What are some examples of arbitrary numbers

Which number ranges are there?

Depending on the type, you can assign numbers to one or more number ranges. Number ranges are quantities that Numbers of a variety contain.

There are these number ranges:

  • Natural numbers $$ NN $$
  • Whole numbers $$ ZZ $$
  • Broken Numbers $$ QQ _ + $$
  • Rational Numbers $$ QQ $$
  • Irrational numbers
  • Real numbers $$ RR $$

What are natural and integers?

Natural numbers $$ NN $$

The number range of the natural numbers $$ NN $$ forms that counting as a natural process.

  • The smallest natural number is the $$ 0 $$.

  • The set of natural numbers contains all successors of the $$ 0 $$ up to infinity:
    $$ NN = {0,1,2,3,4, ..., n, n + 1, ...} $$ .

How can you calculate with natural numbers?

You are allowed without restriction add and multiply.

  • It is said that $$ NN $$ is related to addition and multiplication completed.
  • All other arithmetic operations cannot be carried out without restrictions.

Whole numbers $$ ZZ $$

If you expand the number range of the natural numbers with the negative numbers, do you have the whole numbers:

  • In the set of negative numbers are all positive and negative numbers without comma: $$ ZZ = {…, -3, -2, -1,0,1,2,3,…} $$
  • Now you can also without restrictions subtract.

Successor principle: Is $$ n $$ is any natural number, then $$ n + 1 $$ her successor.

Example: The number $$ n = 73 $$ has the successor $$ n + 1 = 74 $$

Seclusion: The result of the calculation is the same amount, here $$ NN $$.

Example:

  • If you add two natural numbers, the sum is also a natural number. $$ 4 + 3 = 7 $$
  • If you calculate $$ 4: 3 $$, the result is not a natural number, but a fraction $$4/3$$.

What are Fractional and Rational Numbers?

Broken Numbers $$ QQ $$$$+$$

Do you want unlimited to divide, you need the fractions.

  • $$ QQ $$$$+$$ contains all positive fractions
  • $$ QQ $$$$+$$$$ = {$$ $$ a / b | $$ $$ a, b $$ is a natural number and $$ b! = 0} $$

Rational Numbers $$ QQ $$

Do you take the negative fractions in addition, you have the rational numbers.

  • $$ QQ = {$$ $$ a / b | a $$ is an integer, $$ b $$ is a natural number and $$ b! = 0} $$
  • In $$ QQ $$ you can all basic arithmetic run without restriction.
  • $$ QQ $$ contains all positive and negative fractions, as well as all terminating Decimal fractions (e.g. $$ - 3.75 $$) and periodic decimal fractions (e.g. $$ 0.66666 ... $$).
You write a fraction generally $$ a / b $$.
The quotient of two natural numbers is positive.
Division by zero is not permitted in any number range, therefore $$ b! = 0 $$.
$$ a $$ can be negative, so the quotient can also be negative.

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What are irrational numbers?

With the rational numbers only one thing is not completely allowed: that Pulling roots.

You can already pull some roots:

  • $$ sqrt (9) = 3 $$ da $$ 3 * 3 = 9 $$
  • $$ sqrt (0.16) = 0.4 $$ da $$ 0.4 * 0.4 = 0.16 $$
  • $$ sqrt (4/9) = 2/3 $$ da $$ 2 * 2 = 4 $$ and $$ 3 * 3 = 9 $$

Irrational numbers

Some roots are infinitely long decimal numbers and not as a fraction representable. These are irrational numbers.

Examples:

  • $$ sqrt (2) = 1.4142135623730 ... $$
  • $$ sqrt (3) $$, $$ sqrt (5) $$, $$ sqrt (6.12223) $$

What are real numbers?

If you combine the rational and the irrational numbers, you get the real numbers $$ RR $$.

  • In this number range are all positive and negative fractions as all roots.
  • You cannot take a root from negative numbers. $$ sqrt (-4) $$ is not defined. Such numbers are not in the real numbers $$ RR $$ included.

In this figure you can see how the number ranges are interrelated: