**Mathematics** is the science of numbers and there are several different branches of mathematical science including **algebra, geometry, and calculus**. Dictionaries define mathematics as the **science of numbers and their operations**, interrelations, combinations, generalizations, and abstractions and of space configurations and their structure, measurement, transformations, and generalizations.
Mathematics is not an invention. Discoveries and laws of science are not considered inventions. Inventions are material things and processes. However, there is a history of mathematics, a relationship between mathematics and inventions, and mathematical instruments are considered inventions.
Mathematics as an organized science did not exist before the classical Greeks of the period from 600 to 300 BC entered upon the scene.
When civilization began to trade, a need to count was created. When humans traded goods, they needed a way to count the goods and to calculate the cost of those goods. The very first device for counting numbers was the human hand, counting on fingers. To count beyond ten fingers, mankind used natural markers, rocks or shells. From that point, counting boards and the abacus were invented.
**Abacus** -one of the first tools for counting was invented around 1200 A.D. in China.
**Accounting** - The innovative Italians of the Renaissance are widely acknowledged to be the fathers of modern accounting.
**Algebra** comes from the Arabic word al-jabr an ancient medical term meaning "the reunion of broken parts.''
**Archimedes** was a mathematician and inventor from ancient Greece, best known for his discovery of the relation between the surface and volume of a sphere and its circumscribing cyclinder, for his formulation of a hydrostatic principle (Archimedes' principle) and for inventing the Archimedes screw (a device for raising water).
**Differential **Gottfried Wilhelm Leibniz (b. 1646, d. 1716) was a German philosopher, mathematician, and logician who is probably most well known for having invented the differential and integral calculus (independently of Sir Isaac Newton).
**Graph **A graph is a pictorial representation of statistical data or of a functional relationship between variables. William Playfair (1759-1823) is generally viewed as the inventor of most of graphical forms used to display data, including: line plots, bar chart, and pie chart.
**John Napier** was the Scottish mathematician who invented **logarithms and the decimal point.**
**Protractor**
An instrument used to construct and measure plane angles. The simple protractor looks like a semicircular disk marked with degrees, from 0º to180º. The simple protractor is an ancient device. The first complex protractor was created for plotting the position of a boat on navigational charts. Called a three-arm protractor or station pointer, it was invented in 1801, by Joseph Huddart, a U.S. naval captain. The centre arm is fixed, while the outer two are rotatable, capable of being set at any angle relative to the centre one.
**Circular and rectangular slide rules**, an instrument used for mathematical calculations were both invented by mathematician William Oughtred.
**Zero **was invented by the Hindu mathematicians Aryabhata and Varamihara in India around or shortly after the year 520 A.D.
**Math Symbol =**
In 1557, the = sign first used by Robert Record. In 1631, >,< was introduced by T.Harriot.
**Arithmetic**
Arithmetic or arithmetics (from the Greek word αριθμός = number) is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple day-to-day counting to advanced science and business calculations. It involves the study of quantity, especially as the result of combining numbers.
**Arithmetic operations**
The basic arithmetic operations are addition, subtraction, multiplication and division, although this subject also includes more advanced operations, such as manipulations of percentages, square roots, exponentiation, and logarithmic functions.
**History**
The prehistory of arithmetic is limited to a very small number of small artifacts that indicate addition and subtraction. The best-known are the Ishango bone from central Africa, dating from somewhere between 20,000 and 18,000 BC. The earliest written records indicate the Egyptians and Babylonians used all the elementary arithmetic operations as early as 2000 BC. Today, we use decimal notation and Arabic numerals (the ten digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9), which spread throughout Europe in the Middle Ages.
**Basic Arithmetic Operations**
**Addition (+) **1 + 1 = 2 (verbally, "one plus one is equal to two")
Addition is the basic operation of arithmetic. In its simplest form, addition combines two numbers into a single number, the sum of the numbers.
**Multiplication (×, ·, or *) **"2 × 3 = 6" (verbally, "two times three equals six")
Multiplication is the second basic operation of arithmetic. Multiplication also combines two numbers into a single number, the product. The two original numbers are called the multiplier and the multiplicand, sometimes both simply called factors.
**Subtraction (−) **"5 − 2 = 3" (verbally, "five minus two equals three")
Subtraction is the opposite of addition. Subtraction finds the difference between two numbers, the minuend minus the subtrahend. If the minuend is larger than the subtrahend, the difference will be positive; if the minuend is smaller than the subtrahend, the difference will be negative; and if they are equal, the difference will be zero.
**Division (÷ or /) **"6 / 3 = 2" (verbally, "six divided by three equals two")
Division is the opposite of multiplication. Division finds the quotient of two numbers, the dividend divided by the divisor. Any dividend divided by zero is undefined. For positive numbers, if the dividend is larger than the divisor, the quotient will be greater than one, otherwise it will be less than one (a similar rule applies for negative numbers). The quotient multiplied by the divisor always yields the dividend.
**Odd 1, 3, 5, ... and even numbers 2,4,6,...**
**Cardinal Numerals** 2,450,094 two million four hundred and fifty thousand and ninety-four billion, hundreds of... 1987, 1600 B.C. A.D.
**Ordinal Numerals** the fifth, the thirtieth, the twenty-first , 2^{nd }, fifth, twelfth, ninth
once, twice, three times
**Fractions**
**Decimal point **6.89 six point eight nine 0.47 (nought/ zero) point four seven
1/2 one half, 3/2 three halves, 1/3 one third, 1 5/9 one and five ninths 5/21 five over twenty-one
__a__ . __c__ = __ac__
*b d bd* *a* over *b*, this fraction multiplied by *c* over *d *equals *ac* over *bd*
**powers and roots**
*b*^{n }* b* to the *n-th* power / *b* to the *n-th */ the *n-th *power of *b / b *to the power of *n*
*b*^{2 }^{ }the square of *b / *the second power of* b*
*b*^{3 }the cube of* b / *the third power of* b*
^{n}√c the *n-th* root of *c *^{2}√8* * the square root of eight
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