2. Algebra

From Förberedande kurs i matematik 1

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Algebra is the branch of mathematics that is concerned with symbolic expressions and variables, not just calculations with numbers.
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Algebra is needed in many situations. For example, it can be used to represent situations in which one or more of the quantities is unknown, and hence perhaps to find the values of these quantities. It can also be used to represent geometrical structures, such as curves on the plane.
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In some cases, too, we use symbols because it is important that a number be exactly specified; perhaps, for example, a circle is to have a circumference of exactly <math>4\pi</math> units, or the hypotenuse of a triangle is to be <math>\sqrt{3}</math>, or the value of a constant is to be exactly <math>\dfrac{1-\ln 2}{3}</math>.
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| width="220" height="203" |<math>\text{@(a class="image" href="http://smaug.nti.se/temp/KTH/film4.html" target="_blank")@(img src="http://wiki.math.se/wikis/2008/forberedandematte1/img_auth.php/0/00/Lars_och_Elin.jpg" alt="Film om algebra")@(/img)@(/a)}</math>
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'''Varför räknar vi med bokstäver och vem kom på detta?'''
 
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[[Image:grafisk lösning.gif|thumb|250px|A linear equation with two unknowns can be interpreted as a straight line in a coordinate system. The solution (''x'',''y'') of these two simultaneous equations corresponds to the common point of these lines, that is their point of intersection. ]]
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Then it may be easier just to call the number, for example, ''a''. In return you must accept that at the end of a calculation, you may not arrive at a numerical value, but instead may finish up with an expression that contains ''a''.
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''Titta på videon där universitetslektor Lasse Svensson berättar om hur algebran utvecklats och svarar på Elins frågor om Del 2 i kursen.''
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A common situation where algebra may be necessary is simplification. It is often very important to simplify an expression, such as before differentiation, or when an equation is to be solved.
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Simplification reduces the risk of careless mistakes and avoids unnecessary work. To simplify means to transform an expression from one form to another, which exactly equivalent mathematically but which is in some sense simpler. What is meant by &rdquo;simple&rdquo; is sometimes obvious, but it can also depend on what you want to do with the expression.
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When differentiating, it may be advantageous to transform the expression into a sum of a number of terms. On the other hand when we solve an equation, it may be advantageous to reformulate the expression as a product of a number of factors. Therefore, one needs to be proficient in converting expressions into different forms.
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'''It is important to note that the material in this section&mdash; as well as in other parts of the course &mdash; is designed so that one does not use calculators.'''
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''Calculators are not used at all on some degree courses, and on others their use is limited.''
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<div class="inforuta" style="width:580px;">
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'''To do well in Algebra'''
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# Start by reading the section's theory and study the examples.
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#Work through the exercises and try to solve them without using a calculator. Make sure that you have the right answer by clicking on the answer button. If you do not have it, you can click on the solution button to see what went wrong
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#Then go ahead and answer the questions in the basic test of the section.
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# If you get stuck on a point, check to see if someone else has discussed the point in the forum belonging to the section. If not, take up the point yourself. Your teacher (or a student) will respond to your question within a few hours.
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#When you have finished with the exercises and the basic test in a section you should take the final test to get a pass for the section. The requirement here is to answer correctly three questions in a row before you can move on to the next section.
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# When you have answered all the questions correctly in both the basic and the final test of this section you will have a pass for this section and can move on to Part 3 of the course.
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Algebra är den gren av matematiken som behandlar räkning med symboliska uttryck och variabler och inte bara räkning med tal.
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&nbsp;&nbsp;&nbsp;P.S. If you feel that you are very familiar with the contents of a section you can test yourself by going directly to the basic and final tests. You must answer all the questions correctly in a test, but you may do the test several times if you do not succeed at the first attempt. It is your final results that will appear in the statistics.
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Algebra behövs i många situationer, t.ex. kan algebra användas till att beskriva matematiska problem och till att lösa ekvationer. Det går bland annat att beskriva geometriska fakta med hjälp av algebraiska påståenden, och många problem går att lösa med hjälp av algebraiska operationer.
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I en del fall kan man inte räkna ut värdet av ett uttryck till ett numeriskt värde. Anledningen kan vara att uttrycket innehåller obekanta parametrar eller variabler. Det kan också vara så att det är viktigt att ett tal är exakt angivet, t.ex. att en viss cirkel har en omkrets som är exakt <math>4\pi</math>, eller hypotenusans längd för en triangel är <math>\sqrt{3}</math>, eller varför inte att värdet på en konstant är <math>\dfrac{1-\ln 2}{3}</math>.
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[[Bild:grafisk lösning.gif|thumb|250px|En linjär ekvation med två obekanta kan ses som en linje i ett koordinatsystem. Den gemensamma lösningen (''x'',''y'') till dessa ekvationer motsvaras då av den gemensamma punkten för dessa linjer, dvs. skärningspunkten.]]
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Då kan det vara enklast att i uträkningarna kalla talet för exempelvis ''a''. Som svar kan man också acceptera att man inte kommer fram till ett numeriskt värde, utan i stället får ett uttryck som innehåller ''a''.
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En vanlig situation där man kan behöva algebra är förenkling. Det är ofta mycket viktigt att förenkla ett uttryck, t.ex. innan man skall derivera, eller när man löser en ekvation.
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Genom att förenkla minskar man risken för slarvfel och man slipper onödigt arbete. Att förenkla innebär att skriva om ett uttryck från en form till en annan. Vilken form som betraktas som &rdquo;enkel&rdquo; är ibland uppenbart, men det kan också bero på vad man vill göra med uttrycket.
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När man deriverar kan det vara fördelaktigt att formulera uttrycket som en summa av ett antal termer. När man löser en ekvation kan det vara fördelaktigt att formulera det som en produkt av ett antal faktorer. Därför behöver man kunna omvandla uttryck mellan olika former.
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'''Observera att materialet i denna kursdel &mdash; liksom i övriga delar av kursen &mdash; är utformat för att man ska arbeta med det utan hjälp av miniräknare.'''
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''När du kommer till högskolan kommer du nämligen inte att få använda miniräknare på dina "tentor", åtminstone inte på grundkurserna.''
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<div class="inforuta">
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'''Så här lyckas du med Algebran'''
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#Börja med att läsa genomgången till ett avsnitt och tänka igenom exemplen.
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#Arbeta sedan med övningsuppgifterna och försök att lösa dem utan miniräknare. Kontrollera att du kommit fram till rätt svar genom att klicka på svarsknappen. Har du inte det, så kan du klicka på lösningsknappen, för att se hur du ska göra.
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#Gå därefter vidare och svara på frågorna i grundprovet som hör till avsnittet.
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# Skulle du fastna, se efter om någon ställt en fråga om just detta i avsnittets forum. Ställ annars en fråga om du undrar över något. Din lärare (eller en studiekamrat) kommer att besvara den inom några timmar.
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#När du är klar med övningsuppgifterna och grundproven i ett avsnitt så ska du göra slutprovet för att bli godkänd på avsnittet. Där gäller det att svara rätt på tre frågor i följd för att kunna gå vidare.
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#När du fått alla rätt på både grundprov och slutprov, så är du godkänd på den delen och kan gå vidare till Del 3 i kursen.
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&nbsp;&nbsp;&nbsp;PS. Tycker du att innehållet i ett avsnitt känns väldigt bekant, så kan du testa att gå direkt till grundprovet och slutprovet. Du måste få alla rätt på ett prov, men kan göra om provet flera gånger, om du inte lyckas på första försöket. Det är ditt senaste resultat som visas i statistiken.
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Current revision


Algebra is the branch of mathematics that is concerned with symbolic expressions and variables, not just calculations with numbers.

Algebra is needed in many situations. For example, it can be used to represent situations in which one or more of the quantities is unknown, and hence perhaps to find the values of these quantities. It can also be used to represent geometrical structures, such as curves on the plane.

In some cases, too, we use symbols because it is important that a number be exactly specified; perhaps, for example, a circle is to have a circumference of exactly \displaystyle 4\pi units, or the hypotenuse of a triangle is to be \displaystyle \sqrt{3}, or the value of a constant is to be exactly \displaystyle \dfrac{1-\ln 2}{3}.


A linear equation with two unknowns can be interpreted as a straight line in a coordinate system. The solution (x,y) of these two simultaneous equations corresponds to the common point of these lines, that is their point of intersection.
A linear equation with two unknowns can be interpreted as a straight line in a coordinate system. The solution (x,y) of these two simultaneous equations corresponds to the common point of these lines, that is their point of intersection.

Then it may be easier just to call the number, for example, a. In return you must accept that at the end of a calculation, you may not arrive at a numerical value, but instead may finish up with an expression that contains a.

A common situation where algebra may be necessary is simplification. It is often very important to simplify an expression, such as before differentiation, or when an equation is to be solved.


Simplification reduces the risk of careless mistakes and avoids unnecessary work. To simplify means to transform an expression from one form to another, which exactly equivalent mathematically but which is in some sense simpler. What is meant by ”simple” is sometimes obvious, but it can also depend on what you want to do with the expression.

When differentiating, it may be advantageous to transform the expression into a sum of a number of terms. On the other hand when we solve an equation, it may be advantageous to reformulate the expression as a product of a number of factors. Therefore, one needs to be proficient in converting expressions into different forms.


It is important to note that the material in this section— as well as in other parts of the course — is designed so that one does not use calculators.

Calculators are not used at all on some degree courses, and on others their use is limited.


To do well in Algebra

  1. Start by reading the section's theory and study the examples.
  2. Work through the exercises and try to solve them without using a calculator. Make sure that you have the right answer by clicking on the answer button. If you do not have it, you can click on the solution button to see what went wrong
  3. Then go ahead and answer the questions in the basic test of the section.
  4. If you get stuck on a point, check to see if someone else has discussed the point in the forum belonging to the section. If not, take up the point yourself. Your teacher (or a student) will respond to your question within a few hours.
  5. When you have finished with the exercises and the basic test in a section you should take the final test to get a pass for the section. The requirement here is to answer correctly three questions in a row before you can move on to the next section.
  6. When you have answered all the questions correctly in both the basic and the final test of this section you will have a pass for this section and can move on to Part 3 of the course.

   P.S. If you feel that you are very familiar with the contents of a section you can test yourself by going directly to the basic and final tests. You must answer all the questions correctly in a test, but you may do the test several times if you do not succeed at the first attempt. It is your final results that will appear in the statistics.

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