If Only You Knew The Real Definition of Algebra, It Would All Make Sense

Sometimes we don't realize how much of an impact knowing the origin or background of a word will have on our basic understanding of what we are actually saying and doing.

For instance let’s consider the prefix ex-. You know that ex-boyfriend or ex-girlfriend refers to someone who you used to be in a romantic relationship with, but is now out of your life. That word “out” is very important because “out of" is the Latin root meaning of the prefix ex-.

Now, I probably made you think of other words that begin with ex-.

How about:

  • Explore - to go out to travel and learn

  • Extrovert - a person with an outgoing personality

  • Excavate - to dig or scoop out

And here's a math term that my Algebra 2 students always had a hard time saying:

  • Extraneous Roots - the roots of a rational function that are left out of the domain

What we’ve just done is explored the etymology of the prefix “ex-”, that is, the historical and literal meaning based on the original language it came from. When you learn the etymology of words and parts of words, you can apply these root meanings to words you don’t know and predict what they might mean.

Let's look at the etymology of the word Algebra.

I did a little research and stumbled upon some really eye-opening information. First up is Wikipedia. It says,

"The word algebra comes from the Arabic الجبر (al-jabr lit. "the reunion of broken parts") from the title of the book “Ilm al-jabr wa'l-muḳābala” by the Persian mathematician and astronomer al-Khwarizmi.” [sounds like algorithm, right?] “The word entered the English language during the fifteenth century, from either Spanish, Italian, or Medieval Latin."

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Another good article was from MathForum.com:

"In Arabic, al- is the definite article "the." The first noun in the title is jebr "reunion of broken parts," from the verb jabara "to reunite, to consolidate." The second noun is from the verb qabala, with meanings that include "to place in front of, to balance, to oppose, to set equal."

Together the two nouns describe some of the manipulations so common in algebra: combining like terms, transposing a term to the opposite side of an equation, setting two quantities equal, etc. Because the original Arabic title was so long, and because it was in Arabic, Europeans soon shortened it. The result was algeber or something phonetically similar, which then took on the meanings of both nouns and eventually acquired its modern sense."

So let me explain that a little bit More.

The broken parts are the two expressions joined by an equal sign, called “jabr”. This creates an equation. Then there's a filling in of holes/unknowns/variables in our equations by way of "qabala". This means to balance the equation over the equal sign using opposition, also known as opposite operations.

What are opposite operations?

Well think of some common math operations. Then think of their opposites, the operations that undo what the originals are trying to do.

Addition —> Subtraction

Multiplication —> Division

Square —> Square Root

sin(x) —> arcsin(x) [aka inverse sin]

Reuniting broken pieces is so easy to picture in my mind. It makes me think of grafting a branch to a tree with some supportive material.

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Balancing an equation using opposition literally makes me think of balancing a scale. Balancing scales are instruments used to weigh objects. When both sides have different masses then the scale is off balance. When both sides of the scale have the same mass, then the scale is balanced (as shown in the image).

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Since we now know al-Khwarizmi’s definition for Algebra, it makes sense that Algebra is an "analytical balancing scale". What we do to one side of the equal sign has to happen on the other side too in order for the scale (equation) to remain balanced.

So if you add 3 to one side, the only thing you can do to the other side is add 3, to satisfy the balancing. If you add nothing to the other side of the equation that also result in an unbalanced equation.

Equation balancing applies for all operations, including addition, subtraction, multiplication, division, exponents, radicals, inverse trig functions, etc. Any operation that has an opposite can be balanced across an equal sign.

Pause. 🤯 Mind Blown!!!

Algebra had a whole secret hiding in the word. And even more in the title of al-Khwarizmi’s book, “Ilm al-jabr wa'l-muḳābala” as explained above.

When I learned this and made the connections, it blew me away! I felt like knowing the history of the word Algebra, valuable information, would have helped me understand and teach Algebra in a more comprehensive and literal way.

In my opinion, if this root explanation was part of the Algebra curriculum in high school, it would have been much easier to grasp the point of doing it in the first place.

What do you think? Leave a comment below.

Is knowing the etymology of our math words a necessary part to understanding it as a concept? Was this interesting?

Want to start digging deeper with math etymology?

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Use the Vocab Collection Sheet of my printable KOG Math Notes-to-Study Binder Sheets. There you can define the word, draw an example, use it in a sentence and record the etymology.

You’ll also get 6 other templates to help you organize your math notes, and even arrange study groups.

Check out the article If Only You Knew the Real Definition of Algebra, It Would All Make Sense Check out the article If Only You Knew the Real Definition of Algebra, It Would All Make Sense