# What are some negative rational 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 \$\$

### 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\$\$.

### 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 \$\$

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 ... \$\$).
##### \$\$ 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: 