Degrees of numbers: history, definition, basic properties

The simplest mathematical expressions became known to people even in ancient times. At the same time, there was a constant improvement of both the operations themselves and their recording on one or another medium.

In particular, in ancient Egypt, whose scientists made a significant contribution both to the development of elementary arithmetic, and to the creation of the foundations of algebra and geometry, they drew attention to the fact that when a number is multiplied by the same number many times, It spends a lot of unnecessary effort. Moreover, such an operation led to significant financial costs: according to the then applicable settings for the design of any records, each action with a number should be described in detail. If we recall that even the simplest papyrus cost a very impressive amount of money, then it should not be surprising to the efforts that the Egyptians have made to find a way out of this situation.

The solution was found by the famous Diophantus of Alexandria, who came up with a special mathematical sign that began to show how many times it is necessary to multiply this or that number by itself. Subsequently, the well-known French mathematician R. Descartes perfected the writing of this expression, suggesting that when denoting the power of numbers, simply assign it in the upper right corner over the main number.

The final chord in the writing of the degree of numbers was the activity of the notorious N. Schücke, who introduced a negative first and then a zero degree into the scientific revolution.

What does the phrase "build a degree" mean? To begin with, it is necessary to understand that exponentiation itself is one of the most important binary mathematical operations, the essence of which consists in repeatedly multiplying the number by itself.

In general, this operation is denoted by the expression "XY". In this case, "X" will be called the basis of the degree, and "Y" will be called its exponent. In this case, "raise to the power" can be deciphered as "multiply" X "by itself" Y "times."

The degrees of numbers, like most other mathematical elements, have certain properties:

1. When you get to the power zero of any number that is different from zero (both positive and negative), one will be obtained.

X ^^ 0 = 1

2. The degrees of numbers, where the indicators have a negative value, should be converted to an expression with a positive index

X-a = 1 / x ^ a

3. In order to realize the multiplication of numbers with powers, it should be remembered that this operation is possible only if they have the same bases. In this case, the multiplication of numbers with powers is carried out in accordance with the following rule: the base remains unchanged, and to the exponent of one, the value of the exponents of the remaining powers is added.

X ^ yx ^ z = x ^ y + z

4. In the case when the degrees are divided, it is necessary to follow the same rule, but instead of the sum in the exponent there will be a difference.

X ^ y / x ^ z = x ^ yz

5. Another important property of degrees is connected with those situations when it is required to raise the power of the exponent itself. In this case, it is necessary to multiply both of these indicators.

(X ^ y) ^ z = x ^ yz

6. In some cases there is a need to write down the degree of the product in terms of the degree of numbers. In this case it is necessary to bear in mind that the degree of the product is calculated in accordance with this rule:

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

7. If there is a need to write down the degree of the quotient, the first thing to note is that the basis of the denominator can not be zero. In the rest it is necessary to adhere to the following formula:

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

Certain difficulties are encountered when it is required to raise to a power a basis whose expression is less than zero. The result in this case can be either negative or positive. It will depend on the exponent, namely, on what number - odd or even - this indicator was.