Degree numbers: history, definition, basic properties

simple mathematical expression became known to people since ancient times.At the same time constantly going improvement of both the operations and their records in a particular medium.

In particular, in ancient Egypt, whose scientists have made a significant contribution in the development of elementary arithmetic, and in laying the foundations of algebra and geometry, drew attention to the fact that when there is a multiplication of a number by the same number many timesthen it spent a huge amount of unnecessary effort.Moreover, this operation led to significant financial costs: according to the settings in effect at the time of any registration records, each with a number of action was described in detail.If we remember that even the simplest papyrus cost quite a considerable sum of money, then it is no wonder the efforts which the Egyptians have made to find a way out of this situation.

decision to found the famous Diophantus of Alexandria, who invented a special mathematical sign, which was to show how many times you must multiply one or another number by itself.Subsequently, the famous French mathematician Descartes improved the writing of this expression, suggesting the numbers when referring to the degree simply attributing it to the upper right corner above the main number.

final chord in the written form of numbers extent was the work of the notorious N. Shyuke that ushered in the scientific revolution first negative and then the zero degree.

What does the phrase "to build a degree?"First we need to understand that in itself exponentiation is one of the most important binary mathematical operations, the essence of which is repeated multiplication of the number by itself.

In general terms, the operation is indicated by the expression «XY».In this case the «X» is called a base point and «Y» - its index.In this case, the "raised to the power" will be decoded as "multiplied by" X "by itself" Y "time."

Degrees numbers, like most other mathematical elements have certain characteristics:

1. When erecting the zero degree of any number other than zero (both positive and negative) will turn one.

^^ x 0 = 1

2. Degrees of numbers, where the indicators are negative, should be transformed into an expression of a positive indicator

x a = 1 / x and

3. In order to carry out the multiplication of numbers withdegrees, it should be remembered that this operation is only possible if they have the same base.This multiplication of numbers with powers carried out in accordance with the following rule: the base remains unchanged, and is added to the index of the value of the remaining degrees of performance.

x ^ yx ^ z = x ^ y + z

4. In the case when there is a division of powers, it is necessary to adhere to the same rules, but instead in the index is the sum of the difference.

x ^ y / x ^ z = x ^ yz

5. Another important property is largely due to those situations when you need to build in a degree of self exponent.In this case, you need to multiply both ratios.

(x ^ y) ^ z = x ^ yz

6. In some cases, there is a need to paint the degree of the product through the degree numbers.In this case, you must bear in mind that the degree of the product is calculated in accordance with this rule here:

(xyz) ^ a = x ^ ay ^ az ^ a

7. If you need to paint the extent of the private, the first thingshould pay attention to is the fact that the base of the denominator can not be zero.For the rest, you must adhere to the following formula:

(x / y) ^ a = x ^ a / y ^ a

Certain difficulties are encountered when it is required to build a power base, the expression of which is less than zero.The result in this case may be either negative or positive.It will depend on the exponent, namely from what number - odd or even - this figure was.