multiplied does not matter.

$a+b=b+a$
$ab=ba$

Associative Properties

The grouping of real numbers in addition or multiplication does not matter.

$(a+b)+c=a+(b+c)$
$(ab)c=a(bc)$

Distributive Property

$a(b+c)=ab+ac$
$(b+c)a=ab+ac$

Properties of Negatives

1. $(-1)a=-a$
2. $-(-a)=a$
3. $(-a)b=a(-b)=-(ab)$
4. $(-a)(-b)=ab$
5. $-(a+b)=-a-b$
6. $-(a-b)=b-a$

Properties of Fractions

1. $\frac{a}{b}\cdot\frac{c}{d}=\frac{ac}{bd}$
2. $\frac{a}{b}\div\frac{c}{d}=\frac{a}{b}\cdot\frac{d}{c}$
3. $\frac{a}{c}+\frac{b}{c}=\frac{a+b}{c}$
4. $\frac{a}{b}+\frac{c}{d}=\frac{ad+bc}{bd}$
5. $\frac{ac}{bc}=\frac{a}{b}$
6. If $\frac{a}{b}=\frac{c}{d}$ , then $ad=bc$