Equations--Segment01.mp4

Algebra Nspirations

Variables, Equations and Functions: The Foundation of Algebra

From the mathematics of old Babylon and Egypt, to the

mathematics of ancient Greece, China, India, the Islamic

world, and all the way up to the Renaissance and the

present, equations have played a central role in the

development of algebra.

But algebraic equations haven't always been written in

present day symbolic form.

Hello.

My name is Monica Neagoy.

And I will guide your exploration into the world of

variables and equations with the help of the TI-Nspire.

In this program, I'll describe the fundamentals of the

equations and introduce the two most basic types of

equations you'll encounter in a beginning algebra class,

linear and quadratic.

In the first investigation, we'll do a quick historical

overview of equations, examine the building blocks of

equations, and look at the multiple uses of

variables in equations.

In the second investigation, we'll dive into solving

equations, beginning with linear equations and then

quadratic equations.

Two aspects of the history of algebra are of interest here,

the evolution of its notation and the development of its

subject matter.

Namely, what is algebra?

Developed by the Babylonians, rhetorical algebra was

dominant until the 1500s.

Rhetorical algebra was predominantly oral, as writing

materials were limited in most cultures.

Algebraic symbols that are

commonplace today were unknown.

Whenever mathematicians did transcribe algebraic equations

onto clay tablets, for instance, they

wrote out every word.

For example, the equation 3x equals 1/2 x plus 10 would

have looked something like this.

"Three times a certain quantity has the same value as

one half the same quantity increased by ten."

The second stage is called syncopated algebra.

Here we begin to see abbreviations for unknown

quantities and for frequently used operations.

But these varied from country to country, as each country

created its own.

And syncopated algebra was not consistent, as it did not

follow clearly stated rules.

The third and final stage, called symbolic algebra, began

with the Renaissance mathematicians.

From the mid-1500s to about 1700, algebra symbols for

operations, variables, relations, grouping,

exponents, and so on were born, evolved, and matured.

So algebra's symbolic notation used around the world today

has only been used for roughly 300 years.

Regarding algebra's subject matter, it's fair to say that

up until the 18th century, algebra was

about solving equations.

This algebra, called elementary algebra, is a

generalization of arithmetic and is the focus of most high

school algebra courses.

During the 19th century, a new kind of algebra was born.

It's called modern algebra or abstract algebra.

While this advanced algebra is beyond our scope, you can

still remember two important points.

The variables don't always represent generalized numbers,

but other objects or structures.

And therefore, the properties of operations among those

objects don't always hold.

For example, A times B equals B times A is not true if A and

B are matrices.

Even if modern algebra is beyond our scope, it's

important to know that algebra has different meanings

depending on what mathematical objects are being studied.

We're now ready to examine the building blocks of equations.

There are three main components of the basic a

algebraic equation.

Quantities that change the values are called variables

and are usually denoted by letters at the end of the

alphabet, such as x, y, and z.

Quantities that are fixed are called constants and appears

real numbers, namely natural numbers, integers, rational

numbers, or decimals, or irrational numbers.

The main symbol in an equation is the equal sign, indicating

that the left and right sides must be in perfect balance.

Other symbols include operation signs, fraction

lines, parentheses, exponents, et cetera.

Parameters are another component

of generalized equations.

And they are denoted by letters at the beginning of

the alphabet, such as a, b, and c.

The following example illustrates the

meaning of a parameter.

These three equations are different, but

have a common form.

On the left, a constant multiplies a variable.

And then another constant is added on.

On the right, there's a 0.

It can be generalized by the equation, ax plus b equals 0,

where a and b, letters from the beginning of the alphabet,

are

Constants, and x, a letter from the end of the alphabet,

is a variable.

a and b are called parameters, representing constants that

vary from one equation to the next.

If you're asked to solve for the variable, x, in this case,

it's called the unknown.

And ax plus b equals 0 is actually the general form of a

linear equation in one unknown, which

we'll explore later.

Equations can be tricky in algebra, since not all of them

begged to be solved.

Let's take a closer look at the multiple uses of variables

in equations.

There are formulas, some of which may be familiar to you,

like the perimeter of a rectangle or

the area of a triangle.

Formulas provide useful information, but do not need

to be solved per se.

Here, variables are labels.

There are special identities that all algebra students

should know.

These are called the square of the sum of two numbers and the

square of the difference of two numbers.

The variables here stand for all real numbers and not

unknowns to solve for.

You also have special properties of operations,

which you'll also learn, such as the associative property of

addition or the commutative property of multiplication.

Here again, variables generalized real numbers.

Finally, you have equations to be solved

with one or more unknown.

There are lots of different kinds.

This is a linear equation in x and this, a

quadratic equation in w.

You'll learn how to solve for the unknown.

But first, try this.