01_Number_System_Introduction.pptx

Numeration System
Introduction

Numeration Systems
• Reason why numeration exist
• Bases of Numeration System
• Different Representation of Counting Numbers
• One-for-One Correspondence
• The Commonly used Number Systems Today
• Other Number Systems
• Modular Systems
• Conversion of Different Number Systems
• Arithmetic Operations of Different Number
Systems

Numeration Systems
• Structured methods or procedures for counting in order
to determine the total units in a collection.
• Consist of counting bases and some form of
representation.
example: base 2, base 8, base 10, base 16, etc.
• Numerals are symbols or groups of symbols that
represents a number.
example: the symbols 12, twelve, and XII are different
numerals that all represent the same number.

3 Reasons why Numeration
Systems Exist
• To Identify  numeration system identify people and property to
preserve confidentiality, increase security, and minimizes errors.
• To order define a person or unit’s order in a series.(ordinal-
first, second, third, etc. and cardinal- one, two, three, etc.)
• To Tally  used to tally or total
 to find out how many items or units are involved in a
calculation (addition, subtraction , multiplication ,
or division).

The concept of number has
two complementary aspects:
• Cardinal Numbering – It relies only on the principle of
mapping
• Ordinal Numeration – It requires both the technique
of pairing and the idea of succession.

The Bases Of Numeration
Systems
• The base of a numeration system is its frame of
reference or the starting point on which it grounds its
counting method.
• Many bases of numeration systems are founded
upon the most obvious and immediate things in a
person's visual field:(arms, hands, fingers, and toes.
• Common bases of numeration systems are the two
arms of a person (base 2 system), the fingers of one
hand (base 5 system), the fingers of both hands
(base 10 system), or the total of all a person’s
fingers and toes (base 20 system).

Principle Base of 2 (Binary
Principle)
• Was one of the most common numeration systems in
ancient times.
• Primitive people possessed this fundamental arithmetical
rule which allowed them to manipulate numbers far in
excess of four.
• To indicate a number like three or four, the person
says “two-and-one” or “two-and-two.” The number 10 is
indicated with “two-and-two-and-two-and-two-and-
two.” However, as a person counts to higher and higher
numbers in a base 2 system, it becomes harder and harder
to remember one’s place in the long string of twos.
• Have now been replaced by decimal (or base 10) systems.

Continuation..
• Everyone can see the sets of one, of two, and of
three, of four objects in the figure. But that’s about
the limit of our natural ability to numerate.
• Beyond four, quantities are vague, and our eyes
alone cannot tell us how many things they are.

Continuation..
• There are many traces of the ‘limit of four’ in different
languages and cultures.
• For Example: Romans gave ‘ordinary names’ to the
first four of their sons but the fifth and other sons were
named only by a numerical: Quintus (Fifth), Sixtus
(Sixth), Septimus (Seventh), so on.
• Another example is the Roman calendar where the
first four months had names but the fifth to tenth are
referred to by their order names: Quintilis, Sextilis,
September, October, etc.

Base 10 Numeration System
• The base 10 or decimal system has now spread
throughout the world and is the most commonly
used numeration system today.

Different Representation of Counting
Numbers
• People began by counting the first nine numbers by
placing in sequence the corresponding number of
strokes, circles or other similar representing ‘one’ more
or less:

“One-for-One
Correspondence”
• a device in which both the prehistory of
arithmetic and the dominant mode of
operation in all contemporary “hard” science
• allows even the simplest of minds to compare
two collections of beings or things, of the
same kind or not, without calling an ability to
count in numbers.

Mapping/Biunivocal
Correspondence
or Bijection
(Modern Mathematics)

III IIIIII IIII IIIIII IIIII IIII II

30
Number Systems
Base 2: The Binary Number System
Base 8: The Octal Number System
Base 10: The Decimal Number System
Base 16: The Hexadecimal Number System

31
Number Base
• What is a number base?
A number base is a specific collection of
symbols on which a number system can be built.
• The number base familiar to us is base 10, upon
which the decimal number system is built. There
are ten symbols - 0 to 9 - used in the decimal
system.

32
Place Value
• What is the concept of place value?
Place value means that the value of a digit
in a number depends not only on its own natural
value but also on its location in the number.
It is used interchangeably with the term
positional notation.
• Place value tells us that the two 4s in the
number 3474 have different values, that is, 400
and 4, respectively.

33
A Review of the Decimal Number System
• The word “decimal” comes from the Latin word
decem, meaning ten.
• Thus, the number base of the decimal number
system is base 10.
• Since it is in base 10, ten symbols are used in
the decimal number system.
{0,1,2,3,4,5,6,7,8,9}
• This means that only the digits in the above set
can be used for each position in every place
value in a given decimal number.

34
• Note that the highlighted place value can be
filled by the digits in the set {0,1,2,3,4,5,6,7,8,9}.
• Thus, it can be increased by 1 until it reaches -
• At this point, the symbols that can be used to fill
the highlighted position has been exhausted.
Increasing it further causes a shift in place value,
and resets the initial place value to zero. Thus -
2 7 0
2 7 9
2 8 0

35
• Case Study: 3474
• Using place values, the number 3474 is
understood to mean,
3000 + 400 + 70 + 4 = 3474
This can also be expressed as –
(3x1000) + (4x100) + (7x10) + 4 = 3474
Note that each digit is multiplied by powers of
10, so that the above is equal to –
(3x103) + (4x102) + (7x101) + (4x100) = 3474
Note that the rightmost exponent starts from
zero and increases by 1 as the place value
increases.
Hence, the decimal number system is said to be
in base 10.

36
• The word “binary” comes from the Latin word
bis, meaning double.
• Thus, the number base of the binary number
system is base 2.
• Since it is in base 2, two symbols are used in the
binary number system.
{0,1}
value in a given binary number.

37
filled by the digits in the set {0,1}.
1 1 0
1 1 1
1 1 0
0
0
1

38
• To avoid confusion, one should write a binary
number with base 2 as its subscript whenever
necessary.
• Thus, the binary number 10110 should be
written as -
101102
• It should be read as “one-zero-one-one-zero
base two” and NOT “ten-thousand one-hundred
ten” since each phrase denotes an entirely
different number.

39
• We know that the decimal number 3474 can be
expressed as powers of 10 –
(3x103) + (4x102) + (7x101) + (4x100) = 347410
• In the same manner, the binary number 101102
can be expressed as powers of 2 –
(1x24) + (0x23) + (1x22) + (1x21) + (0x20) = 2210
increases.
Hence, the binary number system is said to be in
base 2.

40
• The word “octal” comes from the Greek word
oktõ, meaning eight.
• Thus, the number base of the octal number
system is base 8.
• Since it is in base 8, eight symbols are used in
the octal number system.
{0,1,2,3,4,5,6,7}
value in a given octal number.

41
filled by the digits in the set {0,1,2,3,4,5,6,7}.
3 6 0
3 6 7
3 7 0

42
• To avoid confusion, one should write an octal
number with base 8 as its subscript whenever
necessary.
• Thus, the octal number 72143 should be written
as -
721438
• It should be read as “seven-two-one-four-three
base eight” and NOT “seventy two-thousand
one-hundred forty three” since each phrase
denotes an entirely different number.

43
(3x103) + (4x102) + (7x101) + (4x100) = 347410
• In the same manner, the octal number 721438
can be expressed as powers of 8 –
(7x84) + (2x83) + (1x82) + (4x81) + (3x80) = 2979510
increases.
Hence, the octal number system is said to be in
base 8.

44
• The word “hexadecimal” is a combination of the
Greek word hex, meaning six and the Latin word
decem, meaning ten.
• Thus, the number base of the hexadecimal
number system is base 16.
• Since it is in base 16, sixteen symbols are used
in the hexadecimal number system.
{0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F}
value in a given hexadecimal number.

45
filled by the digits in the set
{0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F}.
A 3 0
A 3 F
A C
3
B
B
0

46
• To avoid confusion, one should write a
hexadecimal number with base 16 as its
subscript whenever necessary.
• Thus, the hexadecimal number B23C
should be written as -
B23C16
• It should be read as “b-two-three-c base
sixteen”.

47
• Case Study: B23C16
(3x103) + (4x102) + (7x101) + (4x100) = 347410
• In the same manner, the hexadecimal number
B23C16 can be expressed as powers of 16 –
(11x163) + (2x162) + (3x161) + (12x160) = 4562810
increases.
Hence, the hexadecimal number system is said
to be in base 16.

48
Comparative Values: Bases 10, 2, 8, 16
Base 10 Base 2 Base 8 Base 16
Decimal Binary Octal Hexadecimal
0 0000 0 0
1 0001 1 1
2 0010 2 2
3 0011 3 3
4 0100 4 4
5 0101 5 5
6 0110 6 6
7 0111 7 7
8 1000 10 8

49
Comparative Values: Bases 10, 2, 8, 16
Base 10 Base 2 Base 8 Base 16
Decimal Binary Octal Hexadecimal
9 1001 11 9
10 1010 12 A
11 1011 13 B
12 1100 14 C
13 1101 15 D
14 1110 16 E
15 1111 17 F
16 10000 20 10
17 10001 21 11

Other Number Systems
• Base 3: The Ternary Number System
• Base 4: The Quaternary Number System
• Base 5: The Quinary Number System
• Base 6: The Senary Number System
• Base 7: The Septenary Number System

Base 3 Number System
• The Ternary numeral system.
• The base is 3.
• a ternary digit is called a trit (trinary digit). One trit is
equivalent to log2 3 (about 1.58496) bits of information.
• There are Three "digits": 0,1, 2.
• Positions correspond to integer powers of three,
starting with power 0 at the rightmost digit, and
increasing right to left.
• The digit placed at a position shows how many times
that power of three is included in the number.

• Called the Quaternary Number System
• The base is four.
• There are four "digits": 0,1, 2, 3 .
• Positions correspond to integer powers
of four, starting with power 0 at the
rightmost digit, and increasing right to left.
• The digit placed at a position shows how
many times that power of four is included
in the number.

• Called the Quinary number system
• The base is five.
• There are five "digits": 0,1, 2, 3, 4 .
of five, starting with power 0 at the
many times that power of five is included
in the number.

• Called the Senary, heximal or seximal number
system
• The base is six.
• There are six "digits": 0,1, 2, 3, 4,5.
• Positions correspond to integer powers of six, starting
with power 0 at the rightmost digit, and increasing right
to left.
• The digit placed at a position shows how many times
that power of six is included in the number.

• Called the Septenary number system
• The base is seven.
• There are seven "digits": 0,1, 2, 3, 4,5,6.
of seven, starting with power 0 at the
many times that power of seven is
included in the number.

Modular Arithmetic
• Modular arithmetic is a
special type of arithmetic
that involves only integers.
• Sometimes, we are only
interested in what the
remainder is when we
divide A by B.
• For these cases there is an
operator called the modulo
operator (abbreviated as
mod).
• Using the same A, B, Q,
and R as sample below,
we would have:
• A mod B = R is same as
A modulo B is equal to R.
Where B is referred to as
the modulus.

Example:
• The remainders start at
0 and increases by 1
each time, until the
number reaches one
less than the number
we are dividing by. After
that, the
sequence repeats.

References
• http://www.ms.uky.edu/~sohum/ma330/files/Ifrah_num
bers.pdf
• http://www.cs.ucr.edu/~ehwang/courses/cs120a/00win
ter/binary.pdf
• https://www.researchgate.net/publication/320677641_
Number_System
• https://www.encyclopedia.com/science-and-
technology/mathematics/mathematics/decimal-system
• https://www.khanacademy.org/computing/computer-
science/cryptography/modarithmetic/a/what-is-
modular-arithmetic

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