Why are some approximations integers

4. The set of real numbers

Characterization of rational numbers

1. As the rational numbers as a summary of the fractions B. and their opposites have been introduced, the set of rational numbers can be characterized as follows:

.

2. The representation of natural numbers is based on one Place value system: Each digit (position) of a natural number must be multiplied by a multiple of 10, depending on how many positions the digit is in. Example:

.

Extension using fractions:

Such a number is called Decimal fraction designated. Every fraction can be converted into a decimal fraction, as the following examples are intended to illustrate.
 









































3. Conversely, decimal fractions can always be converted into fractions. Examples:

breaking decimal fraction:

periodic decimal fraction:

mixed periodic decimal fraction:

The set of rational numbers can therefore also be characterized as the set of all terminating and periodic decimal fractions.

Holes on the number line

A unit square (square with side length 1) is placed on the number line. Its diagonal will be with d designated.

The length of the diagonal d should be determined. To do this, the figure is expanded as follows:

The square ABCD has the area A.1 = 1.

If you halve the square ABCD, you get 2 triangles with the area 1/2.

The square AEFC can be composed of 4 such triangles, so it applies to the area A.2 this square:

A.2 = 2.

On the other hand, the following also applies:

A.2 = d2,

so:

d2 = 2.

Is there a rational number d with this property?

First, an abbreviated notation should be introduced: the positive number d with the property d2 = 2 is referred to as the "root of 2"; Notation:. In general it is determined:

The positive number that by itself multiplies the number a is called the "root of a"; Notation:.
is the number for which applies;
is the number for which applies;
etc.

To see if is a rational number, the following statement about rational numbers is required.
 

Sentence: If is not an integer, then is too not a whole number.

Proof:
 

With the help of the theorem just proved it can now be shown that is not a rational number.

Proof:

It is known that is not an integer because it holds

12 < 2 < 22.

So must lie between the two whole numbers 1 and 2:

.

Accepted, would be a non-integer rational number.

Then the square of also be a non-integer rational number (according to the theorem proved above).

But now it is known: is an integer.

However, this is a contradiction to the assumption that is a non-integer rational number.

So the assumption must be wrong.

So is neither an integer nor a non-integer rational number.

There is no such thing as a rational number whose square is 2.

And it gets even better ... (or worse - depending on the setting ...)

One starts with a square ABCD, whose side is divided by any natural number n > 1 is given, then its area is

A.1 = n2.

If you halve the square ABCD, you get 2 triangles with the area n2 / 2.

The square AEFC can again be composed of 4 such triangles, so this applies to the area A.2 this square:

A.2 = 2n2.

On the other hand, the following applies again:

A.2 = d2,

so:

d2 = 2n2

.

As for can be shown that is not a rational number. That means:

The diagonal of a square whose side is a natural number cannot be given by a rational number.

But now you can take the diagonals of these squares in the circle and turn them onto the number line.

So there are places on the number line for which no rational numbers can be given.

The representability on the number line suggests, however, to assign numbers, in other words “non-rational” numbers, to these places.

Further non-rational numbers result from the following
 

Sentence: For a prime number p isnot a rational number.

Real numbers

The numbers, their decimal representation non-periodic and non-abortive is make up the set of Irrational numbersI.. Irrational numbers and rational numbers are combined to form a set the real numbers.

The following relationships apply:

.

So there are “more” real numbers than rational numbers.

Approximate numbers

An irrational number has no breaking decimal fraction, i.e. the number has an infinite number of decimal places. In addition, there is no period in these decimal places. With a decimal number, an irrational number can only ever approximately can be specified.

There are several methods of finding approximate decimal numbers for irrational numbers. Here is an example Interval halving method to be viewed as.


If a decimal approximate number for the irrational number is to be determined, it can also be expressed as follows:

Find the positive solution of the equation x2 = 2.

First step:

An interval [xLeft ; xright] searched, in which the solution lies.

A suitable interval is xLeft = 1 and xright = 2, because: 12 < 2 < 22.

Second step:

It becomes the center point xm of the interval calculated:

,

so

.

Third step:

It will formed and compared with 2:

.

There it will xm selected as the new right interval limit:

xright = xm = 1,5.

The solution you are looking for is now limited by an interval that is only half as wide as the starting interval:

xLeft = 1 , xright = 1,5.


Repeat from step 2:

Calculation of the center point xm of the halved interval:

Repeat from step 3:

It will formed and compared with 2:

.

There it will xm selected as the new left interval limit:

xLeft = xm = 1,25.

The solution you are looking for is now limited by an interval that is only half as wide as the previous interval:

xLeft = 1,25 , xright = 1,5.


Repeat from step 2:

Repeat from step 3:

new interval:

xLeft = 1,375 , xright = 1,5.

Step 2 and step 3 are repeated until a desired accuracy is achieved.


Laws of Calculation

The rational numbers were identified with points on the number line. The addition of rational numbers was illustrated by placing the corresponding arrows one behind the other:

The real numbers were also introduced as points on the number line. Their addition can also be represented by placing the corresponding arrows one behind the other. For the sum of and So there is also a place on the number line. However, no simplified notation can be given for these. The name of this number is.

As with rational numbers, the subtraction is transferred to real numbers as the addition of the opposite number, the multiplication as the stretching of an arrow and the division as a multiplication with the reciprocal value.

The following laws of calculation apply to the set of real numbers.

For all applies:
 


Exercises

1. Justify as forthat are not rational numbers.

2. Explain why the rationale is not based on the numbers is applicable.

3. Use the interval halving procedure to find decimal approximate numbers for.

4. Application of the distributive law:

(1) Multiply out.

(2) Factor out as much as possible.


You can use the following JavaScript to generate positive numbers r approximate the root with the interval halving method.
For example, choose r = 2; 3; 5; 8; ....
Experiment with different starting approximations for x_left and x_right. (Please note: there is a decimal for decimal numbersPointnot to enter a comma.)
Try Ör to be approximated as closely as possible.