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• • Contributed by David Guichard
• Professor (Mathematics) at Whitman College
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A vista into history from www.ucs.louisiana.edu/ sxw8045/history.htm In the 7th and 8th centuries the Arabs, united by Mohammed, conquered the land from India, across northern Africa, to Spain. In the following centuries (through the 14th) they pursued the arts and sciences and were responsible for most of the scientific advances made in the west. Although the language was Arabic many of the scholars were Greeks, Christians, Persians, or Jews. Their most valuable contribution was the preservation of Greek learning through the middle ages, and it is through their translations that much of what we know today about the Greeks became available. In addition they made original contributions of their own.

They took over and improved the Hindu number symbols and the idea of positional notation. These numerals (the Hindu-Arabic system of numeration) and the algorithms for operating with them were transmitted to Europe around 1200 and are in use throughout the world today.

Like the Hindus, the Arabs worked freely with irrationals. However they took a backward step in rejecting negative numbers in spite of having learned of them from the Hindus. In algebra the Arabs contributed first of all the name. The word "algebra" comes from the title of a text book in the subject, Hisab al-jabr w’al muqabala, written about 830 by the astronomer/mathematician Mohammed ibn-Musa al-Khowarizmi. This title is sometimes translated as "Restoring and Simplification" or as "Transposition and Cancellation." Our word "algorithm" is a corruption of al-Khowarizmi’s name. The algebra of the Arabs was entirely rhetorical.

They could solve quadratic equations, recognizing two solutions, possibly irrational, but usually rejected negative solutions. The poet/mathematician Omar Khayyam (1050 - 1130) made significant contributions to the solution of cubic equations by geometric methods involving the intersection of conics. Like Diophantus and the Hindus, the Arabs also worked with indeterminate equations. Here are other links, if you are interested:

• www.helpalgebra.com/info/algebrahistory.htm
• www.freemathhelp.com/history-algebra.html aleph0.clarku.edu/ djoyce/mathhist/algebra.html
• www.ehow.com/video.4977241_who-invented-algebra.html
• mathforum.org/sum95/ruth/history.html

Think of Algebra basically as Arithmetic using unknown numbers, variables. The variable $$x$$ stands for an unknown, whereas $$3$$ is understood to be three. $$x$$ is a locked $$x$$-box which contains a number

It is important to write algebraic expressions neatly.

Bad Example $$\PageIndex{1}$$

$\dfrac{4+5}{7} + 6$

Did you mean

$\dfrac{4+5 + 6}{7}$

$\dfrac{4+5}{7} + 6$

$\dfrac{45}{7} + 5 + 6$

Bad Example $$\PageIndex{2}$$

$6/2 \cdot 4$

which is interpreted as

$3 \cdot 4=12$

Did you mean

$$\sqrt{ 1 5 5} \cdot \ 4$$

Bad Example $$\PageIndex{3}$$

(0,3)(-5,-1) (0,0)$$\sqrt{1\ 4\ 4.}\ 2\ 5$$

Did you mean

(0,4)(-2,-3) (0,0)$$\sqrt{1\ 4\ 4\ \cdot \ 2\ 5}$$ (8.5,-0.0)or

(0,3)(-13,-7) (0,0)$$\sqrt{1\ 4\ 4}\ \cdot\ 2\ 5$$ (9.5,-0.0)or



(0,3)(-24,-5.3) (0,0)$$\sqrt{1\ 4\ 4.\ 2\ 5}$$ ?

### Some vocabulary

Variable: A symbol (letter like $$x$$) representing a number.

Algebraic Expression: A collection of symbols (other than $$=$$, $$<$$, $$>$$, $$\ge$$, $$\le$$), like $$2x+3$$, $$\displaystyle \frac{x-3}{\sqrt{\pi}}$$

Equation: A mathematical sentence relating two expression with an $$=$$ sign.

## For homework

Study each example in a particular section the text by writing it out on paper.

Get to the point where you can reproduce all the examples without referring to any source. Then you will be ready to start the homework.

While doing a homework exercise, do not copy from an example and substitute homework numbers for example numbers. You cannot consult examples from the text when you take an exam.