Numbers, Properties, and Notation in Algebra

        

This section covers:

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….

Natural Whole Integers Irrationals Rationals Real Imaginary Complex Numbers

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:

Properties of Equality

Commutative and Associative Properties

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

Associative Commutative Property

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”:

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:

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.

Here is another example, and one using a variable:

Distributive Property

Summary of Algebraic Properties

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

Algebraic Properties Chart

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 {1, 2, 3}).  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 {1, 2, 3} and {3, 4, 5} would be {1, 2, 3, 4, 5}, since you include everything in both sets, but don’t repeat numbers.  You write union as , so {1, 2, 3}  {3, 4, 5} = {1, 2, 3, 4, 5}.

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

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:

Math Grammar

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” () is used, we use “[“, and if “greater than or equal to” is used (), we use “]”.  So 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  or  , we only use soft brackets (parentheses) with them.

Remember that  means all real numbers (everything on the number line), and  means natural numbers (1, 2, 3, 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!

On to Solving Algebraic Equations – you are ready!

5 thoughts on “Numbers, Properties, and Notation in Algebra

  1. I’m 64 and have always struggled with math. As a young man I wanted to be an engineer but my poor math skills kept me from it – I ended up with degree a biology! Neither of our daughters had much interest in math but two of my grandchildren do. Both are always full of questions so when they ask me a math question I go to the internet for help – which leads me to my question (Help! they think Papa knows everything!). If the natural numbers are the counting numbers and Zero is not a natural number, how do we count past 9??? AND, I love your website!!!

    • Thanks so much for your comment – you’ve made my day! Think of the counting numbers of all the numbers except 0 and negative numbers – and don’t worry if there’s a 0 in the number. That’s how I look at it (is this too simplistic??) Lisa

  2. Lisa, if I may be permitted to add to your explanation to Jess, it might also be helpful to clarify that it is the VALUE of zero that is not included in the natural numbers and not the DIGIT of zero. The reason for this is that the VALUE of zero, in other words the idea of counting nothing, is more abstract so it was traditionally not included in the first set of numbers that we introduced to preschoolers who are just beginning to count things. The DIGIT of zero, however can be found in the counting/natural number 10 and the multiples of 10. Many ancient number systems (Babylonian, Greek, Egyptian) didn’t even have a representation for the value of zero! Fortunately for us, the Mayans figured it out. By the way Jess, the set of natural numbers may not include the value of zero but the set of whole numbers begins with zero and it is the only element that is in the set of whole numbers that is not in the set of natural/counting numbers – some just consider another way to think of the counting numbers so that the idea of counting nothing is represented. Interesting fact…Using zero to represent a count of nothing is included in the Common Core Standards for Kindergarten.

  3. With questions of this nature, it might help to ignore the artifacts of the base we are using, base-10, currently. If we were using base-16 instead of base-10, we’d represent the value of ten with the letter A (in base-16, or “hexadecimal). The base, or radix, we’re using is somewhat arbitrary. The only reason the number of your fingers is commonly represented with a one and a zero is convenience. Ten can be REPRESENTED in an infinite number of ways. The word “ten” has the exact same meaning as 10 or A or “zehn,” in German. The main point: With questions of this nature, especially number theory, ignore the peculiarities of the radix you’re using and focus on the values. Look at your fingers, or the tiles on your bathroom wall, and forget the representative symbology.

    Beautiful web site. Thank you!!!

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