GLOSSARY
OF NUMBER
Asterisk
Asterisk is a small star * used to mark a
space where something is missing.
Examples:
2 * 3 = 6 means x (multiply)
2 * 3 = 5 means + (add)
3 * 2 = 1 means – (subtract)
3 * 2 = 1.5 means : (divide)
Binary
Operation
Binary operation is a function that works on
ordered pair (a,b) in SxS à S,
where a,b є S, then the result is also in S.
Examples:
2 * 3 = 6
We know that 2 is an element of Z, 3 is an
element of Z, and 6 is also element of Z. * here means multiplication.
Complex
Numbers
Complex
numbers are numbers of the form
, where
and
are real numbers, number
is
a real part of
and number
is
an imaginary part of
, either
or
or
both can be 0 and If
the complex numbers are said to be real
numbers.
For
example:
Composite
Numbers
A
composite number is a positive integer
which has a positive divisor
other than one or itself. In the other words, a composite number is any
positive integer
greater than one
that is not a prime number. Blue number in the figure
below shows composite number from 1 until 100.
Field
Field
is a set together with two operations, usually called addition and
multiplication and denoted by + and x, such that the following axioms hold:
1)
Associativity of addition and multiplication
2)
Commutativity of addition and multiplication
3)
Additive and multiplicative identity
4)
Additive and multiplicative inverse
5)
Distributivity of multiplication over
addition
6)
Closure of F under addition and
multiplication
For
example:
Let
a/b be a rational number, where b is not equal to 0. The additive inverse of
a/b is –a/b and the multiplicative inverse where a is not equal to 0 is b/a.
The
abstractly recquired field axioms reduce to standad properties of rational
number, such as the law of distributivity
Group
In
mathematics, we define group as a set G together with a binary operation on G
that satisfies the following axioms:
1) Associative,
that is for all x, y, z in G, x*(y*z) = (x*y)*z
2) Identity
element, there exists an identity element, denoted as e that satisfies
e*x=x*e=x for all x in G.
3) Inverse
element, for all G there exists an inverse element, denoted by x-1,
so that x*x-1 = x-1*x = e.
For
example:
Consider
a set of natural numbers
N =
{1,2,3,…}
To
determine whether it’s a group or not, we’ve to show whether it’s satisfy
axioms above or not.
1)
For all natural numbers a, b, and c,(a+b)+c =
a(b+c). It means if we add the first and the second number first then we add
third number, we will obtain same value with if we add first number with the
sum of second and the third number.
2)
If a is any natural, then 0+a=a+0=a. Zero is
identity element of addition.
3)
For every natural number a, we must find a
natural number b such that a+b=b+a=0. Since we can’t find b that satisfies the
condition, it fails to satisfy the third axiom.
Because
there is an axiom that is not satisfied, the set of natural numbers is not a group.
Integer
Numbers
Integer
numbers are the set of all natural numbers, their negatives, and zero,
represented by
.
For
example:
Irrational
Numbers
Irrational numbers are numbers that cannot
be expressed in the form (a/b) where a and b are integers and b is not
equal to 0.
For
example:
Both
and
are two of examples of irrational numbers.
Natural
Numbers
Natural numbers are the set of numbers that
is specialization of whole numbers, start from 1, and usually represented by N.
For example:
Rational
Numbers
Rational
Numbers are numbers that can be expressed in the form
, where p and q are integers and
, decimal notation for rational numbers
either terminates or repeats, and usually represented by
.
For
example:
ü -5,
-5
, or
, -5, -1, 1, 5 are integers and
and
.
ü
; 0,75
Terminating
decimal
ü
;
Repeating
decimal
Whole
Numbers
Whole
numbers are the set of all natural numbers and zero, there is no exactly
symbols to denote this numbers.
Illustration:
GLOSSARY OF STATISTICS AND PROBABILITY
Chance
Event
Chance event is an event of
which the outcome is uncertain. For some events, we can predict the result but
we can never be sure.
For example:
Rolling a die, tossing a
coin.
Data
A general term used to
describe a collection of facts, numbers, measurements, and symbols.
Example:
Students’ score in Maths
Test were:
15, 16, 23, 55, 76, 89, 90,
100
Disjoint
Event
Disjoint
events are events that have no outcomes in common. If two events are disjoint,
then the probability of them both occurring at the same time is 0.
P(A and B) = 0
For example:
A six sided die is rolled
once, the events
A: Rolling a 1 or 2
B: Rolling a 5 or 6
Are disjoint since they
can’t be both happen.
Event
Event is subset of sample
space that is the set of possible outcomes resulting from a particular
experiment.
For example:
A possible event when a
single six die is rolled {5,6}, that is the roll could be 5 or 6.
Exhaustive
Event
The two or
more events together form sample space (at least one event must occur).
For
example:
In an
experiment throwing die,
A = {eventof
getting odd number}
B = {event
of getting even number}
Then, A
and B are exhaustive event.
Experiment
Experiment is any controlled
and repeatable process to get certain result. Controlled here means we do a
test carefully, so we don’t change the experimental condition.
For example:
In experiment of tossing a
coin, the second coin tossing is done with same altitude with the first coin
tossing, and so on.
Frequency
The frequency of any item in
a collection of data is the number of times that item occurs in the collection.
For example:
We tossed a die 50 times and
recorded the number of how many times each side is occurred.
Number
|
Frequency
|
1
|
7
|
2
|
12
|
3
|
9
|
4
|
8
|
5
|
6
|
6
|
8
|
Mean
The mean is the average of a
set of scores. It is found by adding up all scores and dividing the sum by the
numbers of scores.
For example:
1,1,2,4,4,6,6,6,8,9
Mean =
Mode
In statistics, the score that
occurs most often in a collection.
Examples:
1,1,2,4,4,6,6,6,8,9
6 is the mode.
Mutually
Event
Two
or more events are said to be mutually
exclusive if the occurrence of any one of them means
the others will not occur (That is, we cannot have 2 events occurring at the
same time). Another word that means mutually exclusive is
disjoint. If two events are mutually exclusive, then the probability of either
occurring is the sum of the probabilities of each occurring.
Only valid when the events
are mutually exclusive.
P(A or B) = P(A) + P(B)
For
example:
When
we tossing a coin, which can result in either heads or tails, but not both.
Mutually
Outcome
A
set of outcomes of an event are said to be mutually if they all have
same chance of happening.
For
example:
If
you toss a fair coin, you are equally likely to get a head as a tail. The
probability of each of these is 0.5.
Outcome
Outcome is a single,
specific, possible result of an experiment.
For example:
For an experiment tossing a
coin, we’ll get outcomes head, tails. Then, for an experiment throwing a
six-sided die, we’ll get outcomes 1,2,3,4,5,6.
Random
Experiment
Random experiment is a result to such an experiment, an experiment, trial, or observation that can be repeated numerous times
under the same conditions, used for a situation of uncertainty about
which we want to have some observations.
For example:
When we toss
a coin, the experiment can yield two possible outcomes, that is heads or tails.
Sample
Point
Sample point is a particular
outcome of an experiment, an element of the sample space.
For example:
When a die is thrown, the
sample space is {1,2,3,4,5,6}. Note that 1, 2, 3, 4, 5, and 6 are the sample
point.
Sample
Space
Sample space is set of all
possible outcomes of an experiment.
For example:
When a coin is flipped, we
have sample space is {head, tails}.
Trial
Trial is an action we do in
a probability experiment that produces only one possible outcome.
For example:
Toss a coin and throwing a
die.
GLOSSARY
OF GEOMETRY
Acute Angle
An
acute angle is an angle that the measure is between 0o and 90o.
For
example:
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