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Cmpe Pt. 1

This document discusses number systems and conversions between them. It covers: - Common number systems like binary, decimal, octal, and hexadecimal - Converting between different bases using positional notation and continuous division/multiplication - Relational, bitwise, and arithmetic operations on different number systems - Complement representations for numbers in diminished and standard radix - Performing addition, subtraction, multiplication, and division on different number systems through examples.

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Kriezhel Kim Tadle
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51 views5 pages

Cmpe Pt. 1

This document discusses number systems and conversions between them. It covers: - Common number systems like binary, decimal, octal, and hexadecimal - Converting between different bases using positional notation and continuous division/multiplication - Relational, bitwise, and arithmetic operations on different number systems - Complement representations for numbers in diminished and standard radix - Performing addition, subtraction, multiplication, and division on different number systems through examples.

Uploaded by

Kriezhel Kim Tadle
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as DOCX, PDF, TXT or read online on Scribd
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Chapter 1: NUMBER SYSTEMS

PART 1
b. base n to Decimal
SOLUTION: using positional notation, it is the
n
total sum of d n x b for n=0 to n−1.
Where,
NUMBER SYSTEM
n – digit position
- Also known as base/radix system
dn – digit representation
b – radix
Radix – pertains to the number of symbols that
is used for counting in a particular number
system.
CONVERT THE FOLLOWING to decimal:
General format: d n−1 d n−2 … d1 d 0 1. 100110 2
2. 234.34 8
3. AB3.8 C 16
COMMON NUMBER SYSTEMS
1. Binary system – base 2 {0,1}
2. Decimal system – base 10 {0-9}
3. Octal system – base 8 {0-7}
4. Hexadecimal system – base 16 {0-9, a-f}

Radix starts from base 2 to base 36. {0-9, a-z}


C. base N 1 to base N 2
SOLUTION:
NUMBER SYSTEM CONVERSION base N 1 → DECIMAL → base N 2
a. Decimal to base n
SOLUTION: USING CONTINUOUS DIVISION
OF BASE N FOR INTEGER PART AND
CONTINUOUS MULTIPLICATION OF BASE N EXAMPLE 3
FOR DECIMAL PART. CONVERT THE FF NUMBER SYSTEMS:
1. 11011002 → base 8
EXAMPLE 1
2. 72 A 16 → base 2
CONVERT THE FOLLOWING:
1. 3810 →binary 3. 4 F .5 K 25 → base 15
2. 156.437510 →octal 4. HP . F 930 → base 20
3. 2739.546875 → hexadecimal 5. 23 AF .10C 16 → base 8
DETERMINING THE RADIX

Solution: using positional notation, both digits


must be converted to decimal then solve for the
unknown base.

SOLVE FOR THE UNKNOWN RADIX:


1. 121R → 144 8
2. 6 AE25 → 301 A R
Chapter 1: NUMBER SYSTEMS
PART 2

NUMBER SYSTEM OPERATIONS

1. Arithmetic operations
2. Bitwise operations
3. Relational operations

RELATIONAL OPERATIONS
RELATIONAL operators are used
to evaluate a condition that's applied to
one or two boolean expressions. The
result of the evaluation is either true or
false.

LISTS OF RELATIONAL OPERATORS:

← LESS THAN !=−¿ NOT


EQUAL
¿−¿ GREATER THAN ¿=−¿
EQUALS
≤−¿ LESS THAN OR EQUAL ¿∧−¿ LOGICAL
AND
≥−¿ GREATER THAN OR EQUAL ¿∨−¿ LOGICAL
OR

BITWISE OPERATIONS

A bitwise operator works with the


binary representation of a number rather
than that number's value. The operand
is treated as a set of bits, instead of as a
single number. Bitwise operators are
similar in most languages that support
them.

LISTS OF BITWISE OPERATORS:

¿−¿ AND OPERATOR ¿−¿ OR OPERATOR ¿−¿ XOR


OPERATOR

!−¿ NOT OPERATOR ≪−¿ SHIFT LEFT ≫−¿ SHIFT


COMPLEMENTS
Two forms of complement
DIMINISHED RADIX: (r −1)' s Complement =( r n−1 ) −N
RADIX: r s Complement =[ ( r −1 )−N ]+1
' n

WHERE,
r – RADIX
N – given number in decimal
n – no. of characters/digit of the given
number

EXAMPLE 1 ADDITION (EXAMPLE 3)


Given: 1. 2374 10+ 899210
1. 100110 2, evaluate 1’s and 2’s 2. 57718 +4273 8
complement 3. A 0 FFC 716+ E 359 B16 + B 11 AF 16
2. 2 F 31C 816, evaluate 15’s and 16’s 4.
complement 100110 2+1111 2+111011 2+1111102 +1011112
3. I 1 BFSR30 , evaluate 29’s and 30’s
complement
C. SUBTRACTION OF NUMBER E. DIVISION OF NUMBER SYSTEM
SYSTEMS
1. 8147 10−258210 1. 25 B7 16 / A 16
2. F 002 C 316 −493 BA16 2. 2 FG 319 /1619
3. 1010012−100110 2

D. MULTIPLICATION OF NUMBER
SYSTEMS
1.35774 8 x 1218
2.6 A 5 F 220 x 15 A 20

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