# 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:

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