Universiteit van Amsterdam
Samenvatting S1, B1, 2020
Introduction to
quantitative methods
Accountancy and Control
,Inhoudsopgave
Week 1. Numbers and Algebra .............................................................................................................................. 2
Different numbers .............................................................................................................................................. 2
Commutative law ............................................................................................................................................... 2
Associative law ................................................................................................................................................... 2
Distributive law .................................................................................................................................................. 2
Factorization....................................................................................................................................................... 2
Multiplication rules ............................................................................................................................................ 4
Adding fractions ............................................................................................................................................. 4
Subtracting fractions ..................................................................................................................................... 4
Multiplication and division of fractions ........................................................................................................ 5
Exponentiation............................................................................................................................................... 5
Summary for all a > 0 ..................................................................................................................................... 5
Order of calculations .......................................................................................................................................... 6
Week 2. Equations, lines and functions ................................................................................................................. 7
General rule ........................................................................................................................................................ 7
Solve system of two linear equations with two variables ................................................................................ 7
General rule ........................................................................................................................................................ 8
First degree function .......................................................................................................................................... 8
From graph to function .................................................................................................................................. 8
The intersection point of two graphs ............................................................................................................ 9
Week 3. Equations, functions .............................................................................................................................. 10
Second degree function (= quadratic function) .............................................................................................. 10
The simplest quadratic function.................................................................................................................. 10
Characteristics of 𝒚 = 𝒂𝒙 + 𝒑𝟐 + 𝒒 ........................................................................................................... 11
Characteristics of 𝒚 = 𝒂𝒙𝟐 + 𝒃𝒙 + 𝒄.......................................................................................................... 12
Intersection of graphs ...................................................................................................................................... 13
Second degree inequalities .............................................................................................................................. 14
‘Two fold’ second degree inequalities ............................................................................................................. 14
From data points to equation .......................................................................................................................... 15
Week 4. Exponential, Power, Root and Logarithmic Functions .......................................................................... 16
Definition .......................................................................................................................................................... 16
Exercises with exponential functions .............................................................................................................. 16
Defintition of 𝒂𝐥𝐨𝐠𝒃 ........................................................................................................................................ 17
Rules of calculation for powers ....................................................................................................................... 18
Rules of calculation for logarithms .................................................................................................................. 18
Definition .......................................................................................................................................................... 19
Definition .......................................................................................................................................................... 19
1
, Week 1. Numbers and Algebra
Different numbers
Natural numbers N are positive numbers (1, 2, 3 … )
Integers numbers Z are the numbers N expanded with the ‘negative’ of the natural number (-1, 0, 1 ..)
𝐴
Rational numbers Q are integers expanded with fractions ( 𝐵 , A ∈ Z, B ∈ Z, 𝐵 ≠ 0 )
Real numbers R are all numbers (N, Z, Q, algebraic numbers (√), transcendental number (π)
Note that: N 𝐶 Z 𝐶 Q 𝐶 R (N is a subset of Z, Z is a subset of Q, Q is a subset or R)
Commutative law
𝐴+𝐵 =𝐵+𝐴 𝐴𝐵 = 𝐵𝐴
Associative law
(𝐴 + 𝐵) + 𝐶 = 𝐴 + (𝐵 + 𝐶) 𝐶(𝐴𝐵) = 𝐴(𝐵𝐶)
Distributive law
𝐴(𝐵 + 𝐶) = 𝐴𝐵 + 𝐴𝐶 𝐶(𝐴 + 𝐵) = 𝐴𝐶 + 𝐵𝐶
𝐴(𝐵 − 𝐶) = 𝐴𝐵 − 𝐴𝐶 𝐶(𝐴 − 𝐵) = 𝐴𝐶 − 𝐵𝐶
(𝐴 + 𝐵)(𝐶 + 𝐷) = 𝐴𝐶 + 𝐴𝐷 + 𝐵𝐶 + 𝐵𝐷
(𝐴 + 𝐵)(𝐶 − 𝐷) = 𝐴𝐶 − 𝐴𝐷 + 𝐵𝐶 − 𝐵𝐷
(𝐴 − 𝐵)(𝐶 + 𝐷) = 𝐴𝐶 + 𝐴𝐷 − 𝐵𝐶 − 𝐵𝐷
Example:
(𝑋 + 𝐴)(𝑋 + 𝐵) = 𝑥 2 + 𝐴𝑥 + 𝐵𝑥 + 𝐴𝐵 = 𝑥 2 + (𝐴 + 𝐵)𝑥 + 𝐴𝐵
(𝑋 + 5)(𝑋 − 2) = 𝑥 2 + (5 − 2)𝑥 + 5 ∗ (−2) = 𝑥 2 + 3𝑥 − 10
For all a, b and c (C≠0):
Remark! For all a, b and c:
Factorization (‘reverse’ distributive law)
𝐴𝐵 + 𝐴𝐶 = 𝐴(𝐵 + 𝐶) 𝐴𝐶 + 𝐵𝐶 = 𝐶(𝐴 + 𝐵)
2
Samenvatting S1, B1, 2020
Introduction to
quantitative methods
Accountancy and Control
,Inhoudsopgave
Week 1. Numbers and Algebra .............................................................................................................................. 2
Different numbers .............................................................................................................................................. 2
Commutative law ............................................................................................................................................... 2
Associative law ................................................................................................................................................... 2
Distributive law .................................................................................................................................................. 2
Factorization....................................................................................................................................................... 2
Multiplication rules ............................................................................................................................................ 4
Adding fractions ............................................................................................................................................. 4
Subtracting fractions ..................................................................................................................................... 4
Multiplication and division of fractions ........................................................................................................ 5
Exponentiation............................................................................................................................................... 5
Summary for all a > 0 ..................................................................................................................................... 5
Order of calculations .......................................................................................................................................... 6
Week 2. Equations, lines and functions ................................................................................................................. 7
General rule ........................................................................................................................................................ 7
Solve system of two linear equations with two variables ................................................................................ 7
General rule ........................................................................................................................................................ 8
First degree function .......................................................................................................................................... 8
From graph to function .................................................................................................................................. 8
The intersection point of two graphs ............................................................................................................ 9
Week 3. Equations, functions .............................................................................................................................. 10
Second degree function (= quadratic function) .............................................................................................. 10
The simplest quadratic function.................................................................................................................. 10
Characteristics of 𝒚 = 𝒂𝒙 + 𝒑𝟐 + 𝒒 ........................................................................................................... 11
Characteristics of 𝒚 = 𝒂𝒙𝟐 + 𝒃𝒙 + 𝒄.......................................................................................................... 12
Intersection of graphs ...................................................................................................................................... 13
Second degree inequalities .............................................................................................................................. 14
‘Two fold’ second degree inequalities ............................................................................................................. 14
From data points to equation .......................................................................................................................... 15
Week 4. Exponential, Power, Root and Logarithmic Functions .......................................................................... 16
Definition .......................................................................................................................................................... 16
Exercises with exponential functions .............................................................................................................. 16
Defintition of 𝒂𝐥𝐨𝐠𝒃 ........................................................................................................................................ 17
Rules of calculation for powers ....................................................................................................................... 18
Rules of calculation for logarithms .................................................................................................................. 18
Definition .......................................................................................................................................................... 19
Definition .......................................................................................................................................................... 19
1
, Week 1. Numbers and Algebra
Different numbers
Natural numbers N are positive numbers (1, 2, 3 … )
Integers numbers Z are the numbers N expanded with the ‘negative’ of the natural number (-1, 0, 1 ..)
𝐴
Rational numbers Q are integers expanded with fractions ( 𝐵 , A ∈ Z, B ∈ Z, 𝐵 ≠ 0 )
Real numbers R are all numbers (N, Z, Q, algebraic numbers (√), transcendental number (π)
Note that: N 𝐶 Z 𝐶 Q 𝐶 R (N is a subset of Z, Z is a subset of Q, Q is a subset or R)
Commutative law
𝐴+𝐵 =𝐵+𝐴 𝐴𝐵 = 𝐵𝐴
Associative law
(𝐴 + 𝐵) + 𝐶 = 𝐴 + (𝐵 + 𝐶) 𝐶(𝐴𝐵) = 𝐴(𝐵𝐶)
Distributive law
𝐴(𝐵 + 𝐶) = 𝐴𝐵 + 𝐴𝐶 𝐶(𝐴 + 𝐵) = 𝐴𝐶 + 𝐵𝐶
𝐴(𝐵 − 𝐶) = 𝐴𝐵 − 𝐴𝐶 𝐶(𝐴 − 𝐵) = 𝐴𝐶 − 𝐵𝐶
(𝐴 + 𝐵)(𝐶 + 𝐷) = 𝐴𝐶 + 𝐴𝐷 + 𝐵𝐶 + 𝐵𝐷
(𝐴 + 𝐵)(𝐶 − 𝐷) = 𝐴𝐶 − 𝐴𝐷 + 𝐵𝐶 − 𝐵𝐷
(𝐴 − 𝐵)(𝐶 + 𝐷) = 𝐴𝐶 + 𝐴𝐷 − 𝐵𝐶 − 𝐵𝐷
Example:
(𝑋 + 𝐴)(𝑋 + 𝐵) = 𝑥 2 + 𝐴𝑥 + 𝐵𝑥 + 𝐴𝐵 = 𝑥 2 + (𝐴 + 𝐵)𝑥 + 𝐴𝐵
(𝑋 + 5)(𝑋 − 2) = 𝑥 2 + (5 − 2)𝑥 + 5 ∗ (−2) = 𝑥 2 + 3𝑥 − 10
For all a, b and c (C≠0):
Remark! For all a, b and c:
Factorization (‘reverse’ distributive law)
𝐴𝐵 + 𝐴𝐶 = 𝐴(𝐵 + 𝐶) 𝐴𝐶 + 𝐵𝐶 = 𝐶(𝐴 + 𝐵)
2
