Investigation 1: Introduction to Equations: Historical Overview of 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.