1.1 Different types of numbers

From Förberedande kurs i matematik 1

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* Rational numbers
* Rational numbers
* Briefly about irrational numbers
* Briefly about irrational numbers
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* Real numbers
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* Real numbers.
}}
}}
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'''Learning outcomes:'''
'''Learning outcomes:'''
-
After this section, you will have learned to:
+
After this section you will have learned to:
* Calculate an expression that contains integers, the four arithmetic operations and parentheses.
* 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.
* Know the difference between the natural numbers, integers, rational numbers and irrational numbers.
-
* Convert fractions to decimals, and vice versa.
+
* 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 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.
* Determine an approximate value to a decimal number and a fraction to a given number of decimal places.
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{{Displayed math||<math>3+4+5=3+5+4=5+4+3=12\,\mbox{.}</math>}}
{{Displayed math||<math>3+4+5=3+5+4=5+4+3=12\,\mbox{.}</math>}}
-
As regards subtraction, the order is important of course.
+
As regards subtraction the order is important of course.
{{Displayed math||<math>5-2=3 \quad \mbox{whereas} \quad 2-5=-3\,\mbox{.}</math>}}
{{Displayed math||<math>5-2=3 \quad \mbox{whereas} \quad 2-5=-3\,\mbox{.}</math>}}
-
When we talk about the difference between two numbers we usually mean the difference between the larger and the smaller . Thus, we understand that, the difference between 2 and 5 is 3.
+
When we talk about the difference between two numbers we usually mean the difference between the larger and the smaller. Thus, we say the difference between 2 and 5 is 3.
-
When numbers are multiplied, their order is not important.
+
When numbers are multiplied their order is not important.
{{Displayed math||<math>3 \cdot 4 \cdot 5=3 \cdot 5 \cdot 4 = 5 \cdot 4 \cdot 3 = 60 \,\mbox{.}</math>}}
{{Displayed math||<math>3 \cdot 4 \cdot 5=3 \cdot 5 \cdot 4 = 5 \cdot 4 \cdot 3 = 60 \,\mbox{.}</math>}}
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-
If several mathematical operations occur in a mathematical expression, it is important to have a standard for the order in which the operations are to be carried out. The following rules apply:
+
If several mathematical operations occur in a mathematical expression it is important to have a standard for the order in which the operations are to be carried out. The following rules apply:
* Parentheses ( brackets, "innermost brackets" first)
* Parentheses ( brackets, "innermost brackets" first)
-
* Multiplication and Division (from left to right)
+
* Multiplication and division (from left to right)
* Addition and subtraction (from left to right)
* Addition and subtraction (from left to right)
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<div class="exempel">
<div class="exempel">
-
'''Exempel 2'''
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'''Example 2'''
<ol type="a">
<ol type="a">
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== Different types of numbers ==
== Different types of numbers ==
-
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:
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The numbers we use to describe quantity, size, etc... are generically called the real numbers and can be illustrated by a straight line real-number axis:
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-
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:
+
The real numbers "fill" the real-number axis, i.e. 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:
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''Rational numbers '' ('''Q''')
''Rational numbers '' ('''Q''')
-
All the numbers that can be written as a ratio of whole numbers (fractions), for example,
+
All the numbers that can be written as a ratio of whole numbers (fractions). For example:
{{Displayed math||<math>-\frac{3}{4},\ \frac{3}{2}, \ \frac{37}{128}, \quad\mbox{etc.}</math>}}
{{Displayed math||<math>-\frac{3}{4},\ \frac{3}{2}, \ \frac{37}{128}, \quad\mbox{etc.}</math>}}
-
Note that even integers count as rational numbers, because
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Note that integers are classed as rational numbers because
{{Displayed math||<math>-1 = \frac{-1}{1},\quad 0 = \frac{0}{1},\quad 1 = \frac{1}{1},\quad 2 = \frac{2}{1},\quad\mbox{etc.}</math>}}
{{Displayed math||<math>-1 = \frac{-1}{1},\quad 0 = \frac{0}{1},\quad 1 = \frac{1}{1},\quad 2 = \frac{2}{1},\quad\mbox{etc.}</math>}}
-
A rational number can be written in various ways since, for example,
+
A rational number can be written in various ways. For example:
{{Displayed math||<math>2 = \frac{2}{1}=\frac{4}{2}=\frac{6}{3}=\frac{8}{4} =\frac{100}{50}=\frac{384}{192}\quad\mbox{etc.}</math>}}
{{Displayed math||<math>2 = \frac{2}{1}=\frac{4}{2}=\frac{6}{3}=\frac{8}{4} =\frac{100}{50}=\frac{384}{192}\quad\mbox{etc.}</math>}}
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<ol type="a">
<ol type="a">
-
<li> Multiplying the numerator and denominator of a rational number with the same factor does not change the value of the number
+
<li> Multiplying the numerator and denominator of a rational number with the same factor does not change the value of the number.
{{Displayed math||<math>\frac{1}{3} = \frac{1\cdot 2}{3\cdot 2}
{{Displayed math||<math>\frac{1}{3} = \frac{1\cdot 2}{3\cdot 2}
= \frac{2}{6} = \frac{1\cdot 5}{3\cdot 5}
= \frac{2}{6} = \frac{1\cdot 5}{3\cdot 5}
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-
The numbers on the real-number axis that can not be written as a fraction are called irrational numbers. Examples of irrational numbers are most roots, for example
+
The numbers on the real-number axis that can not be written as a fraction are called irrational numbers. Examples of irrational numbers are most roots, for example:
<math>\sqrt{2}</math> and <math>\sqrt{3}</math>, but also numbers such as <math>\pi</math>
<math>\sqrt{2}</math> and <math>\sqrt{3}</math>, but also numbers such as <math>\pi</math>
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-
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.
+
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.
<center>{{:1.1 - Figure - Decimal form}}</center>
<center>{{:1.1 - Figure - Decimal form}}</center>
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'''Example 4'''
'''Example 4'''
-
{{Displayed math||<math>1234{,}5678 = 1000 + 200 + 30 + 4 + \frac{5}{10} + \frac{6}{100} + \frac{7}{1000} + \frac{8}{10000}</math>}}
+
{{Displayed math||<math>1234{.}5678 = 1000 + 200 + 30 + 4 + \frac{5}{10} + \frac{6}{100} + \frac{7}{1000} + \frac{8}{10000}</math>}}
</div>
</div>
-
A rational number can be written in decimal form by performing the division. Thus the <math>\textstyle\frac{3}{4} </math> is the same as "3 divided by 4", i.e. 0.75.
+
A rational number can be written in decimal form by performing the division. Thus <math>\textstyle\frac{3}{4}</math> is the same as "3 divided by 4", i.e. 0.75.
Read about [http://en.wikipedia.org/wiki/Long_division long division] on wikipedia.
Read about [http://en.wikipedia.org/wiki/Long_division long division] on wikipedia.
<div class="exempel">
<div class="exempel">
-
'''Exempel 5'''
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'''Example 5'''
<ol type="a">
<ol type="a">
-
<li><math>\frac{1}{2} = 0{.}5 = 0{.}5\underline{0}</math></li>
+
<li><math>\frac{1}{2} = 0.5 = 0.5\underline{0}</math></li>
-
<li><math>\frac{1}{3} = 0{.}333333\,\ldots = 0{,}\underline{3}</math></li>
+
<li><math>\frac{1}{3} = 0.333333\,\ldots = 0.\underline{3}</math></li>
-
<li><math> \frac{5}{12} = 0{,}4166666\,\ldots = 0{,}41\underline{6}</math></li>
+
<li><math> \frac{5}{12} = 0.4166666\,\ldots = 0.41\underline{6}</math></li>
-
<li><math>\frac{1}{7} =0{.}142857142857\,\ldots = 0{,}\underline{142857}</math></li>
+
<li><math>\frac{1}{7} =0.142857142857\,\ldots = 0.\underline{142857}</math></li>
</ol>
</ol>
(underlining signifies that the decimals are repeated)
(underlining signifies that the decimals are repeated)
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-
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.
+
As can be seen the rational numbers above have a periodic decimal expansion, i.e. 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.
-
 
+
-
Conversely it is also true that all numbers with a periodic decimal expansion are rational.
+
<div class="exempel">
<div class="exempel">
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The numbers <math>\pi</math> and<math>\sqrt{2}</math> are irrational and therefore have no periodic patterns in their decimal expansion.
The numbers <math>\pi</math> and<math>\sqrt{2}</math> are irrational and therefore have no periodic patterns in their decimal expansion.
<ol type="a">
<ol type="a">
-
<li><math>\pi=3{,}141 \,592 \, 653 \, 589 \,793 \, 238 \, 462 \,643\,\ldots</math></li>
+
<li><math>\pi=3.141 \,592 \, 653 \, 589 \,793 \, 238 \, 462 \,643\,\ldots</math></li>
-
<li><math>\sqrt{2}=1{,}414 \,213 \, 562 \,373 \, 095 \, 048 \, 801 \, 688\,\ldots</math></li>
+
<li><math>\sqrt{2}=1.414 \,213 \, 562 \,373 \, 095 \, 048 \, 801 \, 688\,\ldots</math></li>
</ol>
</ol>
</div>
</div>
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<ol type="a">
<ol type="a">
-
<li><math> 0{,}600\,\ldots = 0{,}6 = \frac{6}{10} = \frac{3}{5} </math></li>
+
<li><math> 0.600\,\ldots = 0.6 = \frac{6}{10} = \frac{3}{5} </math></li>
-
<li><math> 0{,}35 = \frac{35}{100} = \frac{7}{20} </math></li>
+
<li><math> 0.35 = \frac{35}{100} = \frac{7}{20} </math></li>
-
<li><math> 0{,}0025 = \frac{25}{10\,000} = \frac{1}{400} </math></li>
+
<li><math> 0.0025 = \frac{25}{10\,000} = \frac{1}{400} </math></li>
</ol>
</ol>
</div>
</div>
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'''Example 8'''
'''Example 8'''
-
The number <math>x=0{,}215151515\,\ldots</math> is rational, because it has a periodic decimal expansion. We can write this rational number as a ratio of two integers as follows.
+
The number <math>x=0.215151515\,\ldots</math> 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 10 which moves the decimal point one step to the right.
-
{{Displayed math||<math>\quad 10\,x = 2{,}151515\,\ldots</math>}}
+
{{Displayed math||<math>\quad 10\,x = 2.151515\,\ldots</math>}}
Multiply the number by <math>10\cdot 10\cdot 10 = 1000</math> moving the decimal point three steps to the right
Multiply the number by <math>10\cdot 10\cdot 10 = 1000</math> moving the decimal point three steps to the right
-
{{Displayed math||<math>\quad 1000\,x = 215{,}1515\,\ldots</math>}}
+
{{Displayed math||<math>\quad 1000\,x = 215.1515\,\ldots</math>}}
Now we see that <math>1000\,x</math> and <math>10\,x</math> have the same decimal expansion so the difference between the numbers
Now we see that <math>1000\,x</math> and <math>10\,x</math> have the same decimal expansion so the difference between the numbers
-
{{Displayed math||<math>\quad 1000x - 10x = 215{,}1515\,\ldots - 2{,}151515\,\ldots </math>}}
+
{{Displayed math||<math>\quad 1000x - 10x = 215.1515\,\ldots - 2.151515\,\ldots </math>}}
must be an integer,
must be an integer,
-
{{Displayed math||<math>\quad 990x = 213\mathrm{.}</math>}}
+
{{Displayed math||<math>\quad 990x = 213.</math>}}
-
So that
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So
{{Displayed math||<math>\quad x =\frac{213}{990} = \frac{71}{330}\,\mbox{.}</math>}}
{{Displayed math||<math>\quad x =\frac{213}{990} = \frac{71}{330}\,\mbox{.}</math>}}
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-
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.
+
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.
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Rounding off to 3 decimal places:
Rounding off to 3 decimal places:
<ol type="a">
<ol type="a">
-
<li><math>1{,}0004 \approx 1,000</math></li>
+
<li><math>1.0004 \approx 1.000</math></li>
-
<li><math>0{,}9999 \approx 1{,}000</math></li>
+
<li><math>0.9999 \approx 1.000</math></li>
-
<li><math>2{,}9994999 \approx 2{,}999</math></li>
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<li><math>2.9994999 \approx 2.999</math></li>
-
<li><math>2{,}99950 \approx 3{,}000</math></li>
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<li><math>2.99950 \approx 3.000</math></li>
</ol>
</ol>
</div>
</div>
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Rounding off to 4 decimal places:
Rounding off to 4 decimal places:
<ol type="a">
<ol type="a">
-
<li><math>\pi \approx 3{,}1416 </math></li>
+
<li><math>\pi \approx 3.1416 </math></li>
-
<li><math>\frac{2}{3} \approx 0{,}6667 </math></li>
+
<li><math>\frac{2}{3} \approx 0.6667 </math></li>
</ol>
</ol>
</div>
</div>
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<ol type="a">
<ol type="a">
-
<li>Which is greater <math>\frac{1}{3}</math> or <math>0{,}33</math>?<br/><br/>
+
<li>Which is greater <math>\frac{1}{3}</math> or <math>0.33</math>?<br/><br/>
-
We have that
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We have
-
{{Displayed math||<math>x =\frac{1}{3} = \frac{100}{300}\quad\text{and}\quad y = 0{,}33 =\frac{33}{100} = \frac{99}{300}\mathrm{.}</math>}}
+
{{Displayed math||<math>x =\frac{1}{3} = \frac{100}{300}\quad\text{and}\quad y = 0.33 =\frac{33}{100} = \frac{99}{300}\mathrm{,}</math>}}
-
So <math>x>y</math> as <math>100/300 > 99/300</math>.<br/><br/>
+
so <math>x>y</math> as <math>100/300 > 99/300</math>.<br/><br/>
-
Alternatively, you can see that <math>1/3>0{,}33</math> as <math>1/3 = 0{,}3333\,\ldots > 0{,}33</math>.</li>
+
Alternatively, you can see that <math>1/3>0.33</math> as <math>1/3 = 0.3333\,\ldots > 0.33</math>.</li>
<li>Which number is the larger of <math>\frac{2}{5}</math> and <math>\frac{3}{7}</math>?<br/><br/>
<li>Which number is the larger of <math>\frac{2}{5}</math> and <math>\frac{3}{7}</math>?<br/><br/>

Revision as of 16:06, 21 October 2008

       Theory          Exercises      

Contents:

  • Natural numbers
  • Negative numbers
  • Order of precedence and parenthesis
  • Rational numbers
  • Briefly about irrational numbers
  • Real numbers.

Learning outcomes:

After this section you will have learned to:

  • 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.


Calculations with numbers

Calculating with numbers requires you to perform a series of operations. These are the four basic operations of arithmetic. Here are some concepts that are helpful to know in order to understand a mathematical text:


[Image]


When you add numbers the sum does not depend on the order in which the terms are added together

\displaystyle 3+4+5=3+5+4=5+4+3=12\,\mbox{.}

As regards subtraction the order is important of course.

\displaystyle 5-2=3 \quad \mbox{whereas} \quad 2-5=-3\,\mbox{.}

When we talk about the difference between two numbers we usually mean the difference between the larger and the smaller. Thus, we say the difference between 2 and 5 is 3.


When numbers are multiplied their order is not important.

\displaystyle 3 \cdot 4 \cdot 5=3 \cdot 5 \cdot 4 = 5 \cdot 4 \cdot 3 = 60 \,\mbox{.}

For division the order is of importance.

\displaystyle \frac{6}{3} = 2\quad\mbox{whereas}\quad\frac{3}{6} = 0{,}5 \,\mbox{.}


Hierarchy of arithmetic operations (priority rules)

If several mathematical operations occur in a mathematical expression it is important to have a standard for the order in which the operations are to be carried out. The following rules apply:

  • Parentheses ( brackets, "innermost brackets" first)
  • Multiplication and division (from left to right)
  • Addition and subtraction (from left to right)

Example 1

  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
  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
  3. \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

"Invisible" parentheses

For division the numerator and the denominator must be calculated separately before the division is carried out. One can therefore say that there are "invisible parentheses" around the numerator and denominator.

Example 2

  1. \displaystyle \frac{7+5}{2} = \frac{12}{2} = 6
  2. \displaystyle \frac{6}{1+2} = \frac{6}{3} = 2
  3. \displaystyle \frac{12+8}{6+4} =\frac{20}{10} = 2

This is especially important if calculators are used.

Division

\displaystyle \frac{8+4}{2+4}

must be written as \displaystyle (8 + 4 )/(2 + 4) for a calculator so that the correct answer \displaystyle 2 may be obtained. A common mistake is to write \displaystyle 8 + 4/2 + 4, which the calculator interprets as \displaystyle 8 + 2 + 4 = 14.


Different types of numbers

The numbers we use to describe quantity, size, etc... are generically called the real numbers and can be illustrated by a straight line real-number axis:


[Image]


The real numbers "fill" the real-number axis, i.e. 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:

\displaystyle -\frac{3}{4},\ \frac{3}{2}, \ \frac{37}{128}, \quad\mbox{etc.}

Note that integers are classed as rational numbers because

\displaystyle -1 = \frac{-1}{1},\quad 0 = \frac{0}{1},\quad 1 = \frac{1}{1},\quad 2 = \frac{2}{1},\quad\mbox{etc.}


A rational number can be written in various ways. For example:

\displaystyle 2 = \frac{2}{1}=\frac{4}{2}=\frac{6}{3}=\frac{8}{4} =\frac{100}{50}=\frac{384}{192}\quad\mbox{etc.}

Example 3

  1. Multiplying the numerator and denominator of a rational number with the same factor does not change the value of the number.
    \displaystyle \frac{1}{3} = \frac{1\cdot 2}{3\cdot 2} 
         = \frac{2}{6} = \frac{1\cdot 5}{3\cdot 5}
     = \frac{5}{15}\quad\mbox{etc.}
  2. 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.
    \displaystyle \frac{75}{105} =\frac{75/5}{105/5} 
         = \frac{15}{21} = \frac{15/3}{21/3} = \frac{5}{7}
     \quad\mbox{etc.}

Irrational numbers


The numbers on the real-number axis that can not be written as a fraction are called irrational numbers. Examples 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.

[Image]

Example 4

\displaystyle 1234{.}5678 = 1000 + 200 + 30 + 4 + \frac{5}{10} + \frac{6}{100} + \frac{7}{1000} + \frac{8}{10000}


A rational number can be written in decimal form by performing the division. Thus \displaystyle \textstyle\frac{3}{4} is the same as "3 divided by 4", i.e. 0.75.

Read about long division on wikipedia.

Example 5

  1. \displaystyle \frac{1}{2} = 0.5 = 0.5\underline{0}
  2. \displaystyle \frac{1}{3} = 0.333333\,\ldots = 0.\underline{3}
  3. \displaystyle \frac{5}{12} = 0.4166666\,\ldots = 0.41\underline{6}
  4. \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, i.e. 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.

Example 6

The numbers \displaystyle \pi and\displaystyle \sqrt{2} are irrational and therefore have no periodic patterns in their decimal expansion.

  1. \displaystyle \pi=3.141 \,592 \, 653 \, 589 \,793 \, 238 \, 462 \,643\,\ldots
  2. \displaystyle \sqrt{2}=1.414 \,213 \, 562 \,373 \, 095 \, 048 \, 801 \, 688\,\ldots

Example 7

  1. \displaystyle 0.600\,\ldots = 0.6 = \frac{6}{10} = \frac{3}{5}
  2. \displaystyle 0.35 = \frac{35}{100} = \frac{7}{20}
  3. \displaystyle 0.0025 = \frac{25}{10\,000} = \frac{1}{400}

Example 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.

\displaystyle \quad 10\,x = 2.151515\,\ldots

Multiply the number by \displaystyle 10\cdot 10\cdot 10 = 1000 moving the decimal point three steps to the right

\displaystyle \quad 1000\,x = 215.1515\,\ldots

Now we see that \displaystyle 1000\,x and \displaystyle 10\,x have the same decimal expansion so the difference between the numbers

\displaystyle \quad 1000x - 10x = 215.1515\,\ldots - 2.151515\,\ldots

must be an integer,

\displaystyle \quad 990x = 213.

So

\displaystyle \quad x =\frac{213}{990} = \frac{71}{330}\,\mbox{.}

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.

Example 9

Rounding off to 3 decimal places:

  1. \displaystyle 1.0004 \approx 1.000
  2. \displaystyle 0.9999 \approx 1.000
  3. \displaystyle 2.9994999 \approx 2.999
  4. \displaystyle 2.99950 \approx 3.000

Example 10

Rounding off to 4 decimal places:

  1. \displaystyle \pi \approx 3.1416
  2. \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.

Example 11

  1. Which is greater \displaystyle \frac{1}{3} or \displaystyle 0.33?

    We have
    \displaystyle x =\frac{1}{3} = \frac{100}{300}\quad\text{and}\quad y = 0.33 =\frac{33}{100} = \frac{99}{300}\mathrm{,}

    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.
  2. 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:
    \displaystyle \frac{2}{5} = \frac{14}{35} \quad\text{and}\quad\frac{3}{7} = \frac{15}{35}\mathrm{.}
    Thus \displaystyle \frac{3}{7}>\frac{2}{5} as \displaystyle \frac{15}{35} > \frac{14}{35}.

Exercises


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"

Long division

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.

Listen to the BBC programmes "5 Numbers"

Listen to the BBC programmes "Another 5 numbers"

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