Types of Numbers and Algebraic Properties

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Types of Numbers

Before we get too deep into algebra, we need to talk about the types of numbers there are out there. We saw a few of these earlier, and you may not have seen all these types of numbers yet, but you will have to learn them in school. The letters in parentheses indicate how they are abbreviated sometimes. Sorry, it’s not the most exciting stuff to learn….

The types of numbers we’ll talk about include Whole Numbers, Counting Numbers or Natural Numbers, Integers, Rational Numbers, Irrational Numbers, Real Numbers, Imaginary Numbers, and Complex Numbers:

Type of Number Examples Hints

Natural Numbers

(Counting Numbers)

 (\(\mathbb{N}\))

Numbers you use for counting: \(1, 2, 3,…\) It’s “natural” to count on your fingers: \(1, 2, 3,…\)
Whole Numbers The natural numbers, plus \(0: 0, 1, 2, 3, …\) The word “whole” has an “o” in it, so include 0.
Integers

 (\(\mathbb{Z}\))

Whole numbers, their opposites (negatives), plus \(0: … –2, –1, 0, 1, 2, …\) Integers can be separated into negative, 0, and positive numbers.
Rationals

(\(\mathbb{Q}\))

Integers and all fractions, positive and negative, formed from integers. These include repeating fractions, such as \(\displaystyle \frac{1}{3}\) or \(.33333…\) or \(.\overline{3}\). The word “rational” is a derivation of “ratio”, and rational numbers are numbers that can be written as a ratio of two integers. “Q” stands for quotient.
Irrationals Numbers that cannot be expressed as a fraction, such as \(\pi ,\,\sqrt{2},\,e\).  (We’ll learn about these later). If something is “irrational”, it’s not easy to explain or understand.

Real Numbers

(\(\mathbb{R}\))

Rational numbers and Irrational Numbers. The real number system can be represented on a number line:

If a number exists on a number line that you can see, it must be “real”.

Note that the “smallest” real number is negative \((-)\) infinity \((-\infty)\), and the largest real number is infinity \((\infty) \).

We can never really get to these “numbers” (\(-\infty \) and \(\infty \)), but we can indicate them as the “end” of the real numbers.

Complex Numbers

(\(\mathbb{C}\))

Real numbers, plus imaginary numbers (concept only, such as \(\sqrt{{-2}}=\sqrt{2}i\)). “Imaginary” numbers are difficult to imagine, since they are so “complex”.

 

Here’s a Venn Diagram that shows how the different types of numbers are related. Note that all types of numbers are considered complex. And don’t worry too much about the complex and imaginary numbers; we’ll cover them in the Imaginary (Non-Real) and Complex Numbers section.

Algebraic Properties

Before we get into Algebra, we also need to talk about some of the properties we’ll use to solve equations.  We’ll need these to get the variable all by itself on one side of the equal sign – which is the basis of algebra.

Let me show you this with just plain numbers — since these work with plain numbers, they also work with variables (letters)!

Algebraic Properties of Equality

There are what we call the Algebraic Properties of Equality, since they deal with two sides of an equal sign:

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Example

Property of Equality

 \(\begin{array}{c}5=5\\5+1=5+1\\\,\,\,\,\,\,\,6=6\,\,\,\,\,\surd \end{array}\) We add 1 to both sides and get \(6=6\). So addition works! Try it for other numbers. This is called the Additive Property of Equality, since we are adding the same thing to both sides of the equation.
 \(\begin{array}{c}5=5\\5-3=5-3\\\,\,\,\,\,\,\,\,2=2\,\,\,\,\,\surd \end{array}\) If we subtract 3 from both sides, it works, too! This is called the Subtraction Property of Equality, since we are subtracting the same thing from both sides of the equation.
 \(\displaystyle \begin{array}{c}5=5\\5\times 4=5\times 4\\\,\,\,\,\,\,\,\,\,20=20\,\,\,\,\,\surd \end{array}\) If we multiply both sides by 4, it works, too! This is called the Multiplicative Property of Equality, since we are multiplying the same thing to both sides of the equation.
 \(\displaystyle \begin{array}{c}6=6\\6\div 2=6\div 2\\\,\,\,\,\,\,\,\,3=3\,\,\,\,\,\surd \end{array}\) If we divide both sides by 2, it works too! This is called the Division Property of Equality, since we are dividing the same thing on both sides of the equation.

Commutative and Associative Properties

There are two more properties that will be very useful in solving algebra equations:

Example

Property

 \(\begin{array}{c}(5+4)+2=5+(4+2)\\9+2=5+6\\\,\,\,\,\,\,\,\,\,11=11\,\,\,\,\,\,\surd \end{array}\) If we group numbers differently (using parentheses), we are using the Associative Property.

This only works for addition and multiplication. Try it yourself with subtraction and division to see that it doesn’t work.

I remember this since you associate yourself with different groups.

 \(\begin{array}{c}\,8\times 4\times 2=4\times 2\times 8\\32\times 2=8\times 8\\\,\,\,\,\,\,\,\,\,\,\,\,\,64=64\,\,\,\,\,\,\surd \end{array}\) If we change the order of numbers or variables, we are using the Commutative Property.

This only works for addition and multiplication. Try it yourself with subtraction and division to see that it doesn’t work.

I remember this since the word “commutative” has an “o” for the second letter; this reminds me of “order”.

As an example of why the Associative and Commutative properties are important, we may need to use these to show that “\(5+4+2=2+2+7\)”:

\(\displaystyle \begin{align}5+4+2&=2+2+7\\5+4+2&=7+2+2\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{Commutative}\\\left( {5+4} \right)+2&=\left( {7+2} \right)+2\,\,\,\,\,\,\,\,\,\,\,\,\,\text{Associative}\\9+2&=9+2\\11&=11\end{align}\)

Distributive Property

There is one other property that is used a lot in algebra; this one is a little different.

Let’s say you are trying to take a collection for your sweet algebra teacher for an end of the year gift. You are collecting $10 from 10 girls and 8 boys. You can see that you will collect $180, but there are two different ways to solve this problem:

\($10×(10 + 8)=\$10×18=\$180\)

You add up the girls and boys first to get 18 people, and then multiply by $10.

OR

\((\$10×10)+(\$10×8)=\$100+\$80=\$180\)

You first collect the money from the girls to get $100, and then collect the money from the boys to get $80, and then add the two amounts together to get $180.

This is called the Distributive Property, since we can either leave the 10 on the outside of the parentheses, or distributethrough (or “push it through“) to both the numbers on the inside of the parentheses. We can do this when there is addition or subtraction inside the parentheses.

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Here is another example, and one using a variable:

Example

Distributive Property

\(\displaystyle \begin{align}4\times (2+3)&=4\times 5\,\,=4+4+4+4+4\\&=(4+4)+(4+4+4)\\&=4\times 2+4\times 3\,\,\,\,\,\surd \end{align}\) I just randomly divided up 5 into 2 and 3. When you multiply 4 times 5 (\(2+3\)), it is just like adding 4 five times. Because of the associative property above, you can lump together the first two 4‘s first and the last three 4‘s second.

 

This “proves” the Distributive Property.

 \(\displaystyle 2n+3n=n(2+3)=n(5)=5n\) Distributive Property again; you will see this again a lot in algebra. This is an example of “combining like terms”, or putting together the same variables.

Summary of Algebraic Properties

Since these are pretty important, here’s another table with these properties (and a couple more) with new examples:

Name Hints Examples Notes

Associative

“Grouping”

You “associate” with different groups.

\(\begin{array}{l}5+\left( {15+4} \right)\\\,\,\,=\left( {5+15} \right)+4\end{array}\)

Works with addition and multiplication, not subtraction or division.

Commutative

“Ordering”

Since Commutative has an “o” in it, think “order”.

\(\begin{array}{l}5+4+3\\\,\,=4+3+5\end{array}\)

Works with addition and multiplication, not subtraction or division.

Distributive

“Distributing or Pushing Through Parentheses”

Think of “distributing” something to your friends.

\(\displaystyle \begin{array}{*{20}{l}} \begin{array}{l}5\times \left( {3+4} \right)\\\,\,\,=\left( {5\times 3} \right)+\left( {5\times 4} \right)\\\,\,\,=15+20=35\end{array} \end{array}\)

 

\(\begin{array}{l}5-2\left( {x-3} \right)\\\,\,\,=5-2x+6\\\,\,\,=11-2x\end{array}\)

 

\(\begin{array}{l}5x+7x\\\,\,\,=\left( {5+7} \right)=12x\end{array}\)

When negatives are on the outside of the parenthesis, make sure you distribute the negatives to second number, too.

 

Remember that multiplying two negatives results in a positive.

Identity

“Staying the Same”

You always come back to your “identity”.

\(\begin{array}{c}9+0=9\\9\times 1=9\end{array}\)

Additive identity is 0.

 

Multiplicative identity is 1.

Inverse

“Undoing”

When you put your car in “inverse”, you go backwards.

\(\displaystyle \begin{align}9+-9=0\\9\times \frac{1}{9}=1\\\frac{8}{9}\times \frac{9}{8}=1\end{align}\)

Additive inverse is \(-a\), since \(-a+a=0\).

 

Multiplicative inverse is \(\displaystyle \frac{1}{a}\), since \(\displaystyle \frac{1}{a}\times \frac{a}{1}=1\); the multiplicative inverse of \(\displaystyle \frac{a}{b}\) is \(\displaystyle \frac{b}{a}\), since \(\displaystyle \frac{a}{b}\times \frac{b}{a}=1\). Multiplicative inverses are reciprocals.

There are actually other properties used in algebra that you’ll be learning, but these are the main ones you’ll be using to solve algebra problems. Remember that your goal in solving algebra problems is to get the variable or unknown to one side all by itself! We’ll see this in the Solving Algebraic Equations section.

Proper Algebraic Notation

One more boring thing we must talk about before we solve equations is proper algebraic notation, or “grammar”.

Just like English has proper “grammar”, math does to!  The proper way to write the solutions of equations (and inequalities, which we’ll learn shortly) is shown below.

Sets

A set of numbers (or anything!) is a collection of items that are called elements. A set can be finite, such as the numbers 1, 2, and 3 (written as \(\left\{ {1,2,3} \right\}\)). A set can also be infinite (with an unlimited set of numbers), such as the set of real numbers (including all the fractions) between 0 and 1.

Union and Intersection

The union of two or more sets includes everything in either of the sets. For example, the union of the sets \(\left\{ {1,2,3} \right\}\) and \(\left\{ {3,4,5} \right\}\) would be \(\left\{ {1,2,3,4,5} \right\}\), since you include everything in both sets, but don’t repeat numbers. You write union as \(\cup\), so \(\left\{ {1,2,3} \right\}\cup \left\{ {3,4,5} \right\}=\left\{ {1,2,3,4,5} \right\}\).

The intersection of two or more sets includes only those things that are in both sets.  For example, the intersection of the sets \(\left\{ {1,2,3} \right\}\) and \(\left\{ {3,4,5} \right\}\) would be \(\left\{ 3 \right\}\), since you include only the numbers in both sets, but don’t repeat the numbers. You write intersection as \(\cap \), so \(\left\{ {1,2,3} \right\}\cap \left\{ {3,4,5} \right\}=\left\{ 3 \right\}\).

The way these are written (with the brackets) are called roster notation, since you have a “roster” or list of numbers.

Set Builder, Inequality, and Interval Notation

Other notations are more useful and will be used by your teacher. These include set builder notation, inequality notation, and interval notation, as shown with examples. You can see how there may be many ways to show set builder notation.

Example Words or Equation

Set Builder/Inequality Notation

Interval Notation

 \(x=4\)

 \(\left\{ {x|x=4} \right\}\) or \(\left\{ {x:x=4} \right\}\)

This is read “the set of all \(x\) such that \(x\) equals 4”.

 \(\left[ 4 \right]\)

(Not typically used)

\(x\ne 6\)

 

All real numbers except 6

\(\left\{ {x|x\ne 6} \right\}\)

or

\(\left\{ {x\in \mathbb{R}|\,\,x\ne 6} \right\}\)

or

\(\left\{ {x|x\in \mathbb{R},\,\,\,x\ne 6} \right\}\)

This is read as “the set of all numbers \(x\) such that \(x\) is not 6 and \(x\) is a real number”.

\(\begin{array}{c}\left( {-\infty ,6} \right)\cup \left( {6,\infty } \right)\\\left( {-\infty ,6} \right)\,\,\text{or}\,\,\left( {6,\infty } \right)\end{array}\)

 

(Either way can be used)

All Real Numbers

 \(\left\{ {x|x\in \mathbb{R}} \right\}\)  \(\left( {-\infty ,\infty } \right)\)

No Solution

{ }

Ø (symbol for null set or “nothing”)

 \(x=4,5,6\)

\(\{4,5,6\}\)

  \(\left[ 4 \right]\cup \left[ 5 \right]\cup \left[ 6 \right]\)

(Not typically used)

 \(\displaystyle -2\le x<4\)

\(\left\{ {x|-2\le x<4} \right\}\)

or

\(\left\{ {x\in \mathbb{R}|-2\le x<4} \right\}\)

(Inequality notation)

\([2,4)\)

(The 2 is included, but the 4 is not)

 \(\displaystyle x\le -2\text{ or}\,\,\text{ }x>2\text{ }\)

\(\left\{ {x\in \mathbb{R}|\,\,x\le -2\,\,\text{or}\,\,x>2} \right\}\)

or

\(\left\{ {x|x\le -2\,\,\text{or}\,\,x>2} \right\}\)

(Inequality notation)

\(\left( {-\infty ,-2} \right]\text{ }\cup \text{ }\left( {2,\infty } \right)\)

\(\left( {-\infty ,-2} \right]\text{ or }\left( {2,\infty } \right)\)

Positive even numbers

\(\left\{ {x\in \mathbb{N}|\,\,x=2n} \right\}\)

or

\(\left\{ {x|\,\,x=2n\,\,\,\text{and}\,\,\,x\in \mathbb{N}} \right\}\)

Not applicable; not an interval

Note that for interval notation, if a “less than” (\(<\)) is used, we use “(“, if a greater than (\(>\)) is used, we use “)”. If a “less than or equal to” (\(\le \)) is used, we use “[“, and if “greater than or equal to” is used (\(\ge \)), we use “]”. Just remember that we used brackets (also called hard brackets) if we are including the endpoint (the actual number at the end), and we use parentheses (also called soft brackets) if we are not including the endpoint.

Also note that since we never actually get to \(\infty \) or \(-\infty \), we only use soft brackets (parentheses) with them.

Remember that \(\mathbb{R}\) means all real numbers (everything on the number line), and \(\mathbb{N}\) means natural numbers (1, 23, and so on). Also remember that Ø means “no answer” or “no solution”; this happens sometimes in algebra.

We’re going over this now, since we’ll be talking about inequalities soon, and it will get a little more complicated on how to write our answers. Don’t worry too much if you’re overwhelmed with all this (it’s like learning a new language!); you’ll probably give your answers in simple inequality notation or more likely interval notation.

Learn these rules, and practice, practice, practice!


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