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# elementary algebra

## summary

Elementary algebra is the mathematical field of study of the properties of operations on the real number system and complex number system and the relations between real and complex numbers which involve a finite number of additions and muliplications. These can be extended to solutions of equations that may require an infinite number of additions and multiplications.

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Professors are invited to give feedback on both the proposed contents and the propsed order of this text book. Send commentary to Milo, PO Box 1361, Tustin, California, 92781, USA.

# elementary algebra

**Elementary algebra** is the mathematical field of study of the properties of operations on the real number system and complex number system and the relations between real and complex numbers which involve a finite number of additions and muliplications. These can be extended to solutions of equations that may require an infinite number of additions and multiplications.

The ancient Greeks attributed the development of algebra to the ancient Egytpians. There is archaeological evidence to show that the Babylonians independently developed algebra (algorithmic methods for solving linear equations, quadratic equations, and indeterminate linear equations) and some scholars argue that the Egyptians learned algebra from the Babylonians. At this time the Greeks, Egyptians, and Chinese were primarily using geomtric solutions.

The ancient Greeks made significant advances using geometric algebra, that is solving number problems using geometric processes. These processes involved classical construction (straight edge and compass) and the sides of geometric objects. German mathematician Johann Carl Friedrich Gauss demonstrated in *Disquisitioned arithmeticae* in a discussion of the problem of constructing regular polygons that there was a one to one correspondence between classical geometric construction and repeated use of four rational operations and square root extraction.

Hero or Heron of Alexandria (1st Century C.E.) combined the methods of the Egyptians and the Babylonians to make advances in algebra. Alexandrian mathematician Diophantus (3rd Century C.E.) wrote a series of books entitled *Arithmetic* dealing with the solution of algebraic equations. Note that the Library of Alexandria combined both the Greek and Egyptian cultures and religions in the worship of Serapis and Isis.

Indian mathematician and astronomer Brahmaupta (7th Century C.E.) wrote *Brahmasphutasiddhanta,* which is often considered to be the foundational work of modern algebra. In this work Brahmaupta first introduced a complete arithmetic solution to quadratic equations (including both zero and negative solutions).

The Hindu advances in algebra were introduced to Europe by the Islamic mathematician Muhammad ibn Musa Al-Khawarizmi (8th Century C.E.), who studied mathematics in India. Al-Khowarizmi (variant spelling) wrote *Book on Addition and Subtraction after the Method of the Indians,* in which he argued in favor of the use of Hindu-Arabic numerals, and *The Compendious Book on Calculation by Completion and Balancing,* which established algebra as a separate branch of mathematics from arithmetic and geometry. The Enlgish words “algorithm” and “algorism” are corruptions of Al-Khawarizmi’s name. The Arabic word *al-jabr* means “restitution”, a reference to the transposition of subtracted terms to the other side of an equation or the cancellation of like terms on opposite sides of the equation.

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