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Translations of Encyclopedia about Mathematics

 

Real Numbers

Besides fractions and their decimal expression (rational numbers), there exist also numbers which cannot be expressed as fractions of two whole numbers, which are also numbers with an unlimited number of decimal numbers but which do not have any repetition. Such numbers are referred to as irrational numbers.

Rational and irrational numbers together make up the set of real numbers, referred to as IR.

Many roots have an unlimited number of long, non-periodic decimal numbers and belong to the family of irrational numbers. For example, the square foot of 2 is:

 = 1,414213562373095048801688724209

Furthermore, many constants are also irrational numbers, such as the number p ( = 3,14159…) or Euler’s number e ( = 2,71828…).

A notable phenomenon of mathematics is the fact that there exist many more irrational numbers than irrational numbers.

An irrational number cannot be counted (such as 1, 2, 3, etc.) in that there would always be some other number we could say (following the decimal point) which would more closely describe the actual number. With such sequences of numbers, we are never able to define an end point.

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Prime and Composite Numbers

Prime numbers are special subsets of natural numbers and include all numbers which are greater than or equal to 2 and which are not divisible by nothing other than themselves and the number 1.

For example:

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31…

Composite numbers are all natural numbers besides the number one which are not prime numbers.

For example:

4, 6, 8, 9, 10, 12, 14, 15, …

We can also consider each composite number as a multiple of two different prime numbers, a procedure which we also refer to as dissolution to prime number.

Examples:

9 = 3.3 = 32 (refer to exponents)
15 = 5.3
45 = 3.3.5 = 3
2.5

If during some dissolution we end up with an even number (divisible by 2), it makes sense to first determine how many times it is divisible by the number 2 before we start testing whether it is a prime number, starting with the least in value.

For example:

36 : 2 = 18 divide by 2
18 : 2 = 9 divide by 2
9 : 3 = 3 divide by 3
3 : 3 = 1 divide by 3

From which we can determine that: 36 = 2 . 2 . 3 . 3 = 22 . 32

When dissolving odd numbers to prime numbers, we must first determine whether it is the second power of some prime number (refer to exponents). If not, then we try further, starting with the number 3, to find out into what prime number it can be dissolved.

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Exponents and Roots

The mathematical functions of exponents and roots relate very much to one another.

Exponents

An exponent is a shortened way how to describe multiplication.

For example:

4 .4 .4 = 43 = 64

The superscript indicates how many times the number preceding it multiplies against itself. Therefore, the statement 43 is only a shortened version of the more detailed statement.

We write an and say "a to the power of n".

An exponent is based on the base (in this case "a") and an exponent (in this case "n"). The effect of exponentiation is also referred to as the exponent/power or the exponent value.

We refer to the calculation of exponents as exponentiation. If we exponentiate some base by the power (exponent) of 2, we refer to that as the second power. We often run into the second power when calculating surface areas, such as square metres (m2).

Even numbers (2, 4, 6, 8 etc.) and odd exponents (1, 3, 5, 7 etc.) have a particular significance when exponentiating negative bases.

When exponentiating negative bases by even exponents, the result is always positive:

(-2)2 = (-2) . (-2) = +2

If we exponentiate a negative base with an odd exponent, the end value is always negative:

(-3)2 = 9 (-5)3 = -125

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