Algebra - slimstuderen.nl SlimStuderen... · the three dots indicate that the digit 3 is repeated indefinitely. This is also called a real number. When applied to a real number, the

Overview Study Material Business Mathematics 1 2015-2016

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1

Algebra

The Real Numbers

The basic numbers are 1,2,3,4…, these numbers are natural numbers and also called

positive integers. The positive integers, together with the negative integers 0,-1,-2,-3,…,

make up the integers. The integers can be represented on a number line (showed

below), where the arrow provides the direction in which the numbers increase.

Rational numbers are all numbers that are written in the form c/d (where c and d are

both integers). They can also be presented on a number line. An integer (say n) is also a

rational number, because n = n/1. There are also Irrational Numbers like √3, this is

because there are no integers p and q such that √3 = p/q.

The way people write numbers today is called the decimal system or base 10 system.

This is a positional system with 10 as the base number, where every natural number can

be written by using only the symbols: 1,2,3,4 (…) 9, which are called digits. An example

of how the decimal system works:

This enables us to also express rational numbers other than natural numbers, for

example:

Rational numbers that can be written exactly, by using only a finite number of decimal

places, are called finite decimal fractions. If this is not the case, we call them infinite

decimal fractions. An example of an infinite decimal fraction is: 20/6 = 3,3333…, where

the three dots indicate that the digit 3 is repeated indefinitely. This is also called a real

number.

When applied to a real number, the four arithmetic operations (addition, subtraction,

multiplication and division) always result in a real number. The only exception is that

they cannot devide by 0.

Integer Powers

When b is any number and n is any natural number, the following applies:

This expression is called the nth power of b: where b is the base and n the exponent.

A special case is , where b can be any number except 0. Powers with negative

exponents can be defined as:

0123 1071021031011327

43210 10/510/310/710/21042735,4

bn = b×b×... ×bn factors

10 b

nn bb /1



2

Properties of Powers

There are several rules that apply to Powers. The most important two are:

The division of two powers with the same base goes like:

Finally, note that:

One of the most common errors committed is:

is NOT (in general).

Compound Interest

Suppose a company deposits an amount of K euros in a bank account paying p% of

interest per year. After t years the amount of money in the account will be:

Where is the growth factor for a growth of p%.

If you want to calculate the original amount of money deposited in the bank, t years ago.

Where the bank pays an interest rate of p% per year, use the following formula, where K

is the amount of money now and X is the original amount:

This formula can also be used for calculating the depreciation of an asset. Suppose a car

worth K euros, with a depreciation rate of p%. After t years, the value of the car will be:

Here is called the growth factor for a decline of p%.

Rules of Algebra

If a, b, and c are arbitrary numbers, the following rules are the most important:

These rules can be combined.

lklk aaa lklk aa )(

lklk aaa /

kkk cbbc )(kkkkk cbcbcb /)/(

kcb )( bk +ck

tpK )

1001(

1001

p

tpKX )

1001/(

tpK )

1001(

1001

p

abba )()( cbacba

aa 00)( aa

baab )()( bcacab

aa 1

011 avooraa

abbaba )()(

abba ))((

acabcba )(

bcaccba )(



3

Quadratic Identities

There are 3 important quadratic identities, which are very important to remember:

The last one is called differences-of-squares-formula. This is the proof for the formula:

Algebraic Expressions

The expression is an algebraic expression, because it involves

letters. We call the terms in the expression. The numbers 5, 3 and 7

are the numerical coefficients of the terms. Terms where only the numerical coefficient is

different, are called terms of the same type. For example:

Factoring

When we write 81 as 9x9, we have factored this number. Algebraic expressions can be

factored in a similar way:

With Factoring, we mean: to express it as a product of simpler factors. However, most

algebraic expressions can not be factored.

Fractions

In the equation , c is the numerator and d is the denominator. The fraction is a

proper fraction because the numerator is smaller than the denominator. An improper

fraction is: , because the numerator is larger (or equal to) the denominator. This

fraction can be re-written as: which is called a mixed number.

222 2)( bababa 222 2)( bababa

22))(( bababa

2222))(( babababababa

322 735 abaab

5ab,3a2and7b2a3

4242 63 zxenzx

)3(103010 2 xxxx

d

c

7

4

10

15

2

11



4

The most important properties of fractions are listed below:

Fractional Powers

An example of a power with a fractional exponent is , when and x = ½. We

define as √b, the square root of b.

√a is a nonnegative number because when multiplied by itself, the answer gives a.

If a and b are nonnegative numbers, then two rules apply:

(i)

(ii)

For the second rule, b ≠ 0.

In addition:

It is known that and have the same solution, which is 4. When solving the

equation , the solution must be written like . Note however, that √4

means only 2, not -2.

Nth Roots

The following rule applies to :

We can expand this rule:

Here a must be positive (a>0), in addition p must be an integer and q must be a natural

number.

0,0

cb

b

a

cb

ca

b

a

b

a

b

a

)1()(

)1()(

b

a

b

a

b

a

b

a

)1()1(

c

ba

c

b

c

a

db

cbda

d

c

b

a

c

bca

c

ba

c

ba

c

ba

db

ca

d

c

b

a

cb

da

c

d

b

a

d

c

b

a

xb 0b2/1b

b

a

b

a

baba

22 22

42 x 24 x

a1/n

nn aa /1

pqpqqp aaa )()( /1/

baab



5

Inequalities

Properties:

a>0 and b>0, imply a+b>0 and a x b > 0;

a>b means that a-b>0;

a≥b means that a-b≥0;

If a > b, then a + c > b + c for all c.

To deal with more complicated inequalities involves using the following properties:

According to these properties, the following rules apply:

If the two sides of the inequality are multiplied by a positive number, the direction

of the inequality is preserved;

If the two sides of an inequality are multiplied by a negative number, the direction

of the inequality is reversed.

Sign Diagrams

For the solution of an equation, for example , a sign diagram might be

useful. For an example of a sign diagram, review the compulsory study material.

Double Inequalities

An example of a double inequality is:

It is natural to write:

Intervals and Absolute Values

Let y and z be any two numbers on the real line. All the numbers that lie between y and

z, are called an interval. There are four different intervals that all have a and b as

endpoints (see the figure below).

Notation Name Interval x

(a, b) Open interval

[a, b] Closed interval

(a,b] Half-open interval

[a,b) Half-open interval

All four intervals have the same length (b –a), only the endpoints differ. The open

interval for example uses neither of the endpoints. All the intervals mentioned above are

bounded intervals. An example of an unbounded interval is:

[a,∞) = for all values of x, with

Here, ∞ stands for infinity. This symbol is not a number at all, the symbol only indicates

that we are considering the collection of all numbers larger than a, without any upper

bound on the size of the number. Similarly, (-∞,b) has no lower bound. When all real

numbers are permitted, we say: (-∞, ∞).

a> 0 and b> 0; a+b> 0; a×b> 0

a> bthen a+c> b+c

a> b and b> c®a> c

a> b and c> 0®ac> bc

a> b and c< 0®ac< bc

a> b and c> d®a+c> b+d

0)2)(5( xx

bdendc

bdc

bxa

bxa

bxa

bxa

ax



6

Absolute value

The absolute value of b is denoted |b|, and:

Suppose two numbers, a en b. The distance between a and b on the number line is a-b if

and –(a-b) if a<b. Therefore, we have the distance between a and b on the

number line defined as:

Absolute values can also be part of an inequality:

b = bif b³ 0 and -bif b< 0{ }

ba

abba

x < a means-a< x < a

x £ a means-a£ x £ a



7

A. Equations

How to solve simple equations

To solve an equation means to find all values of the variables for which the equation is

satisfied. If any value of a variable makes an expression in an equation undefined, that

value is not allowed. An example is: z/(z-5), here z can not be 5 because it makes the

expression undefined, because it is 5/0.

When two equations have exactly the same solution, we call them equivalent. To get

equivalent equations, you have to do the following on both sides of the equality sign (=):

Add or subtract the same number;

Multiply or divide by the same number (≠0).

Equations with Parameters

These equations have a common structure, which makes it possible to write down a

general equation covering all the special cases:

Here a and b are real numbers, also called parameters. The variables in the equation

are x and y.

Consider the basic macroeconomic model:

Where Y is the GDP (Gross Domestic Product), C is consumption, and is total

investment (which is treated as fixed). Here, a and b are the positive parameters, with

b<1. These equations represent the structural form of the model.

Substituting C = a + bY + I gives us:

Next, rearrange the equation so that all the terms containting Y are on the left-hand

side:

The left-hand side is equal to Y(1-b), so devide both sides by 1-b so that the coefficient Y

becomes 1. This gives us the answer:

This is one part of the reduced form, that expresses endogenous variables as

functions of exogenous variables.

Quadratic Equations

The general quadratic equation (also called second-degree equation) has the form:

Where x is the unknown variable, and a, b, and c are given constants.

baxy

ICY

bYaC

I

IbYaY

IabYY

I

bb

aY

1

1

1

)0(02 acbxax



8

The Quadratic formula

To solve a quadratic equation like: , it is possible to use the quadratic

formula.

If: Then: If and only if:

When < 0, the square root of a negative number appears and no real solution

exists. The solutions of the equation are often called the roots of the equation.

Because the Quadratic formula contains a ± sign, there are two solutions, and ,

that make sure the quadratic equation equals 0. This can be written as:

then:

Linear Equations in Two Unknowns

This section will review some methods for solving two linear equations with two

unknowns:

We must find the values of x and y that satisfy both equations. There are two methods

for this:

Method 1

(1) Solve one of the equations for one of the variables in terms of the other, then (2)

substitute the result into the other equation:

1)

2)

The solutions should always be checked by direct substitution.

Method 2

This method is based on eliminating one of the variables by adding or subtracting a

multiple of one equation from the other. In the following equation, we will eliminate x.

We multiply the second equation with -2/3, then we will add the transformed equations.

The term x disappears and we obtain:

-------------------------

Hence, y = 4 which gives us x=3 (through substitution of 4 for y). To solve a general

linear system, we use the following two equations and two unknowns:

02 cbxax

0042 aacb 02 cbxaxa

acbbx

2

42

b2 - 4ac

1x 2x

0))(( 21 xxxx ))(( 21

2 xxxxacbxax

3664 yx

1486 yx

xy 4366

xy 3/26

14)3/26(86 xx

143/16486 xx343/111 x

43 yx

3664 yx

-4x+5 1/ 3y= 9 1/ 3

3664 yx

3/193/154 yx

3/1453/1110 y

cbyax



9

Here a, b, c, d, e, and f are arbitrary given numbers, whereas x and y are the unknowns.

Using Method 2, we multiply the first equation by e and the second by –b to obtain the

following, which gives the value for x:

By substituting it back into or

, we get y.

Nonlinear Equations

An example of a nonlinear equation is: . People tend to calculate the solution

in the following way:

However, this is an easy way to make a serious mistake, because here a factor is

cancelled which might be zero (x=0 is a solution too!).

A safer method is . When the equation is in this form, it is easier to see that

x=0 is also a solution.

In general:

is equivalent to a=0 of b=c.

feydx

)/()( bdaebfcex

)/()( bdaecdafy

cbyax feydx

06 23 xx

23 6xx

6x

0)6(2 xx

acab



10

B. Summation

The capital Greek letter ∑ is used as a summation symbol, and the sum is written as:

This reads ‘the sum from i=1 to n of ’. If there are n regions, then:

The symbol i is called the index of summation. It is a ‘dummy variable’ that can be

replaced by any other letter (which has not already been used for something else).

The lower and upper limits of the summation can both vary, for example:

The solution to the first summation:

In order to summarize the overall effect of price changes, a number of price indices

have been suggested. A price index for year t, in which year 0 as the base year, is

defined as:

Here q(i) is the number of goods (in the basket), p0(i) is the price per unit of good i in

year 0, and pt(i) is the price per unit of good i in year t.

In the case where the quantities are levels of consumption in the base year 0, this is

called: the Laspeyres price index. But if the quantities are levels of consumption

in year t, this is called the Paasche price index.

A few aspects of logic

Propositions are assertions that are either true or false. This is an example of a

mathematical proposition: .

Implications

Suppose P and Q are two propositions, such that whenever P is true, then Q is

necessarily true. If this is the case, we usually write:

The arrow is an implication arrow and it points in the direction of the logical

implication.

n

i

iN1

iN

nNNN ...21

r

qi

i

i

beni5

1

2

5554321 222225

1

2 i

i

100

1

)()(

0

1

)()(

n

i

ii

n

i

ii

t

qp

qp

)(iq)(iq

042 x

BA



11

In some cases, where the previous implication is valid, it may also be possible to draw a

logical conclusion in the other direction:

In such cases, we can write both implications together in a single logical equivalence,

as shown below:

The symbol is called an equivalence arrow.

Necessary and Sufficient Conditions

There are other ways of expressing that proposition P implies proposition Q, or that P is

equivalent to Q.

To emphasize this point, consider the following two propositions:

Breathing is a necessary condition for a person to be healthy;

Breathing is a sufficient condition for a person to be healthy.

The first proposition is obviously true. The second proposition is false, because a sick

person is still breathing.

BA

AÛB

Û



12

C. Functions of one variable

One variable is a function of another if the first variable depends upon the second, for

instance . Here, when x is given, y can be determined, y is a function of x. In this

situation x can be any number. However, mostly there are some restrictions like: only

the values 1 to 10 are relevant. Sometimes a graph is preferable to a formula, for a

better overview of the situation.

Basic Definitions

A function of real variable x with domain is a rule that assigns a unique real number

to each real number x in D. As x varies over the whole domain (D) the set of all possible

values f(x) is called the range of . Functions are given letter names like f, z, g, F, or

φ.

If f is a function, we often let y denote the value of f at x, so:

We call x the independent variable or the argument of f, whereas y is the dependent

variable, because the value y depends on the value of x (in general). The domain of f is

then the set of all possible values of the independent variable, whereas the range is the

set of corresponding values of the dependent variable. In economics, x is called the

exogenous variable, which is supposed to be fixed OUTSIDE the economic model,

whereas for each given x the equation y = f(x) serves to determined the endogenous

variable y INSIDE the economic model.

Functional Notation

Consider the following function:

Now suppose all values of x increase with 1. A common mistake is:

The right answer is:

Substituting a for x in the formula of f gives:

Whereas (if a a + 1):

Domain and Range

The definition of a function is not really complete unless its domain is either specified

explicitly or obvious. So: if a function is defined using an algebraic formula, the domain

consists of all values of the independent variable for which the formula gives a unique

value (unless another domain is explicitly mentioned).

xy 2

fD

fR

)(xfy

2)( xxf

1)1( 2 xxf

2)1()1( xxf

2)( aaf

1212)1)(1()1()()1( 22222 aaaaaaaaaafaf



13

For example, we are going to find the domain of:

For x=2, the formula reduces to a meaningless expression (6/0). For all other values of

x, the formula makes f(x) a well-defined number. Thus, the domain consists of all

numbers except x = 2.

Another example, find the domain:

The function is uniquely defined for all x such that 6 +x is nonnegative. Thus, the domain

of the function can be written as the interval: [-6,∞). The range of the function can also

be calculated. The range of f is all the numbers we get as output, using all numbers in

the domain as imputs. The range of the function is in this case[0,∞).

A function f is called increasing if implies . A function is strictly

increasing if implies . Decreasing and strictly decreasing are

defined in the opposite way.

Graphs of Functions

A rectangular or Cartesian coordinate system is obtained by first drawing two

perpendicular lines, called coordinate axes. The two axes are respectively the y-axis (the

vertical axis) and the x-axis (the horizontal axis). The intersection point O is called the

orgin. This rectangular coordinate system is also called the x y-plane. We measure the

real numbers along of each line, although the unit distance on the x-axis is not

necessarily the same as on the y-axis.

Any point P in the plane can be represented by a unique pair of real numbers (c,d). The

point represented by (a, b) lies at the intersection of x=c and y=d. We call (c, d) the

coordinates of P, (c, d) is also called an ordered pair because the order of the two

numbers in the pair is important. For instance, (4, 5) and (5, 4) represent two

completely different points.

Each function of one variable can be represented by a graph in such a rectangular

coordinate system. The graph of a function f is simply the visual representation of the set

of all points (x, f(x) ), where x belongs to the domain of f.

Linear Functions

Linear functions occur a lot in economics, and they are defined as:

where a and b are constants.

The graph of this equation is a straight line. Suppose x increases with 1, then:

This shows that a measures the change in the value of the function when x increases by

1 unit, for this reason a is the slope of the line or the function.

x

xxf

2

3)(

)6( x

21 xx )()( 21 xfxf

21 xx )()( 21 xfxf

baxy

abaxbxaxfxf )1()()1(



14

If the slope of a is positive, the line goes upward to the right, and the larger the value of

a, the steeper is the line. On the other hand, if the value of a is negative, then the line

slants downward to the right, and again the absolute value of a measures the steepness of the line. For example, when a is -6 the steepness is 6.

In the special case, when a (the slope) equals 0, the steepness is zero because the line is

horizontal. Algebraically, we have y = ax + b =b for all x. Here, b is called the y-

intercept.

The Point-Slope and Point-Point Formulas

Suppose point S(x1, y1). Let us find the straight line equation passing through point P

with slope a. If is any other point on the line, then the slope a is given by the

formula:

where

Graphical solutions of Linear Equations

Earlier, we have dealt with algebraic methods for solving a system of two linear

equations in two unknowns. The equations are linear, so naturally their graphs are

straight lines. The coordinates of any point on the line satisfy the equation of that line.

Thus, the coordinates of any point of the intersection of these two lines will satisfy both

equations. This means that any point where these lines intersect will satisfy the equation

system.

Linear inequalities

Suppose the linear inequality . The inequality can be written as y ≤ 8 - 4x. The

set of points (x, y) that satisfy the inequality y ≤ 8 - 4x must have y-values below those

points on the line y = -4x + 8, because y = -4x +8 is a straight line.

Linear Models

In Keynesian macroeconomic theory, total consumption expenditure (on goods and

services) is C. C is assumed to be a function of national income Y:

This is called the Consumption Function.

The slope of b is called the marginal propensity to consume. This tells us by how

many consumption changes if national income increases with 1. C, Y, and b can also be

measured in billions of dollars for example.

The point where demand is equal to supply, represents an equilibrium. The price P* at

which this occurs is the equilibrium price and the corresponding quantity Q* is the

equilibrium quantity.

Consider the following simple example of demand and supply functions:

Equilibrium occurs at: P*= 36 and Q*= 164.

These are the general linear demand and supply schedules:

Here a, b, , and are the positive parameters of the functions.

),( 22 yxT

)(

)(

12

12

xx

yya

21 xx

84 yx

bYaCYfC )(

PD 200

PS 420

bPaD

PaS



15

The formula for the equilibrium Price is:

The formula for the equilibrium Quantity is:

Quadratic Functions

The general quadratic function is:

Where a, b, and c are constants and a ≠ 0.

In general, the graph of is called a parabola.

The shape of a parabola roughly resembles an upside down U when a < 0, and a U when

a > 0. Parabolas are symmetric about the axis of symmetry, for examples review the

compulsory literature.

To calculate for which value of x, f(x) is equal to 0, you can use the quadratic formula.

Suppose a≠0 and then:

Where stands for the two possible values where y=0.

Furthermore:

To find the maximum/minimum of the function use the following formulas:

When a>0, has a minimum of at .

When a<0, has a

maximum of at .

Polynomials

Polynomials or cubic functions have a general

form:

Here a, b, c, and d are constants and a≠0. On

the right is an example of a cubic function.

General Polynomials

Cubic, quadratic and linear functions are all examples of polynomials. The function P

defined for all x by:

b

aP

*

b

baQ

*

cbxaxxf 2)(

cbxaxxf 2)(

042 acb

a

acbbx

2

42

2,1

2,1x

))(( 21

2 xxxxacbxax

cbxaxxf 2)(a

bc

4

2

a

bx

2

cbxaxxf 2)(

a

bc

4

2

a

bx

2

dcxbxaxxf 23)(



16

Here, all a’s are constants and .

This function is called the general polynomial of degree n with coefficients

.

Numerous problems in mathematics and its applications involve polynomials. Often one is

very interested in finding the number of zeros in P(x). This means, the values of x such

that P(x) = 0. This equation:

is called the general equation of degree n. This equation has at most n real solutions

or roots, but it doesn't need to have any.

According to the fundamental theorem of algebra, every polynomial of the form

can be written as a product of polynomials of degree

1 or 2. For example, we can re-write as .

Factoring Polynomials

Let P(x) and Q(x) be two polynomials for which . Then there always exist

unique polynomials q(x) and r(x) such that:

where the degree of r(x) is less than the degree of Q(x). This fact

is called the remainder theorem.

When x is such that . Then can be written in the form:

An important and special case is when Q(x)=x-a. Then Q(x) is of the degree 1, so the

remainder r(x) must have degree 0, and is therefore a constant. In this special case, for

all x:

In particular, when x=a , P(a)=r. Hence, x – a devides P (x) if and only if P(a) = 0.This

can be formulated as:

Polynomial division

You can devide polynomials in much the same way as you use long division to devide

numbers. It is possible that a devision gives a solution but leaves a remainder. For an

example consider the compulsory literature.

Rational functions

A rational function is a function: that can be expressed as the ratio

of two polynomials P(x) and Q(x). This function is defined for all x where . The

ratio is proper if the degree of P(x) is less than the degree of Q(x). When the degree of

P(x) is more than the degree of Q(x), it is called an improper rational function.

01

1

1 ...)( axaxaxaxP n

n

n

n

0na

naaa ,...,, 10

01

1

1 ...)( axaxaxaxP n

n

n

n

123 xxx )1)(1( 2 xx

)()( xQxP

)()()()( xrxQxqxP

0)( xQ )()()()( xrxQxqxP

)(

)()(

)(

)(

xQ

xrxq

xQ

xP

raxxqxP ))(()(

0)( aPax

)(/)()( xQxPxR

0)( xQ



17

Power Functions

The general power function f is defined by the formula:

(Where x>0 and r and A are constants).

Graphs of Power Functions

The shape of the graph depends crucially on the

value of r. See the example below:

Exponential Functions

A quantity that increases/decreases by a fixed

factor per unit of time is said to

increase/decrease exponentially. If the factor

is a, this leads to the following exponential

function:

(A and a are positive constants)

Beware of the fundamental difference between:

en .

The first one is an exponential function where

the exponent varies. The second one is an

example of a power function where the base

varies and the exponent is a constant.

The doubling time is the time it takes for the result of an exponential function to

double. Suppose the function:

The value at t=0 is A. The doubling time t* is given by the equation or

after cancelling A; . Thus, the doubling time of the exponential function

is the power to which a must be raised in order to get 2.

The Natural Exponential Function

The most important base for an exponential function is denoted by the letter e

(=2,7182818284590…).

Given the base e the natural exponential function is:

Logarithmic Functions

Like we said before, the doubling time of an exponential function was

defined as the time it takes for f(t) to become twice as large:

To solve this, we use the natural logarithm. Therefore, we introduce the following

useful definition: If , we call u the natural logarithm of a, and we write u=ln a.

Hence, we have the following definition of the symbol ln a:

Here, a can be all positive values. When , then x is: ln 7.

rAxxf )(

tAatf )(

xaxf )(axxg )(

tAatf )(

AAatf t 2*)( *

2* tatAatf )(

xexf )(

**)( tAatf

2* ta

aeu

ae a ln

7xe



18

There are a couple of rules for the natural logarithmic function ln:

(x and y are positive);

(x and y are positive);

(x is positive);

;

There are no simple formulas for en .

The Function g(x)=ln(x)

For each positive number x, the number ln x is defined by: . So, the u =ln x is

the solution to the equation e^u=x. We call the resulting function the natural

logarithm of x:

(x>0)

The graph of the function can be found in the compulsory study material.

Logarithms with Bases other than e

Suppose a (a>1). If , then we call u the logarithm of x to base a, and we write:

where is also true.

By taking the ln on each side of , we obtain , so that:

Log a obeys the same rules as ln:

;

;

;

.

yxxy lnln)ln(

yxy

xlnln)ln(

)ln()ln( xpx p

xeexe xx ln,,1ln,01ln ln

)ln( yx )ln( yx

xe x ln

xxg ln)(

xau

xu alog xaxa

log

xaxa

log

xa

xa lnln

1log

yxxy aaa loglog)(log

yxy

xaaa loglog)(log

)(log)(log xpx a

p

a

1log,01log aaa

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Algebra - slimstuderen.nl SlimStuderen... · the three dots indicate that the digit 3 is repeated indefinitely. This is also called a real number. When applied to a real number, the