Mathematics essay
1. The emergence of agriculture changed the whole way of living in human society. Agriculture provided a dependable food source, and therefore people could stay at the same places, develop new technologies and expand their knowledge. The very science of mathematics evolved due to the emergence of agriculture: people traced the movements of the moon and the sun to make calendars and to forecast seasons, developed measurements of seed and grain amounts. Later, people learned how to measure land, introduced different measures of value, and, eventually, money. Therefore, the development of early mathematics started when agriculture emerged.
2a. Simple grouping systems are based on small numbers (which have special names) and their powers going up to the numbers which are actually needed for use. For example, early Egyptians used the hieroglyphs and wrote their numbers on stones using a simple grouping system.
2b. In multiplicative grouping systems, not only the numbers used for the base of the system (b) but also all numbers preceding the number b (2..b-1) receive special names. The symbols used to denote numbers 2..b-1 are used to indicate the repetitions of the numbers 1, b, b^2, etc. The traditional national Chinese system was a multiplicative grouping system.
2c. Positional numeral systems are based on giving special names to numbers 1..b-1 (where b is the base) and the numbers are written as sequences of these numbers multiplied by the base in the appropriate power: N = anbn + an − 1bn − 1 + ⋯ + a1b + a0. Modern decimal system which is widely adopted nowadays is the example of positional numeral system.
3. The Babylonian number system was called a “hybrid” system because it involved both the operations of multiplication and addition, and included the symbols for numbers and the symbols for powers.
4a. The Babylonians did not have a notation (digit) for zero, and viewed it rather as an absence of the number. The Babylonians used a space in the places where the place value of zero should be inserted (i.e. used placeholder instead of zero). Later, they introduced a placeholder symbol but it also denoted the absence of a number instead of a number.
4b. In Hindu (Indian) position numeral system, there was a separate symbol for zero, which was included in the list of ordinal numerals and had an own name. Indian mathematician, Brahmagupta, who first described the concept of zero, also outlined the rules and procedures of operating with zero. Therefore, the Indian/Hindu culture used zero as a value.
5a. A regular hexagon inscribed in a circle of radius one can be divided into six equilateral triangles with side length equal to the radius – 1 (by drawing the diagonals of the hexagon). Bisecting the inner angles (starting in the center of the circle), one can get 12 equal right-angled triangles. The length of the bisected side will be half of the radius (1/2), and the length of the hypotenuse is 1. Therefore, the length of the other side will be Therefore, the area of one triangle will be equal to . Therefore, the area of the regular hexagon will be equal to 12 triangle areas: .
5b. For a regular hexagon, circumscribed around a circle, one can also divide it into six equilateral triangles, and then divide these triangles into halves by bisecting the inner angles of these triangles. In this case, the length of the height of each of these triangles will be equal to the radius (i.e. 1), and the other side will constitute one half of the hypotenuse. Denoting hypotenuse as x and the remaining side as x/2 and applying Pythagorean theorem, one can obtain the following: . The area of the small circle in this case is: , and the area of the circumscribed regular hexagon will be equal to 12 such areas.
