If 0 3 is a positive rational number

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: