Mathematics for students Contents Anna Strzelewicz October 6, 2015

Mathematics for students
Anna Strzelewicz
October 6, 2015
Contents
I
Introduction
1
1 Sets and relations
2
2 Real numbers
3
3 Absolute value
4
4 Intervals
4
5 Algebra and geometry
5
6 Greek alphabet
7
7 Scientific notation
7
8 Mathematical induction
7
1
Part I
Introduction
1
Sets and relations
x∈A
x∈
/A
∅
A⊆B
A⊂B
the object x is in the set A, or x is an element of A
the object x is not in the set A
empty set
A is a subset of B (with A possibly equal to B)
A is a proper subset of B (i.e., A ⊆ B but A 6= B), every element
contained in set A is also contained in set B, and there is at least
one element in B that is not contained in A
A ∪ B denotes the union of A and B
A ∩ B denotes the intersection of A and B
A \ B the set of all elements of A that are not in B
A ∩ B = ∅ two sets A and B are disjoint
A = {x : P } the set A is a set of all x for which property P holds
A = {x : x = 2n − 1, n ∈ N} A is a set of all odd natural numbers
A × B Cartesian product
Note that ⇐⇒ , iff, and if and only if, all mean the same thing.
2
2
Real numbers
Classification
Natural numbers
(positive integers)
N
1, 2, 3, 4, 5, etc.
Integers
Z
0, 1, -1, 2, -2, 3, -3, etc.
Rational numbers
Q
numbers that can be expressed in the form of a fraction pq ,
where p (numerator) and q (denominator) are integers, q 6= 0
The rational number can be expressed as:
a) terminating decimal - having a finite number of digits after the decimal point (terminating expansion can be padded
In each
with infinitely many zeros), for example, 41 = 0.25000
. . . case
the that
three
b) repeating decimal - ending with a string of digits
dots
...
repeats over and over, for example, 131
=
1.323232
.
.
.
99
indicate
√ √
that
the
2, 3 ,π = 3.1415926 . . . ,e,
sequance
of decimal
numbers that can be expressed as decimals
1
digits goes
=
0.3333
.
.
.
3
√
on forever.
2 = 1.4142 . . .
Between any two real numbers there is a rational number
and an irrational number.
Irrational numbers
Real numbers
R
A prime number is an integer greater than 1 whose only positive factors are 1
and the integer itself i.e. 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, . . .
Operations on numbers
• Addition
• Substraction
• Multiplication
• Division
Exponential notation
3
1. x0 = 1
2. xn xm = xn+m
3.
xn
xm
= xn−m
4. (xn )m = xnm
5. (xy)n = xn y n
1
( xy )n =
xn
yn
6. x m means the m-th root of x i.e.
n
7. x m
3
√
m
x
√
√
means the m-th root of xn i.e. m xn = ( m x)n
Absolute value
Definition
|a| =
a
if a ≥ 0
−a if a < 0
The notation a ≥ 0 means that a is eiter greater than zero or equal to zero.
oher characterizations
√
• |a| = | − a| = a2 ≥ 0
• |a| - the nonnegative distance between real number a and zero
• |a − c| - the distance between the real numbers a and c
• |ab| = |a||b|
• |a| < b if and only if −b < a < b
• |a + b| ≤ |a| + |b| - triangle inequality - the absolute value of the sum
of two numbers cannot exceed the sum of the lengths of the other two
sides
4
Intervals
Suppose that a < b. The open interval (a, b) is the set of all real numbers
between a and b:
(a, b) = {x : a < x < b}.
4
The closed interval [a, b] is the open interval (a, b) together with the endpoints a and b :
[a, b] = {x : a ≤ x ≤ b}.
We also use the half-open intervals:
[a, b) = {x : a ≤ x < b} and (a, b] = {x : a < x ≤ b}.
Unbounded intervals, can have a form:
[a, +∞) = {x : x ≥ a}
(−∞, a] = {x : x ≤ a}
The symbols +∞ and −∞ , denoting ”plus infinity” and ”minus infinity”.
We use a bracket to include an endpoint; otherwise a parenthesis.
5
Algebra and geometry
Fractions:
How to express fractions in English?
1
one half, a half
2
1
3
one third
1
4
one quarter, a quarter
2
3
two thirds
2
5
two fifths
3
4
three quarters
3
7
three sevenths
5
6
five sixths
2 13
two and one third
9
5 10
five and nine tenths
5
a
b
+
c
d
=
ad+bc
bd
When adding or subtracting fractions, you must find a common denominator
a c
ac
·
=
Multiplying fractions multiply the numerators together and
b d
bd
then multiply the denominators together
The quadratic equation:
ax2 + bx + c = 0,a 6= 0
has solution:
x=
√
−b± b2 −4ac
2a
Factoring
a2 − b2 = (a − b)(a + b)
a3 − b3 = (a − b)(a2 + ab + b2 )
The square of a binomial:
(a + b)2 = a2 + 2ab + b2 The Square of the binomial is the square of the first term
plus the square of the second term plus twice the product of
two terms.
2
2
2
(a − b) = a − 2ab + b
The square of the difference of two binomials (two unlike
terms) is the square of the first term plus the second term
minus twice the product of the first and the second term.
The cube of a binomial:
(a + b)3 = a3 + 3a2 b + 3ab2 + b3 The sum of a cubed of two binomial is equal to the cube of
the first term, plus three times the square of the first term
by the second term, plus three times the first term by the
square of the second term, plus the cube of the second term.
3
3
2
2
3
(a − b) = a − 3a b + 3ab − b
The difference of a cubed of two binomial is equal to the cube
of the first term, minus three times the square of the first
term by the second term, plus three times the first term by
the square of the second term, minus the cube of the second
term.
Binomial formula:
n−2 b2 + n(n−1)(n−2) an−3 b3 +· · ·+nabn−1 +bn
(a+b)n = an +nan−1 b+ n(n−1)
1·2 a
1·2·3
if n is a positive integer.
Factorials
0! = 1
1! = 1
2! = 1 · 2
3! = 1 · 2 · 3
·
6
·
·
n! = 1 · 2 · 3 · . . . (n − 2) · (n − 1) · n
Pythagorean theorem:
In a right triangle with legs a and b and hypotenuse c, a2 + b2 = c2 .
circle
sphere
6
α
β
γ
δ
ζ
η
θ
ι
κ
λ
µ
area = πr2 , circum ference = 2πr
volume = 43 πr3
Greek alphabet
A
B
Γ
∆
E
Z
H
Θ
I
K
Λ
M
alpha
beta
gamma
delta
epsilon
zeta
eta
theta
iota
kappa
lambda
mu
ν
ξ
o
π
ρ
σ
τ
υ
φ
χ
ψ
ω
N
Ξ
O
Π
P
Σ
T
Υ
Φ
X
Ψ
Ω
7
Scientific notation
8
Mathematical induction
nu
xi
omicron
pi
rho
sigma
tau
upsilon
phi
chi
psi
omega
d|n means there is an integer k such that n = dk.
Note that d|n 6= d/n. Recall that d/n represents the fraction nd .
The expression d|n may be read in any of the following ways:
• d divides n
• d is a divisor of n
• d is a factor of n
• n is a multiple of d
Example: The following statements are equivalent:
7
• 2|10
• 2 divides 10
• 2 is a divisor of 10
• 2 is a factor of 10
• 10 is a multiple of 2
Definition: 2|10 ⇐⇒ 10 = 2k for some k , where k is a positive integer
8