Download Module 2 – Number Systems, Operations and Codes

Introduction to Digital Electronics, Module 1: Intro to Digital Concepts
Module 2 – Number Systems, Operations and Codes
Introduction:
A digital systems designer must be familiar with arithmetic number
manipulation of decimal, binary, and hexadecimal (HEX) numbers.
This module will show you how each of these number systems are
related and will show you how to manipulate the data. You will learn
how to convert between systems, how to do basic arithmetic in any
number base, and how to perform addition and subtraction of signed,
unsigned and floating point binary numbers. You will learn how to
handle negative numbers and how to perform 2’s complement
subtraction of binary numbers. Finally you will learn about codes, and
how data is formatted and sent from one digital circuit to the next.
CONCEPT 2-1: NUMBER SYSTEMS
The difference between number systems is the base or number
where a carry takes place.
In decimal, the base is ten, and each digit at each position in the
number can have a value of 0 through 9. At the number 10, a carry is
generated. The weight of each position is equal to the base raised to
the power of the position. The first position to the left of the decimal
point is the 0th position, the next is position 1, the next is position 2,
position 3, and so on. The weight of the 0 position is 10 to the 0th
power (100=1). Any number raised to the power of zero is equal to 1,
so the weight of the first digit is one. The weight of the next digit is
10 to the power of 1 (101=10), the next is 10 to the positional power
of 2 (102=100), and so on.
Consider the number 1987. Seven is in the zero position, eight is in
the 1 position, nine is in the 2 position and one is in the 3 position.
Each digit is multiplied by its positional weight of 1000, 100, 10, 1.
The decimal number 1987 is equivalent to 1000 + 900 + 80 + 7.
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The same process works for all number systems. The positions are the
same, but the weight for each position is the base raised to power of
the position. In binary, the number count is 0, 1 with a carry
generated at two; the base is equal to 2. Two raised to the power of
zero is still one. Two to the power of 1 is 2. Two to the power of 2 is
4. Two raised to the power of 3 is 8. In binary, the weight of each
digit position is twice the value of the previous position, or it doubles
as 1, 2, 4, 8, 16, 32, 64, 128, and so on.
We can use the same method to evaluate a binary number as we did
to evaluate a decimal number. There can only be a 0 or a 1 value for
each digit.
Evaluate the number:
0 1 1 1 1 1 0 0 0 0 1 1.
There are 12 digits. Starting with the right most digit, it is worth 1.
The next is 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, and 2048.
We add up the values at those positions where the digit is equal to 1,
omitting those places where the digit is 0 (0 times anything = 0). The
above binary number has a decimal value of:
1024 + 512 + 256 + 128 + 64 + 2 + 1 = 1987
Definition 2.1: Base
The base of a number system is the numeric value that is reached
when a carry takes place. In a decimal system this is ten, in a binary
system it is two, in a hexadecimal system it is 16. All number systems
have single symbols, which have a counting number value up to the
base minus one. In decimal the numbers go from 0 through 9 with a
carry at 10, in binary, the numbers are 0 or 1 with a carry at 2, and in
hexadecimal, the numbers run from 0 through 9 corresponding to 0
thru 9 in decimal, then A thru F (where F is worth 15 in decimal). The
carry takes place at a count of 16.
Definition 2.2: Digit Position
Number systems must record values much greater than the value of
the base, so several digits or numbers, side by side in succession, are
needed to express a value. Each digit in the number system is
weighted according to its position. The position is the location of the
digit relative to the decimal point. The decimal point is the location
immediately to the right of the first whole value in the number. The
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first number is in position 0, the next is in position 1, the next is in
position 2, and so on.
Definition 2.3: Positional Weight
The weight of a digit is the value assigned to its position in the
number. The weight is calculated as the base of the number system
raised to the positional power. The first, or 0th position is worth 1 in
all number systems; the next, or 1 position, is worth the value base in
all number systems. In decimal this is 10, in binary this is 2, and in
hexadecimal this is 16. The value of any digit in a number is equal to
its counting value times its positional weight
CONCEPT 2-2: Conversion from Binary to Hexadecimal
Each hexadecimal digit can represent, or be represented by
four binary bits. To convert from binary to HEX, divide the
binary number into four bit slices, and substitute the equivalent
hexadecimal digit.
Four binary bits result in sixteen combinations of ones and zeros.
Each combination has a decimal and a hexadecimal value. Conversion
between binary and hexadecimal requires substitution from the table
below:
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CONCEPT 2.3: Conversion from Hexadecimal to Binary
Conversion from HEX to binary is the opposite process.
Substitute the four binary bit equivalent for each HEX digit.
Convert the four hexadecimal numbers into their equivalent 16 binary
number. 7 C 3H = 0 1 1 1 1 1 0 0 0 0 1 1 B. Note the "H" and "B" at
the end of each number tells us the numbers are hexadecimal (HEX),
and binary numbers respectively.
CONCEPT 2-4:
Conversion from Hexadecimal to Decimal
Hexadecimal is a base 16 number system. Each digit has a
counting value from zero to 15. Each positional weight is equal
to 16 to the power of the position.
As an example, we will convert the number 7 C 3 H to its decimal
equivalent. The 0 at the front is a convention to designate the
E 3 9 value as a number and not a string of letters. The H on the end
means it is a HEX number or base 16.
The
The
The
The
3 position is worth 3 x 160 = 3 x 1
C position is worth 12 x 161 = 12 x 16
7 position is worth 7 x 162 = 7 x 256
three values are added together
CONCEPT 2-5:
=
3
=
192
= 1792
= 1987 decimal
Conversion from Decimal to Binary
One method of converting from decimal to binary subtracts the
binary positional weights from the decimal number being
converted starting with the largest value that will subtract
leaving 0 or a positive value.
Example: Convert 1987 to binary.
1.
Create a binary weighted progression:
2048
1024
512
256
128
64
32
16
8
4
2
1
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Progressively subtract the binary position weights from
the decimal number, putting a “1” by each positional
weight that will subtract leaving a zero or a positive
remainder:
1987
- 1024
963
2048
0
963
- 512
451
1024
1
512
1
451
- 256
195
256
195
- 128
67
128
1
67
- 64
3
64
1
32
1
3
- 2
1
16
0
8
0
1
-1
0
4
0
2
0
1
1
1
After 64 is subtracted from the previous remainder of 67, a remainder
of 3 is left. Therefore 32, 16, 8, and 4 will not subtract, thus these
positions get a zero.
CONCEPT 2-6:
Addition in any Number System
When doing addition in any number system, a carry is
generated if the value of the number system base is reached.
The carry is added to the next higher digit. In binary, a carry is
generated every time that 1+1 is added.
The base of a number system is the number where a carry occurs. The
manipulation of the numbers is the same for all number systems. In
binary, if you add two bits, there are three possible results. 0 + 0 = 0,
0 + 1 = 1, and 1 + 1 = 0 and a carry, or 1 0. The carry takes place
when the value of two is reached, and therefore two is the base
number.
CONCEPT 2-7:
Multiplication by the Number System Base
Any number system can be multiplied by its base by moving the
decimal point to the right one place, or by adding a zero to the
least significant digit end of the number (right most position.)
In decimal, 19 x 10 = 190. In binary, 1101 x 2 = 11010
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CONCEPT 2-8:
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Division by the Number System Base
Any number system can be divided by its base by moving the
decimal point to the left one place. In decimal , 192/10 = 19.2.
In binary, 11010/2 = 1101 .
CONCEPT 2-9:
Getting the 2’s Complement of a Binary Number
The 2’s complement of a binary number allows a computer to
perform addition and subtraction of signed numbers, and to get
a result that keeps track of a negative or positive sign or result.
To get the 2’s complement you must first find the 1’s
complement.
The 1's complement is found by inverting each binary digit:
1 1 0 1 0 -- 1's complement = 0 0 1 0 1.
The 2's complement can be realized by adding a binary 1 to the
1's complement:
1 1 0 1 0 - 2's complement = 0 0 1 0 1 + 1 = 0 0 1 1 0.
The same result can be achieved by taking the original number,
1 1 0 1 0, and starting from the least significant digit, move to the left
of the first 1. Leave the first 1 you come to, and invert everything else
to the left. Using our number 1 1 0 1 0, the 2's complement becomes
0 0 1 1 0, using the "invert everything left of the first 1" method.
CONCEPT 2-10:
2’s Complement Subtraction
Two binary numbers can be subtracted by taking the 2’s
complement of the subtrahend (number being subtracted), and
adding it to the minuend. If a carry bit is generated by the
addition of the 2’s complemented subtrahend, and the uncomplemented minuend, the sign is positive, and the resulting
sum is the correct answer. If no carry is generated, the result
is a negative sign, and an inverted result. To get the correct
negative value, the inverted result must be 2’s complemented.
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CONCEPT 2-11: Signed Numbers
Numbers that can either be positive or negative, such as
voltage, can use the most significant bit as a sign bit. A "0"
means the value is positive, and a "1" means the value is
negative. This relates to the 2's complement system because
negative numbers are the 2's complement of the corresponding
positive value. Just remember that if the most significant bit is
being used as a sign bit, it cannot also be part of the number
magnitude.
CONCEPT 2-12: Arithmetic Operations with Signed Numbers
When performing arithmetic operations using signed numbers,
the result will give you the sign of the result, but the
magnitude of the answer is also dependant on a resulting
carry, or 2's complement overflow. This is the carry bit in the
previous example.
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CONCEPT 2-13: Floating Point Numbers
A floating point number can have a decimal point or a fractional
value. When a number in any number system has digits to the
right of the decimal point, the positions have negative values
starting with –1, then –2, -3, and so on. The weight of each
position is the base raised to the negative power of the
position. In binary, the number 1101.011 is equal to 13 from
the left side + .25 + .125 for the digits on the right side of the
decimal point. 1101.011 = 13.375 .
CONCEPT 2-14: Binary Coded Decimal (BCD)
In binary coded decimal, four binary bits are used to represent
each decimal digit in a decimal number. Four bit binary
combinations above 9 are not allowed. Many calculators use
BCD systems to perform their mathematical operations directly.
CONCEPT 2-15: Digital Codes and Parity
Digital codes are often used to represent everything from the
alphabet to strange instrument weightings or number
sequences. The most common use of codes is in error detection
and correction.
A binary sequence of bits can tell you if data has been corrupted.
Some codes look for a repeated pattern and send a key to verify that
the pattern was not changed during transmission. Parity sends an
additional 1 or 0 at the end of a sequence of binary bits. The 1 or 0 is
sent to keep the number of 1's being transmitted either an odd or an
even count. If the data received has an odd number of 1's, like 1, 3, 5,
7, 9, etc. ...., to maintain even parity, the parity bit would be 1. For all
even numbers of 1 in the binary sequence, 2, 4, 6, 8, etc. ..., the
parity bit would be 0. This allows for detection of a single bit error.