Algebra.

For thousands of years, algebra has improved humanity in innumerable ways. Algebra comes from the Arabic word “al-jabr” that means “reunion of broken parts.” It is part of mathematics along with the subjects of number theory, geometry, and analysis. Algebra focuses on solving problems, using mathematical symbols, and its rules are relevant in handling the composition of such symbols. Algebra is the glue of other forms of mathematics. It holds everything together. Basic algebra is elementary algebra, and more abstract forms of it have been called abstract algebra or modern algebra. Elementary algebra is used by scientists and engineers including by teachers plus students. It is used in medicine and economics. Abstract algebra used advanced math, and it is studied by professional mathematicians. At the core of algebra is the embrace of abstractions. That means that letters are used to stand for numbers that are either unknown or allowed to take on many values. One example is X + 2 = 5. The letter  X refers to the unknown. So, you have to use the law of inverses to find the value of X. The value of X is, of course, the number 3. Writing and solving equations are key methods to understand algebra.

Linear algebra and algebraic topology are part of the overall algebra subject too. Algebra has roots from ancient Babylonia where they created math with algorithmic parts. They calculated linear equations, quadratic equations, and indeterminate linear equations. Ancient Egyptians, Greek, and Chinese mathematics used geometric methods to solve equations too. This process has been found in works like the Rhind Mathematical Papyrus, Euclid's Elements, and The Nine Chapters on the Mathematical Art. The geometric work of the Greeks, typified in the Elements, provided the framework for generalizing formulae beyond the solution of particular problems into more general systems of stating and solving equations, although this would not be realized until mathematics developed in medieval Islam. During the time of Plato, Greek mathematics used geometric algebra. Diophantus (3rd century AD) was an Alexandrian Greek mathematician and the author of a series of books called Arithmetica. These texts deal with solving algebraic equations and have led, in number theory to the modern notion of the Diophantine equation. 

The Persian mathematician Muḥammad ibn Mūsā al-Khwārizmī (c. 780–850) and the Indian mathematician named Brahmagupta developed an understanding of algebra too. Later, Persian and Arabic mathematicians developed algebraic methods to a much higher degree of sophistication. Although Diophantus and the Babylonians used mostly unique ad hoc methods to solve equations, Al-Khwarizmi's contribution was fundamental. He solved linear and quadratic equations without algebraic symbolism, negative numbers or zero; thus he had to distinguish several types of equations. Al-Khwarizmi and Diophantus are the fathers of algebra. Another Persian mathematician Omar Khayyam is credited with identifying the foundations of algebraic geometry and found the general geometric solution of the cubic equation. His book Treatise on Demonstrations of Problems of Algebra (1070), which laid down the principles of algebra, is part of the body of Persian mathematics that was eventually transmitted to Europe. Yet another Persian mathematician, Sharaf al-Dīn al-Tūsī, found algebraic and numerical solutions to various cases of cubic equations. He also developed the concept of a function. The Indian mathematicians Mahavira and Bhaskara II, the Persian mathematician Al-Karaji, and the Chinese mathematician Zhu Shijie, solved various cases of cubic, quartic, quintic and higher-order polynomial equations using numerical methods. In the 13th century, the solution of a cubic equation by Fibonacci is representative of the beginning of a revival in European algebra. Abū al-Ḥasan ibn ʿAlī al-Qalaṣādī (1412–1486) took "the first steps toward the introduction of algebraic symbolism". He also computed ∑n2, ∑n3 and used the method of successive approximation to determine square roots. Italian mathematician Girolamo Cardano and other scholars developed algebra concepts. Abstract algebra was formed in the 19th century. Josiah Willard Gibbs built algebra of vectors in a three-dimensional space, and Arthur Cayley developed an algebra of matrices (this is a noncommutative algebra). 

The basic form of algebra is called elementary algebra. Basic algebra uses arithmetical operations (such as +, −, ×, ÷) greatly. In algebra, numbers are often represented by symbols called variables (such as a, n, x, y or z). This is useful because it promotes functional relationships, it helps to solve equations, and it promotes arithmetical laws. In the algebraic expression notation of 3x2-2xy+c, you have the exponent, the coefficient being 3, 2, and 1, there is the term (or 3x2, and 2xy), the operator (or the subtraction and addition sign), and the constant term which is c. The variables or constants in the notation include the letters of x, y, and c. A polynomial is an expression that is the sum of a finite number of non-zero terms, each term consisting of the product of a constant and a finite number of variables raised to whole number powers. For example, x2 + 2x − 3 is a polynomial in the single variable x. A polynomial expression is an expression that may be rewritten as a polynomial, by using commutativity, associativity, and distributivity of addition and multiplication. For example, (x − 1)(x + 3) is a polynomial expression, that, properly speaking, is not a polynomial. A polynomial function is a function that is defined by a polynomial, or, equivalently, by a polynomial expression. The two preceding examples define the same polynomial function. You can factor polynomials, add them, subtract them, multiply them, and divide them. To factor them, you have to find the polynomial greatest common divisors. The example polynomial above can be factored as (x − 1)(x + 3). A related class of problems is finding algebraic expressions for the roots of a polynomial in a single variable. Algebra is readily taught to children as young as 11 years old.

American schools have algebra classes in the seventh or eighth grades. Some schools start algebra in the ninth grade. Abstract algebra uses sets or numbers found in matrices, polynomials, and two-dimensional vectors in a plane. It has binary operations and identity elements. Inverse elements are negative numbers. The inverse of a is written −a, and for multiplication, the inverse is written a−1. A general two-sided inverse element a−1 satisfies the property that a ∗ a−1 = e and a−1 ∗ a = e, where e is the identity element. Many integers have associativity too. That is, the grouping of the numbers to be added does not affect the sum. For example: (2 + 3) + 4 = 2 + (3 + 4). In general, this becomes (a ∗ b) ∗ c = a ∗ (b ∗ c). This property is shared by most binary operations, but not subtraction or division or octonion multiplication. Numbers include natural numbers, integers, and rational numbers too. The quadratic formula, which is the solution to the quadratic equation ax2 + bx + c =0 where a doesn’t equal to 0. Here the symbols a, b, and c  represent arbitrary numbers, and  x is a variable which represents the solution of the equation.  The quadratic formula is  x = (-b+/-square root of b2-4ac)/(2a). The Pythagoras rule is a similar formula that means a2 + b2 = c2 which refers to right triangles. Linear equations with one variable deal with the expression of ax + b = c. Linear equations form a straight line in a graph. One simple equation in algebra is about how if you double the age of a child and add 4, the resulting answer is 12.

How old is the child? The equation will be 2x + 4 =12. When you solve it, the X will equal to 4. A quadratic equation is one which includes a term with an exponent of 2, for example, x2, and no term with higher exponent. The name derives from the Latin quadrus, meaning square. In general, a quadratic equation can be expressed in the form  ax2 + bx + c=0 where is not zero (if it were zero, then the equation would not be quadratic but linear). Because of this, a quadratic equation must contain the term ax2, which is known as the quadratic term. Hence, a not equal to 0 and so we may divide by and rearrange the equation into the standard form. Quadratic equations look at parabolas on the graph. Complex numbers, exponential equations, and logarithmic equations are all part of algebra. Radical equations use square roots, cube roots, and nth roots. The system of linear equations and the substitution method are great to understand mathematics. 


By Timothy