Introduction to Real Numbers. Algebra II is an intermediate algebra course
that is designed to refresh and refine the skills and concepts studied in your introductory
algebra class. With a greater emphasis on synthesizing the concepts with a higher level
of critical thinking. In addition, you will be introduced to new functions and techniques
to solve more complicated equations. Before we refine our skills let’s make sure you
have the basics down. Let’s start with the real number system.
Mathematicians like to place objects with
similar properties in sets. A set is a collection of objects. For example a set of a standard
deck of cards contains 52 cards. Each of those cards or objects of the set are called the
elements of the set so a standard deck of cards contains 52 elements. Sets are denoted
by placing braces around the elements in the set. Just like this.
The numbers that we use to count things are called the natural numbers and are represented
by the following set. These include the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 and so on.
Obviously there are more numbers than 11 but it is impossible to list them all so we use
ellipsis to denote that the elements of the set continue on indefinitely forever and ever
with the implied pattern. This means that the number 12, 17, 100, and 1000 are also
part of the set even though they are not explicitly enclosed within the braces.
Each natural number is either a prime number or a composite number. A Prime number is
a number greater than 1 that is divisible (evenly) only by itself and 1. For example
the numbers 2, 3, 5, 7, 11 and 13 are the first 6 prime numbers.
A natural number that is not prime is called a composite number. The numbers 4, 6, 8, and
9 are the first 4 composite numbers notice that composite numbers are divisible by itself
and other numbers, for example the number 8 is divisible by itself evenly, you can also
divide 8 by 2, and 4 which also divides evenly hence it is not a prime number but a composite number.
If you look back at the set of the natural
numbers notice that the set does not contain a symbol to denote the concept of none. When
you add zero into the set of natural numbers we obtain the set of whole numbers.
The whole numbers alone do not provide all the numbers that are useful in various applications.
We also need numbers below zero and above zero.
The Set of integers include negative integers such as -5, -4, -3, -2, and -1 the number,
zero, and the positive integers such as 1, 2, 3, 4, and 5.
Note that the natural numbers and the positive integers are the same set of numbers. The
integer zero is neither a positive nor a negative integer. So keep that in mind.
We still need other numbers to solve other types of problems especially those that require
fractions. The numbers that include fractions are called rational numbers. Rational numbers
have the form p over q where p and q are integers and q cannot equal 0 because division by zero
is undefined in algebra. In more advance mathematics courses division by zero can be interpreted
in other ways. For all intended purposes, in your algebra class division by zero is
undefined. Examples of rational numbers include the number one third, the number
negative 8 over 5, and the number 5 over 1. Note that 5 over 1 reduces to 5, so this means that
all integers are rational numbers. A number like 4 over pi is not a rational number because
pi is not an integer. Remember rational numbers have the form p over q where both the numerator
and denominator have to be integers. In addition, remember that rational numbers written as
a fraction can be rewritten in decimal notation by dividing the numerator with the denominator
for example 3/8 as a decimal is equal 0.375 or .375 and 2/15 as a decimal is equal to 0.13333333…
or written differently as 0.13 with a bar on top the bar over the 3 indicates that this digit repeats indefinitely.
Likewise the fraction 11/13 can be written as 0.846153 and all those numbers with a bar on top.
Notice that there are two types of rational numbers when you write them in decimal form.
Either the decimal representation terminates like the rational number 3/8 or its decimal
representation goes indefinitely like 2/15 and 11/13. Another characteristic of rational
numbers whose decimal representation go indefinitely is the fact that they have a repeating pattern
that’s why they contain a bar on top on those numbers that repeat over and over again.
It turns out that there are actually some numbers whose decimal point representation
do not terminate or have a repeating pattern these numbers are called irrational numbers
they cannot be written as a rational number of the form p over q, these are numbers
like pi, the square root of 7, and the square root of 3. The decimal representation of these
numbers do not have a nice pattern that they follow and for the most part they are rounded
of when we need to use them In calculations. Together the set of rational and irrational
numbers form what we call the real numbers.
So the real numbers include the set of rational numbers and irrational numbers.
The graph of a real number is made by placing
a bold dot on a number line directly above the number. The graph of some real numbers
are shown below. Alright let’s recap and visually show
the relationship among various sets and subsets of the real numbers.
The real numbers represented by this symbol are divided into rational and irrational numbers,
furthermore the rational numbers are broken up into integers and non integers, in addition,
the integers are broken up into positive integers also known as natural numbers, negative integers
and the number zero, when you include zero into the set of positive integers we have
the set of whole numbers. Lastly, the natural numbers can be divided into prime and composite
numbers. And there you have It the real numbers at
a glance, throughout your intermediate algebra course you will learn about imaginary numbers
and numbers that combine both real and imaginary numbers called complex numbers and this graph
will look a little different very soon. For now be aware that these are the numbers
that you will be playing around with throughout your studies of intermediate algebra.
Alright in our next video we will go over Inequalities and the absolute value of a number.