1.1 Verschiedene Zahlen
Aus Online Mathematik Brückenkurs 1
Contents:
- Natürliche Zahlen
- Negative Zahlen
- Operatorrangfolge und klammern
- Rationale Zahlen
- Irratiolale Zahlen (übersichtlich)
- Reelle Zahlen
Lernziele
Nach diesem Abschnitt sollst Du folgendes können:
- Calculate an expression that contains integers, the four arithmetic operations and parentheses.
- Know the difference between the natural numbers, integers, rational numbers and irrational numbers.
- Convert fractions to decimals, and vice versa.
- Determine which of two fractions is the larger, either by a decimal expansion or by cross multiplication.
- Determine an approximate value to a decimal number and a fraction to a given number of decimal places.
Berechnungen mit Zahlen
Berechnungen mit Zahlen bestehen aus mehreren Schritten. Diese Schritte bestehen aus den vier Grundrechnungsarten der Arithmetik. Volgende Begriffe sind wichtig in der Mathematik:
Bei der Addition ist die Reihenfolge der Zahlen egal
Vorlage:Displayed math
Bei der Subtraktion im gegensinn, ist die Reihenfolge bedeutend.
Mit der Differenz zwischen zwei Zahlen, meint man meistens die größte Zahl subtrahiert mit der kleineren Zahl. Die Differenz zwischen 2 und 5 ist also 3.
Bei der Multiplikation ist die Reihenfolge der Zahlen auch egal
Vorlage:Displayed math
Bei der Division im gegensinn, ist die Reihenfolge bedeutend.
Operatorrangfolge
In den Fällen wo ein mathematischer Ausdruck mehrere Rechnungsarten enthält, ist es wichtig die Operatorrangfolge zu kennen. Ein Ausdruck soll in folgender Reihenfolge berechnet werden:
- Klammern (die inneren klammern zuerst)
- Multiplikation und Division (von links nach rechts)
- Addition und Subtraktion (von links nach rechts)
Beispiel 1
- \displaystyle 3-(2\cdot\bbox[#FFEEAA;,1pt]{(3+2)}-5) = 3-(\bbox[#FFEEAA;,1pt]{\vphantom{()}2\cdot 5}-5) = 3-\bbox[#FFEEAA;,1pt]{(10-5)} = 3-5 = -2
- \displaystyle 3-2\cdot\bbox[#FFEEAA;,1pt]{(3+2)}-5 = 3-\bbox[#FFEEAA;,1pt]{\vphantom{()}2\cdot 5}-5 = \bbox[#FFEEAA;,1pt]{\vphantom{()}3-10}-5 = -7-5 = -12
- \displaystyle 5+3\cdot\Bigl(5- \bbox[#FFEEAA;,1pt]{\frac{-4}{2}}\Bigr)-3\cdot(2+ \bbox[#FFEEAA;,1pt]{(2-4)}) = 5+3\cdot\bbox[#FFEEAA;,1pt]{(5-(-2))} -3\cdot\bbox[#FFEEAA;,1pt]{(2+(-2))} \displaystyle \qquad{}=5+3\cdot\bbox[#FFEEAA;,1pt]{(5+2)} -3\cdot\bbox[#FFEEAA;,1pt]{(2-2)} = 5+\bbox[#FFEEAA;,1pt]{\vphantom{()}3\cdot 7} - \bbox[#FFEEAA;,1pt]{\vphantom{()}3\cdot 0} = 5+21-0 = 26
"Unsichtbare" Klammern
Bei der Division soll der Zähler und der Nenner zuerst berechnet werden, bevor man dividiert. Man kann also sagen dass um den Zähler und Nenner "unsichtbare klammern" gibt.
Beispiel 2
- \displaystyle \frac{7+5}{2} = \frac{12}{2} = 6
- \displaystyle \frac{6}{1+2} = \frac{6}{3} = 2
- \displaystyle \frac{12+8}{6+4} =\frac{20}{10} = 2
Dies muss man besonders beachten wenn man einen Taschenrechner benutzt.
Division Vorlage:Displayed math
muss als \displaystyle (8 + 4 )/(2 + 4) geschrieben werden, sodass der Taschenrechner die Richtige Antwort \displaystyle 2 ergibt. Ein häufiger Fehler ist das man stattdessen \displaystyle 8 + 4/2 + 4 schreibt. Dies interpretiert der Rechner als \displaystyle 8 + 2 + 4 = 14.
Verschiedene Zahlen
The numbers we use to describe the “how many” and size, etc.., are called generically the real numbers and can be illustrated by a straight line real-number axis:
The real numbers "fill" the real-number axis:, ie. there are no holes or spaces along the real-number axis. Each point on the real-number axis can be specified by a decimal. The set of real numbers are all the decimals , and is denoted by R. The real-number axis also shows the relative magnitude of numbers; a number to the right is always greater than a number to the left. It is standard to classify the real numbers into the following types:
Natural numbers (usually symbolised by the letter N)
The numbers which are used when we calculate “how many”: 0, 1, 2, 3, 4, ...
Integers (Z)
The natural numbers and their negative counterparts: ..., -3, -2, -1, 0, 1, 2, 3, ...
Rational numbers (Q)
All the numbers that can be written as a ratio of whole numbers (fractions), for example, Vorlage:Displayed math
Note that even integers count as rational numbers, because Vorlage:Displayed math
A rational number can be written in various ways since, for example,
Vorlage:Displayed math
Beispiel 3
- Multiplying the numerator and denominator of a rational number with the same factor does not change the value of the number Vorlage:Displayed math
- Dividing the numerator and denominator of a rational number with the same factor, is called reducing and does not change the value of the number. Vorlage:Displayed math
Irrational numbers
The numbers on the real-number axis that can not be written as a fraction are called irrational numbers. Beispiels of irrational numbers are most roots, for example
\displaystyle \sqrt{2} and \displaystyle \sqrt{3}, but also numbers such as \displaystyle \pi
Decimal form
All types of real numbers can be written in decimal form, with an arbitrary number of decimal places. Decimal integers written to the right of the decimal point specify the number of tenths, hundredths, thousandths, and so on., In the same way as the integers to the left of the decimal point indicate the number of units, tens, hundreds, and so on.
Beispiel 4
A rational number can be written in decimal form by performing the division. Thus the \displaystyle \textstyle\frac{3}{4} is the same as "3 divided by 4", i.e. 0.75.
Read about long division on wikipedia.
Exempel 5
- \displaystyle \frac{1}{2} = 0{.}5 = 0{.}5\underline{0}
- \displaystyle \frac{1}{3} = 0{.}333333\,\ldots = 0{,}\underline{3}
- \displaystyle \frac{5}{12} = 0{,}4166666\,\ldots = 0{,}41\underline{6}
- \displaystyle \frac{1}{7} =0{.}142857142857\,\ldots = 0{,}\underline{142857}
(underlining signifies that the decimals are repeated)
As can be seen the rational numbers above have a periodic decimal expansion, ie. the decimal expansion, ends up with a finite block of digits that is repeated endlessly. This applies to all rational numbers and distinguishes them from the irrational numbers, which do not have a periodic pattern in their decimal expansion.
Conversely it is also true that all numbers with a periodic decimal expansion are rational.
Beispiel 6
The numbers \displaystyle \pi and\displaystyle \sqrt{2} are irrational and therefore have no periodic patterns in their decimal expansion.
- \displaystyle \pi=3{,}141 \,592 \, 653 \, 589 \,793 \, 238 \, 462 \,643\,\ldots
- \displaystyle \sqrt{2}=1{,}414 \,213 \, 562 \,373 \, 095 \, 048 \, 801 \, 688\,\ldots
Beispiel 7
- \displaystyle 0{,}600\,\ldots = 0{,}6 = \frac{6}{10} = \frac{3}{5}
- \displaystyle 0{,}35 = \frac{35}{100} = \frac{7}{20}
- \displaystyle 0{,}0025 = \frac{25}{10\,000} = \frac{1}{400}
Beispiel 8
The number \displaystyle x=0{,}215151515\,\ldots is rational, because it has a periodic decimal expansion. We can write this rational number as a ratio of two integers as follows.
Multiply the number by 10 which moves the decimal point one step to the right.
Multiply the number by \displaystyle 10\cdot 10\cdot 10 = 1000 moving the decimal point three steps to the right
Now we see that \displaystyle 1000\,x and \displaystyle 10\,x have the same decimal expansion so the difference between the numbers
must be an integer,
Vorlage:Displayed math So that
Rounding off
Since it is impractical to use long decimal expansions one often rounds off a number to an appropriate number of decimal places. The standard practise is that the numbers 0, 1, 2, 3 and 4 are rounded down while 5, 6, 7, 8 and 9 are rounded up.
We use the symbol \displaystyle \approx (is approximately equal to) to show that a rounding off has taken place.
Beispiel 9
Rounding off to 3 decimal places:
- \displaystyle 1{,}0004 \approx 1,000
- \displaystyle 0{,}9999 \approx 1{,}000
- \displaystyle 2{,}9994999 \approx 2{,}999
- \displaystyle 2{,}99950 \approx 3{,}000
Beispiel 10
Rounding off to 4 decimal places:
- \displaystyle \pi \approx 3{,}1416
- \displaystyle \frac{2}{3} \approx 0{,}6667
Comparing numbers
To indicate the relative size between numbers one uses the symbols > (is greater than), < (is less than) and = (is equal to). The relative size between two numbers can be determined either by giving the numbers in decimal form or by representing rational numbers as fractions with a common denominator.
Beispiel 11
- Which is greater \displaystyle \frac{1}{3} or \displaystyle 0{,}33?
We have that Vorlage:Displayed math So \displaystyle x>y as \displaystyle 100/300 > 99/300.
Alternatively, you can see that \displaystyle 1/3>0{,}33 as \displaystyle 1/3 = 0{,}3333\,\ldots > 0{,}33. - Which number is the larger of \displaystyle \frac{2}{5} and \displaystyle \frac{3}{7}?
Write the numbers with a common denominator, e.g. 35: Vorlage:Displayed math Thus \displaystyle \frac{3}{7}>\frac{2}{5} as \displaystyle \frac{15}{35} > \frac{14}{35}.
Study advice
Basic and final tests
After you have read the text and worked through the exercises, you should do the basic and final tests to pass this section. You can find the link to the tests in your student lounge.
Remember
Be careful! Many solutions are wrong because of copying errors or other simple errors, and not because your understanding of the question is wrong.
Reviews
For those of you who want to deepen your studies or need more detailed explanations consider the following references
Learn more about arithmetic in the English Wikipedia
Who discovered zero? Read more in "The MacTutor History of Mathematics archive"
Did you know that 0,999... = 1?
Useful web sites
How many colours are needed to colour a map? How many times does one need to shuffle a deck of cards? What is the greatest prime number? Are there any "lucky numbers"? What is the most beautiful number? Listen to the famous writer and mathematician Simon Singh, who among other things, tells about the magic numbers 4 and 7, about the prime numbers, about Keplers piles and about the concept of zero.
