algebra history ...

As i posted a while back ... i was wondering about the earliest forms of algebra. (Though i used the alternate Nydronderian spelling, really.)

Anyway

Like most of mathematics as we have it today, algebra is not just one thing. At least one primary component, however, linear equations, is as old as ancient Egypt, nearly 2000 bce.

What i found remarkable ... (what the author [Victor Katz] found remarkable, and pointed out, and with which i agreed) ... is that the thought process is so similar.

We have papyrus with math lessons on them. They'll say things like this ...

Take a number times 3 and add to it 4. The result is 10. What is the number?
Our linear equation for this is 3x + 4 = 10.
Now, they did not use the notion, of course, but the papyrus talks the student through solving the equation.
First, just like we do, they subtracted the 4 from 10 ... 3x = 6
Second, they divided both sides by 3 ... x = 2
Answer = 2

Aside from using linear equation type thinking, they also seemed to have a robust notion of what a linear equation is ... that is, a linear relationship between two quantities.

Given a problem such as ... x + (1/4)x = 15 [again, they described it in words] ... like a good SAT taker they would try a quick and easy "false position", for example ...

x + (1/4)x = ?
x=4 (because the math is easy that way)
so, substituting ...
4 + (1/4) * 4 = ?
... = 5

Now, here is the key concept ...
realizing the relation of 5 to 15 (the original quantity on the right side of the equation) is 3, they know the unknown on the left side of the equation needs to be 3 times larger than the number they tried. That is ... 4 * 3 ... that is ... 12

12 + (1/4) *12 = ?
15 of course
We have used a linear equation AND demonstrated the basic knowledge of the relationship between the 2 quantities.


So ... while it is not as old as counting (the Summerians did not have linear equations), it is still very old.