Introduction to Algebra -

On this page:

Al’Khwarizmi
Completing and Balancing
Why Letters?
The Language of Algebra

Al’Khwarizmi

Algebra gets its name from the title of a book: Hisab al-jabr w’al-muqabala. Muhammad Al-Khwarizmi, a ninth century Arabic mathematician was the author of this book. Al-Khwarizmi worked in the House of Wisdom. The House of Wisdom was a center of knowledge, collaboration, translation, innovation and learning in Baghdad from about 800 to 1200ACE. About the book, MacTutor History of math states:

Al’Khwarizmi’s … treatise Hisab al-jabr w’al-muqabala gives us the word algebra and can be considered as the first book to be written on algebra.

Interesting fact #1:

Al’Khwarizmi wrote his book to help the regular person solve day to day number problems that went beyond simple arithmetic. He intended to teach:

what is easiest and most useful in arithmetic, such as men [people] constantly require in cases of inheritance, legacies, partition, lawsuits, and trade, and in all their dealings with one another, or where the measuring of lands, the digging of canals, geometrical computations, and other objects of various sorts and kinds are concerned. (MacTutor, Al-Khwarizmi)

Interesting fact #2:

Al’Khwarizmi did not use $x$ and $y$ like we do now. In fact, he wrote all of his number problems using full sentences. Symbolic algebra took a long time to develop. Descartes had the idea to use $x$ and $y$ for unknowns 800 years later in 1637. (MacTutor, Earliest uses of Symbols for Variables)

Completing and Balancing

The full title of the treatise written by Al-Khwarizmi is: Al-Kitāb al-mukhtaṣar fī hīsāb al-ğabr wa’l-muqābala, which translates to ‘The Compendious book on Calculation by Completing and Balancing’. (https://islam.fandom.com/wiki/Al-Jabr)

Math is integral to human thinking and we can figure out a lot of math using our own methods.

Completing and balancing is a method derived for those problems where there are several steps involved. Algebra breaks complex number problems down into a sequence of small steps. In this way algebra is like knitting: to knit a complex pattern it must be done one stitch at a time. While some of us like to follow instructions one step at a time, others like to have the whole picture and create the complex design and study how and what other people have created.

Completing:

This is the process of adding a negative term. eg

$5x-2=18$

To complete the $5x$ we need to add two to both sides:

$\begin{align*}5x-2+2&=18+2\\[10pt] 5x&=20\end{align}$

Balancing

Balancing is to remove a positive term. eg

$8x-2=18+3x$

We can keep this equation in balance when we remove the positive term $3x$ from both sides:

$\begin{align*}8x-3x-2&=18+3x-3x\\[10pt] 5x-2&=18\end{align}$

Why Letters?

We use a letter to represent a number if we don’t know its value.

Many mathematical processes are calculations. A calculation is a set of operations done on known quantities. No letters are required unless you are following formulas.

For example, to calculate the price to pay for a hoodie we need to add tax to the ticket price. Given that the ticket price is 49.99 and the tax is 12% we can calculate the price to pay as follows:

$\begin{align*}49.99 \times 1.12 &=?\\[10pt]49.99 \times 1.12 &= 55.99\end{align}$

On the other hand, algebra helps us out with equations. An equation is a set of operations on one or more unknown quantities.

For example, suppose the price after tax of a hoodie is 72.80. What was the pretax price, given that tax is 12%.

First, introduce a letter to hold the place of the unknown number. Let $h$ stand for the price of the hoodie. Lets write the same calculation, however this time we don’t know what the ticket price of the hoodie is:

$h\times 1.12 = 72.80$

To solve this equation, we divide both sides by 1.12.

$\begin{align*} h\times 1.12 &= 72.80\\[10pt] \frac{h\times 1.12}{1.12}&=\frac{72.80}{1.12}\\[10pt] h&= 65.00\end{align}$

Note that using the algebraic process in this example is overkill. There was only one calculation to undo, not several.

The language of algebra

In order to understand the math processes of algebra, its helpful to understand what the writing means. Here are some things to consider:

$x$	Your algebra $x$ should look different from your multiplication sign $\times$ to avoid confusion. Practice writing an algebra ‘x’.
Add	Add is written in exactly the same way as arithmetic.
Subtract	Subtract is written in exactly the same way as arithmetic. However, remember that $-3$ means $3$ less than zero, and it means $3$ multiplied by $-1$ . That is, $-3=0-3=(-1)\times(3)$ .
Multiply	Multiply is rarely written with the $\times$ symbol. To multiply numbers, use brackets or a dot: $3\times 4 = (3)(4)=3\cdot 4$ . To multiply a number by a letter, letters by letters, just squish them together. $3 \times x = 3x$ .
Divide	Divide is rarely written with the $\div$ symbol. To divide numbers, use a fraction line. $20 \div 5 = \frac{20}{5}$ , also $x \div 5 = \frac{x}{5}$ .
$3x$	Multiply is repeated adding. $3x$ means $x + x + x$ , or, the number $x$ multiplied by the number 3.
$xy$	In math, one letter is one number. $xy$ means the number $x$ multiplied by the number $y$ .
coefficient	The coefficient of a term is the numerical multiplier. The coefficient of $3x$ is $3$ .
$1x$	When the coefficient of a term is positive 1, we don’t write the 1. In other words, write $1x$ as just $x$ .
$-1x$	Similarly, when the coefficient of a term is negative 1, we don’t write the 1. In other words, write $-1x$ as $-x$ .
Expression	An expression is one or more numbers added together, however the ‘numbers’ may have a letter component. Eg, $3x+2xy-y+6$ .
Term	A term is a number that might have a letter component. A term must be positive, negative or zero. The expression $3x+2xy-y+6$ has four terms. The terms $3x$ , $2xy$ and $6$ are all positive. The term $-y$ is a negative term.
Zero	Zero needs to be written when it is the only term on one side of an equation.
Leading term	The leading term is the first term in an expression. If the first term is positive, we don’t need to write the $+$ , we just assume it.
Like terms	Two terms are like when they have the same letter combination. You can add like terms. Eg, $3xy+2xy$ is the same as $5xy$ .
Calculation	Calculations are familiar to everybody. I calculate when I apply a math operation – eg, add, subtract, multiply, divide etc to one or more known numbers. Eg, $5 \times 2$ . The answer to this calculation is $10$ .
Equation	An equation is like a calcuation, except that the math operations are applied to on one or more unknown numbers. The result is equal to a known value or equal to another set of operations done to one or more unknown numbers. Eg $3x+2=4x-8$
Exponent	$x^3$ means $x \times x \times x$ .
Exponent	$5x^3$ means $5\times x \times x \times x$ .
Exponent	$(5x)^3$ means $5x \times 5x \times 5x=5\times 5\times 5\times x\times x \times x = 125x^3$ .
Exponent	$-x^3$ means $-1 \times x \times x \times x$ .
Exponent	$(-x)^3$ means $(-x) \times (-x) \times (-x)$ .