Algebra—Everyday and Extraordinary

 

Learning Objective(s)

·         Define algebra and distinguish it from arithmetic.

·         Understand algebraic symbols.

 

Introduction

 

Algebra is used to model the world around us. In ancient times, people began to develop powerful mathematical tools to describe important events and activities. The Egyptians learned to predict the time of year when the Nile would flood by making careful measurements of the stars in the sky.

 

The Chinese invented financial mathematics to determine how to trade goods fairly, and the Babylonians came up with a method for solving very complex equations that allowed them to decide how much land to assign to different farmers.

 

In our modern world, math is just as important. For example, understanding slope helps us design everything from football stadiums to skate parks to ramps for the disabled.  It sent us to the moon and back again, it runs our computers and our cars, it tells us when it's going to rain and how early to get up to get to school on time.  Behind both the everyday and the extraordinary, there’s algebra.

 

Arithmetic vs. Algebra

 

So what exactly is algebra?  A textbook would say something like "algebra is the branch of mathematics that deals with operations on sets of numbers and relationships between them.” Okay—but what does it really mean? Let's compare algebra to arithmetic, the kind of math we already understand.

 

Often, we can use simple numbers to describe things. A pair of apples can be represented mathematically with the number 2. We can also describe actions mathematically. If I cut an apple pie into six pieces and then put one slice on a plate, I can describe that process as 6 1.

 

We call this type of mathematics, math that uses only numbers and nothing else to represent values, arithmetic. It is a great tool for figuring some things out. But sometimes it takes too long.  And sometimes, it just can’t give us the answers we need. Many times we have a lot of complex information to consider, or we don’t know the exact value that describes a situation or action. That's when algebra comes in.

 

Remember those 2 apples? Let’s say that instead of making a pie, I decide to use them to start an orchard in the backyard. How would I describe the number of trees that I grow? I don't know how many seeds there are, or how many of those will sprout. I could wait a year and then go out and count the trunks. Or I could write a mathematical description right now: if s is the number of seeds in the apples, and the average germination rate for apples is 50%, then I can expect to grow 0.5s apple trees:

 

number of trees = number of seeds times the percentage of seeds that will sprout

 

t = s • 50%

 

t = 0.5s

 

I have my answer, and I didn't even get my hands dirty.

 

In math, we call unknown amounts like s and t variables. A variable is a letter or symbol that stands for an unknown amount which may vary. Latin letters like x, y and z are commonly used as variables, as are Greek letters like α, β and λ. But any letter will do.

 

The branch of mathematics that involves variables as well as numbers is called algebra.

 

Algebra Rules

 

Algebra does rule. It also has rules, and conventions, and procedures. Not every mathematician and math teacher and math book writes the symbols for variables and operations the exact same way. Even in math, which can seem so rigid, different people express themselves in different ways and there is room for individuality. In this course, we also have our own way of describing algebraic activity. Check out these conventions, and you'll know exactly what we mean from here on out:

 

Math Symbol

Description

+

plus sign

minus sign

multiplication sign

x

multiplication sign in scientific notation

 

÷

division sign

 

(both of these division signs will be used in the course)

=

equals

does not equal

almost equals

> 

greater than

greater than or equal to

< 

less than

less than or equal to

|  |

absolute value

root

 

 

Summary

 

Algebra is a very powerful, very practical kind of mathematics. Because it includes variables, symbols that represent unknown values, algebra lets us describe and think about not just what we know, but also about what we want to find out.

 

More than just a set of rules to follow, algebra is an abstract way of thinking. As you learn it, you’ll be using numbers and symbols to represent things in the real world, manipulating those numbers and symbols, and then applying your answers back to the real world. This kind of thinking actually opens up new pathways in your brain, which will make solving all sorts of problems—even those that don’t involve numbers—easier.