Unit 1: Binary Systems [6 Hours]

Binary systems form the numerical and logical foundation of digital computers and electronic systems. This unit explains how digital systems represent information using two states, how numbers are written and converted among binary, octal, decimal, and hexadecimal bases, and how complements and signed representations support arithmetic. It also covers decimal codes such as Binary-Coded Decimal, weighted codes, Excess-3 and Gray code, followed by binary storage, registers, and the basic logic operations used to process binary information.

Tested Topics

1. Gray Code — 2x (2083, 2081)
2. Number base conversion — 2x (2083, 2081)
3. BCD — 1x (2081)
4. Binary addition — 1x (2078)
5. Complement arithmetic — 1x (2080)
6. Digital and analog systems — 1x (2078)

1.1 Digital Systems

Definition

A digital system is a system that represents, processes, stores, or transmits information using a finite set of discrete values, usually the binary values 0 and 1.

A digital system works with clearly separated states. A switch can be OFF or ON, a stored bit can be 0 or 1, and a logic circuit can treat a voltage range as LOW or HIGH. Instead of following every possible value continuously, the system works with selected discrete levels.

Detailed Explanation

The word digital comes from the idea of representing information using distinct symbols or digits. Modern digital electronics normally uses the binary number system, where only two symbols, 0 and 1, are required.

A digital system normally performs one or more of the following functions:

• Input: receives information from a keyboard, sensor, switch, network, or another system.
• Processing: performs arithmetic, comparison, logical decisions, encoding, or other operations.
• Storage: keeps binary information in memory cells, registers, disks, or other storage devices.
• Output: presents or transmits the processed information.

For example, when a student enters the number 25 into a calculator, the calculator converts the input into an internal binary representation, performs operations using digital circuits, stores temporary results, and converts the final result into a form that can be displayed.

Discrete Values

A discrete value is a separate and distinct value rather than a continuously changing one. A digital signal therefore changes among defined levels.

In a typical binary digital system:

• Logic 0 represents one state, often called LOW, FALSE, OFF, or inactive.
• Logic 1 represents the other state, often called HIGH, TRUE, ON, or active.

The exact electrical voltage corresponding to 0 or 1 depends on the electronic technology being used. The logical meaning is more important than the exact voltage value.

Analog Systems

An analog system represents information using continuously varying physical quantities. Its value can change smoothly through an entire range rather than moving between fixed digital levels.

Examples include:

• the continuously changing voltage produced by an analog microphone,
• the position of the pointer in a traditional analog voltmeter,
• the continuous motion of the hands of a mechanical clock,
• physical quantities such as temperature, pressure, speed, and sound pressure.

If temperature changes from 20°C to 21°C, the real physical temperature can pass through 20.1°C, 20.25°C, 20.753°C, and infinitely many values in between. This continuous behavior is characteristic of analog information.

Digital System and Analog System Compared

Digital Computer and Analog Computer

A digital computer processes information represented by discrete binary digits. General-purpose computers, laptops, smartphones, and most modern computing systems are digital computers.

An analog computer represents and solves problems by using continuously varying physical quantities such as voltage, current, rotation, or fluid motion to model another physical system.

Why Digital Representation Is Widely Used

• Reliable distinction between states: small signal variations often do not change whether a value is interpreted as 0 or 1.
• Easy storage: binary information can be stored in memory elements having two stable states.
• Easy processing: arithmetic and decision operations can be implemented using logic circuits.
• Reproducibility: digital data can be copied without gradually accumulating the same type of degradation associated with repeated analog copying.
• Programmability: the same hardware can perform different tasks under program control.

1.2 Binary Numbers

Definition

A binary number is a number written in the base-2 number system using only the digits 0 and 1.

Decimal numbers use ten digits from 0 to 9. Binary needs only two. Each position represents a power of 2, so the position of a 1 tells us which power of 2 contributes to the total value.

Detailed Explanation

The binary number system is the natural numerical system for digital hardware because electronic circuits can represent two logical states efficiently.

Each binary digit is called a bit, short for binary digit. A sequence such as 101101 contains six bits.

Positional Value in Binary

Binary is a positional number system. Starting from the right side of the binary point, the place values are powers of 2:

... 2⁵, 2⁴, 2³, 2², 2¹, 2⁰ . 2^-1, 2^-2, 2^-3 ...

For example:

(101101)₂

= 1 × 2⁵ + 0 × 2⁴ + 1 × 2³ + 1 × 2² + 0 × 2¹ + 1 × 2⁰

= 32 + 0 + 8 + 4 + 0 + 1

= (45)₁₀

Binary Fractions

Positions to the right of the binary point use negative powers of 2.

For example:

(0.101)₂

= 1 × 2^-1 + 0 × 2^-2 + 1 × 2^-3

= 1/2 + 0/4 + 1/8

= 0.5 + 0.125

= (0.625)₁₀

Common Binary Groupings

• 1 bit: one binary digit.
• 4 bits: one nibble.
• 8 bits: one byte.
• Word: a processor-related group of bits treated as one unit. Word size depends on the computer architecture.

Binary Addition

Binary addition follows the same column-by-column idea as decimal addition, but every digit is either 0 or 1.

The basic addition rules are:

The rule 1 + 1 = 10₂ means the current column receives 0 and a carry of 1 is transferred to the next higher position.

Worked Example: Binary Addition

Add (11001)₂ and (11011)₂.

       1 1 0 0 1
     + 1 1 0 1 1
     -----------
     1 1 0 1 0 0

Working from right to left:

• 1 + 1 = 10 → write 0, carry 1.
• 0 + 1 + carry 1 = 10 → write 0, carry 1.
• 0 + 0 + carry 1 = 1 → write 1, no carry.
• 1 + 1 = 10 → write 0, carry 1.
• 1 + 1 + carry 1 = 11 → write 1, carry 1.
• Write the remaining carry 1.

Therefore:

(11001)₂ + (11011)₂ = (110100)₂

A decimal check confirms the result:

25 + 27 = 52, and (52)₁₀ = (110100)₂.

Basic Binary Subtraction Rules

Direct subtraction is possible, but digital systems commonly use complement methods because subtraction can then be performed by an addition circuit. That method is developed in 1.5 Complements.

1.3 Number Base Conversion

Definition

Number base conversion is the process of changing the representation of a number from one number system to another without changing its actual value.

The written form changes, but the quantity does not. Decimal 10, binary 1010, octal 12, and hexadecimal A are different representations of the same numerical value.

Detailed Explanation

A radix or base tells how many distinct digit symbols are used by a positional number system. The common bases in digital systems are:

General Positional Value Formula

For a number written in base r:

(d_nd_n-1...d₁d₀.d_-1d_-2...)_r

its decimal value is:

d_nrⁿ + d_n-1r^n-1 + ... + d₁r¹ + d₀r⁰ + d_-1r^-1 + d_-2r^-2 + ...

Each digit is multiplied by the positional weight associated with its position.

Conversion from Any Base to Decimal

To convert a positional number from base r to decimal:

1. Write the power of the base associated with every position.
2. Multiply each digit by its positional weight.
3. Add all products.

Worked Example: Hexadecimal to Decimal

Convert (F4B.B)₁₆ to decimal.

In hexadecimal:

• F = 15
• B = 11

Therefore:

(F4B.B)₁₆

= 15 × 16² + 4 × 16¹ + 11 × 16⁰ + 11 × 16^-1

= 15 × 256 + 4 × 16 + 11 × 1 + 11/16

= 3840 + 64 + 11 + 0.6875

= (3915.6875)₁₀

Decimal Integer to Another Base

To convert a decimal integer to base r, use repeated division by r.

1. Divide the decimal integer by the target base.
2. Record the remainder.
3. Divide the quotient again by the target base.
4. Continue until the quotient becomes 0.
5. Read the remainders from bottom to top.

Worked Example: Decimal 110 to Binary

Reading the remainders upward:

(110)₁₀ = (1101110)₂

Decimal Fraction to Another Base

To convert a decimal fraction to base r, use repeated multiplication by r.

1. Multiply the fractional part by the target base.
2. Record the integer part of the result.
3. Take the new fractional part and multiply it again by the target base.
4. Continue until the fractional part becomes zero or the required precision is reached.
5. Read the recorded integer parts from top to bottom.

Worked Example: Decimal 0.625 to Binary

Reading the integer parts downward:

(0.625)₁₀ = (0.101)₂

Combining the integer and fraction:

(110.625)₁₀ = (1101110.101)₂

Worked Example: Decimal 110.625 to Octal

For the integer part:

Thus:

(110)₁₀ = (156)₈

For the fraction:

0.625 × 8 = 5.0

Therefore:

(0.625)₁₀ = (0.5)₈

Hence:

(110.625)₁₀ = (156.5)₈

Binary to Octal Conversion

Since 8 = 2³, one octal digit corresponds exactly to three binary bits.

Procedure:

1. Starting from the binary point, divide binary digits into groups of three.
2. For the integer portion, group toward the left.
3. For the fractional portion, group toward the right.
4. Add leading or trailing zeros when a group is incomplete.
5. Replace each three-bit group with the corresponding octal digit.

Example:

(110001)₂

= 110 001

= 6 1

= (61)₈

Octal to Binary Conversion

Replace each octal digit with its three-bit binary equivalent.

Example:

(572)₈

5 → 101
7 → 111
2 → 010

Therefore:

(572)₈ = (101111010)₂

Binary to Hexadecimal Conversion

Since 16 = 2⁴, one hexadecimal digit corresponds to four binary bits.

Example:

(1011011110)₂

Pad and group:

0010 1101 1110

0010 = 2
1101 = D
1110 = E

Therefore:

(1011011110)₂ = (2DE)₁₆

Hexadecimal to Binary Conversion

Replace every hexadecimal digit with four binary bits.

Convert (A57)₁₆ to binary:

A → 1010
5 → 0101
7 → 0111

Therefore:

(A57)₁₆ = (101001010111)₂

Octal and Hexadecimal Conversion Through Binary

Because both octal and hexadecimal have direct power-of-two relationships with binary, conversion between them is conveniently performed through binary.

Octal → Binary → Hexadecimal

or

Hexadecimal → Binary → Octal

Example:

(73)₈

7 → 111
3 → 011

Thus:

(73)₈ = (111011)₂

Group into four bits:

0011 1011

0011 = 3
1011 = B

Therefore:

(73)₈ = (3B)₁₆

1.4 Octal and Hexadecimal Numbers

Definition

An octal number is a base-8 number using digits 0 to 7, while a hexadecimal number is a base-16 number using digits 0 to 9 and letters A to F.

Binary strings can become long and difficult to read. Octal compresses every three binary bits into one digit, while hexadecimal compresses every four binary bits into one digit.

Detailed Explanation

Octal Number System

The octal number system has radix 8 and therefore uses eight symbols:

0, 1, 2, 3, 4, 5, 6, 7

Its positional weights are powers of 8:

... 8³, 8², 8¹, 8⁰ . 8^-1, 8^-2 ...

Example:

(357)₈

= 3 × 8² + 5 × 8¹ + 7 × 8⁰

= 192 + 40 + 7

= (239)₁₀

Octal-to-Binary Relationship

This direct relationship follows from 8 = 2³.

Hexadecimal Number System

The hexadecimal number system has radix 16. It requires sixteen symbols:

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F

The letters represent decimal values:

• A = 10
• B = 11
• C = 12
• D = 13
• E = 14
• F = 15

Its positional weights are powers of 16:

... 16³, 16², 16¹, 16⁰ . 16^-1, 16^-2 ...

Example:

(2AF)₁₆

= 2 × 16² + 10 × 16 + 15

= 512 + 160 + 15

= (687)₁₀

Hexadecimal-to-Binary Relationship

The four-bit relationship follows from 16 = 2⁴.

Octal and Hexadecimal Compared

Why Octal and Hexadecimal Are Useful

Computers internally work with binary, but humans can make mistakes when reading long binary strings. Octal and hexadecimal provide compact representations while preserving an easy conversion path back to binary.

For example:

Binary: 1111011010111100

In hexadecimal:

1111 0110 1011 1100 → F 6 B C

So:

(1111011010111100)₂ = (F6BC)₁₆

1.5 Complements

Definition

A complement is an alternative representation of a number used mainly to perform subtraction through addition and to represent negative values in digital systems.

Instead of building a separate arithmetic process for every subtraction, a system can transform the number being subtracted into a complement and add it. Binary systems commonly use 1's complement and 2's complement, while decimal complement arithmetic uses 9's complement and 10's complement.

Detailed Explanation

For a number system of radix r, two general complements are used:

• (r − 1)'s complement: also called the diminished radix complement.
• r's complement: also called the radix complement.

Diminished Radix Complement

For an n-digit number N in base r:

(r − 1)'s complement = (rⁿ − 1) − N

For binary, r = 2, so the diminished radix complement is 1's complement.

For decimal, r = 10, so the diminished radix complement is 9's complement.

Radix Complement

For an n-digit number N:

r's complement = rⁿ − N

It can also be found by:

r's complement = (r − 1)'s complement + 1

Thus:

• 2's complement = 1's complement + 1
• 10's complement = 9's complement + 1

1's Complement

The 1's complement of a binary number is obtained by changing every 0 to 1 and every 1 to 0.

Example:

Original number: 101001

1's complement: 010110

2's Complement

The 2's complement of a binary number is obtained by taking its 1's complement and adding 1.

Example:

Number: 101001

1's complement: 010110

Add 1:

010110
+    1
------
010111

Therefore the 2's complement of 101001 is 010111.

9's Complement

The 9's complement of a decimal number is obtained by subtracting every digit from 9.

Example:

Number: 58324

9's complement:

• 9 − 5 = 4
• 9 − 8 = 1
• 9 − 3 = 6
• 9 − 2 = 7
• 9 − 4 = 5

Therefore:

9's complement of 58324 = 41675

10's Complement

The 10's complement is found by adding 1 to the 9's complement.

For 58324:

9's complement = 41675

10's complement = 41675 + 1 = 41676

General Complement Comparison

Subtraction Using 2's Complement

To calculate M − N using 2's complement:

1. Use the same number of bits for M and N.
2. Find the 2's complement of N.
3. Add this complement to M.
4. If an end carry appears, discard the carry. The remaining result is positive.
5. If no end carry appears, the result is negative. Find the 2's complement of the obtained sum to find its magnitude and attach a minus sign.

Worked Example: (011101)₂ − (110011)₂ Using 2's Complement

Let:

M = 011101

N = 110011

First find the 2's complement of N.

1's complement of 110011:

001100

Add 1:

001100 + 1 = 001101

Now add it to M:

  011101
+ 001101
--------
  101010

There is no end carry, so the result is negative.

Find the 2's complement of 101010 to obtain the magnitude:

1's complement:

010101

Add 1:

010101 + 1 = 010110

Therefore:

(011101)₂ − (110011)₂ = −(010110)₂

In six-bit 2's complement representation, the stored result is 101010. Its decimal value is −22.

Subtraction Using 1's Complement

To compute M − N using 1's complement:

1. Find the 1's complement of N.
2. Add it to M.
3. If a carry is produced, add that carry back to the least significant bit. This is called an end-around carry.
4. If no carry is produced, complement the result and mark it negative.

Example: calculate 10110 − 01101.

1's complement of 01101 = 10010.

  10110
+ 10010
-------
1 01000

Add the end-around carry:

01000
+   1
-----
01001

Therefore:

10110 − 01101 = 01001

Decimal check: 22 − 13 = 9.

Subtraction Using 9's Complement

For decimal subtraction M − N using 9's complement:

1. Use the same number of decimal digits in both numbers.
2. Find the 9's complement of N.
3. Add it to M.
4. If an end carry appears, add it to the least significant digit.
5. If no end carry appears, take the 9's complement of the sum and attach a minus sign.

Worked Example: 89344 − 98654 Using 9's Complement

Find the 9's complement of 98654:

9 − 9 = 0
9 − 8 = 1
9 − 6 = 3
9 − 5 = 4
9 − 4 = 5

Therefore:

9's complement of 98654 = 01345

Add it to 89344:

  89344
+ 01345
-------
  90689

No end carry is produced, so the answer is negative.

Find the 9's complement of 90689:

9's complement of 90689 = 09310

Therefore:

89344 − 98654 = −9310

Subtraction Using 10's Complement

For M − N using 10's complement:

1. Find the 10's complement of N.
2. Add it to M.
3. If there is an end carry, discard the carry and keep the remaining digits.
4. If there is no end carry, find the 10's complement of the sum and attach a negative sign.

1's versus 2's Complement

1.6 Signed Binary Numbers

Definition

A signed binary number is a binary representation that includes information about whether the represented number is positive or negative.

Ordinary unsigned binary bits represent only non-negative values. To represent values below zero, computers use a defined encoding method such as sign-magnitude, 1's complement, or 2's complement.

Detailed Explanation

An n-bit unsigned binary number can represent values from:

0 to 2ⁿ − 1

For example, an 8-bit unsigned number represents values from:

0 to 2⁸ − 1 = 255

Signed representations reserve part of the bit pattern to distinguish negative values from positive values.

Sign-Magnitude Representation

In sign-magnitude representation, the most significant bit acts as a sign bit:

• 0 indicates a positive number.
• 1 indicates a negative number.

The remaining bits store the magnitude.

Using 8 bits:

+13:

00001101

−13:

10001101

The major weakness is that zero has two representations:

+0 = 00000000

−0 = 10000000

1's Complement Representation

In 1's complement signed representation, a positive number is written normally, while the negative form is obtained by inverting every bit of its positive form.

For 8-bit 13:

+13 = 00001101

Invert every bit:

−13 = 11110010

1's complement also has two representations of zero:

+0 = 00000000

−0 = 11111111

2's Complement Representation

In 2's complement signed representation, a negative value is produced by taking the 1's complement of its positive magnitude and adding 1.

For 8-bit 13:

+13 = 00001101

1's complement = 11110010

Add 1:

−13 = 11110011

Unlike sign-magnitude and 1's complement, 2's complement has only one representation of zero.

Ranges of Signed Representations

For n bits:

For 8 bits:

• Sign-magnitude: −127 to +127.
• 1's complement: −127 to +127.
• 2's complement: −128 to +127.

Finding the Decimal Value of a Negative 2's Complement Number

If the most significant bit of a fixed-width 2's complement number is 1, the number is negative.

Example: interpret 11110101 as an 8-bit 2's complement number.

1. Invert every bit: 00001010.
2. Add 1: 00001011.
3. 00001011 = 11 decimal.
4. Therefore the original number is −11.

Alternative Positional Interpretation of 2's Complement

In an n-bit 2's complement number, the most significant bit has weight −2ⁿ⁻¹. All other bits retain positive binary weights.

For 8-bit 11110101:

= −128 + 64 + 32 + 16 + 0 + 4 + 0 + 1

= −11

Sign-Magnitude, 1's Complement, and 2's Complement Compared

Overflow in Signed Binary Arithmetic

Overflow occurs when the true arithmetic result cannot be represented with the available number of bits.

In 2's complement addition, overflow occurs when:

• two positive numbers produce a result whose sign bit indicates negative, or
• two negative numbers produce a result whose sign bit indicates positive.

Example using 4-bit 2's complement:

0111 represents +7.

0011 represents +3.

  0111
+ 0011
------
  1010

The mathematical result is +10, but 4-bit 2's complement represents only −8 to +7. The bit pattern 1010 would normally mean −6, so the operation has overflowed.

1.7 Decimal Codes

Definition

A decimal code is a binary coding scheme used to represent decimal digits or decimal information with defined groups of binary bits.

A decimal digit such as 7 can be represented using several different four-bit patterns depending on the code being used. The pattern 0111 means 7 in standard Binary-Coded Decimal, but another coding system may assign a different pattern to the same decimal digit.

Detailed Explanation

Digital circuits operate with binary values, while many human applications use decimal numbers. Decimal codes provide a controlled way to encode decimal digits in binary form.

The codes in this unit include:

• Binary-Coded Decimal (BCD), usually the 8-4-2-1 code,
• 2-4-2-1 code,
• 8-4-−2-−1 code,
• Excess-3 code,
• Gray code.

Weighted and Non-Weighted Codes

A weighted code assigns a fixed numerical weight to every bit position. The represented decimal digit can be determined by adding the weights of bit positions containing 1.

Examples include:

• 8-4-2-1 Binary-Coded Decimal,
• 2-4-2-1 code,
• 8-4-−2-−1 code.

A non-weighted code does not assign fixed positional weights that directly produce the represented value.

Examples include:

• Excess-3 code,
• Gray code.

Binary-Coded Decimal

Binary-Coded Decimal (BCD) is a coding method in which each decimal digit is represented separately by its four-bit binary equivalent.

The most common BCD form is the 8-4-2-1 code. Its bit positions have weights 8, 4, 2, and 1.

Only ten of the sixteen possible four-bit combinations are valid BCD digit codes. The combinations 1010 through 1111 do not represent decimal digits in standard 8421 BCD.

Converting Decimal to BCD

Convert each decimal digit separately into its four-bit BCD equivalent.

Example: convert (10)₁₀ to BCD.

Decimal digit 1 → 0001

Decimal digit 0 → 0000

Therefore:

(10)₁₀ = 0001 0000 in BCD

Notice that this is not the same as the ordinary binary representation of decimal 10:

Decimal 10 in pure binary = 1010

Decimal 10 in BCD = 0001 0000

Example: Decimal 59 in BCD

5 → 0101

9 → 1001

Therefore:

(59)₁₀ = 0101 1001 in BCD

BCD Compared with Pure Binary

2-4-2-1 Code

The 2-4-2-1 code is a weighted four-bit decimal code whose bit positions carry weights 2, 4, 2, and 1 from left to right.

A common 2-4-2-1 assignment is:

The 2-4-2-1 code is self-complementing. Complementing all four bits of a valid code word produces the code word for the 9's complement of the corresponding decimal digit.

Example:

2 → 0010

Bit complement → 1101

1101 represents 7, and 9 − 2 = 7.

8-4-−2-−1 Code

The 8-4-−2-−1 code is a weighted decimal code with bit weights 8, 4, −2, and −1.

The sum of the positional weights is:

8 + 4 − 2 − 1 = 9

This property allows the code to be self-complementing.

Excess-3 Code

Excess-3 code is a non-weighted decimal code obtained by adding 3 to each decimal digit and then expressing the result as a four-bit binary number.

The encoding rule is:

Excess-3 code for decimal digit d = four-bit binary representation of d + 3

Example: encode decimal 47 in Excess-3.

4 + 3 = 7 → 0111

7 + 3 = 10 → 1010

Therefore:

47₁₀ = 0111 1010 in Excess-3

Decoding Excess-3

To decode an Excess-3 digit:

1. Convert the four-bit group to its ordinary binary numerical value.
2. Subtract 3.

Example:

1001 = 9

9 − 3 = 6

Therefore 1001 in Excess-3 represents decimal digit 6.

Excess-3 is also a self-complementing code. Bitwise complementing the Excess-3 code of a decimal digit produces the Excess-3 code for its 9's complement.

Gray Code

Gray code is a non-weighted binary code in which two consecutive code words differ in only one bit position.

In ordinary binary counting, several bits can change at the same time. For example, moving from 0111 to 1000 changes all four bits. Gray code arranges adjacent values so that only one bit changes, reducing ambiguity when transitions are sensed physically.

Unit-Distance Property

The defining property of Gray code is its unit-distance property: adjacent code words have a Hamming distance of one, meaning exactly one bit changes between neighboring values.

Binary to Gray Code Conversion

Gray conversion uses the Exclusive-OR (XOR) operation. XOR produces 1 when its two input bits are different and 0 when they are equal.

To convert binary to Gray code:

1. Copy the most significant binary bit directly as the most significant Gray bit.
2. For every following Gray bit, XOR the current binary bit with the binary bit immediately to its left.

For binary bits B and Gray bits G:

G_MSB = B_MSB

Each next G = previous B XOR current B

Worked Example: Binary 1010 to Gray Code

Binary:

1 0 1 0

• First Gray bit = 1.
• 1 XOR 0 = 1.
• 0 XOR 1 = 1.
• 1 XOR 0 = 1.

Therefore:

(1010)₂ = (1111)_Gray

Gray Code to Binary Conversion

To convert Gray code to binary:

1. Copy the most significant Gray bit directly as the most significant binary bit.
2. For each following position, XOR the previously obtained binary bit with the current Gray bit.

Worked Example: Gray 1111 to Binary

• First binary bit = 1.
• 1 XOR 1 = 0.
• 0 XOR 1 = 1.
• 1 XOR 1 = 0.

Therefore:

(1111)_Gray = (1010)₂

Generating Reflected Gray Code

Gray code is also known as reflected binary code because an n-bit sequence can be generated from an (n−1)-bit sequence by reflection.

For two-bit Gray code:

00
01
11
10

To create three-bit Gray code:

1. Write the two-bit sequence in its original order.
2. Write the same sequence in reverse order.
3. Prefix 0 to the original half.
4. Prefix 1 to the reflected half.

000
001
011
010
110
111
101
100

Uses of Gray Code

• Rotary and shaft encoders: reduces ambiguous readings when position changes between adjacent values.
• Position sensing: only one code bit changes between adjacent mechanical positions.
• Digital communication and conversion systems: useful where minimizing transition errors is desirable.
• Karnaugh-map ordering: adjacent cells are arranged using Gray-code ordering so neighboring combinations differ by one variable.

BCD and Gray Code Compared

Comparison of Decimal Codes

1.8 Binary Storage and Registers

Definition

Binary storage is the storage of information as binary bits, while a register is a group of binary storage elements used to hold a binary word temporarily inside a digital system.

One storage element can remember one 0 or 1. Several such elements placed together can remember a complete binary value. An eight-bit register, for example, can hold a pattern such as 10110110 at one moment and a different eight-bit pattern later.

Detailed Explanation

Digital systems need to preserve information between operations. If every intermediate value disappeared immediately after being calculated, a processor could not execute a sequence of instructions. Binary storage elements provide memory of previous states.

Binary Storage Element

A binary storage element has two stable logical states:

• one state represents binary 0,
• the other state represents binary 1.

A bistable digital circuit such as a flip-flop can store one bit. The detailed internal construction of flip-flops belongs to sequential logic, but at this level the essential fact is that one storage element preserves one binary state until an operation changes it.

Register Structure

A register combines multiple one-bit storage elements so that several bits can be stored together.

If a register contains n storage elements, it stores an n-bit word.

For example:

• 4 storage elements → 4-bit register.
• 8 storage elements → 8-bit register.
• 16 storage elements → 16-bit register.
• 32 storage elements → 32-bit register.

An 8-bit register storing decimal 45 may contain:

00101101

Because:

32 + 8 + 4 + 1 = 45.

Register Operations

Depending on the design, a register can support operations such as:

• Load: place a new binary value into the register.
• Clear: set stored bits to 0.
• Read: use the stored value without necessarily changing it.
• Transfer: copy a stored value from one register or unit to another.
• Shift: move bit positions left or right in registers designed for shifting.

Parallel Storage

When all bits of a register are loaded at the same time through separate input lines, the operation is called parallel loading.

For a 4-bit register:

D₃ D₂ D₁ D₀ → Register → Q₃ Q₂ Q₁ Q₀

If the inputs are 1011 when a load operation occurs, the register stores 1011.

Register Capacity

The capacity of a register is normally stated by its number of bits.

An n-bit register has:

• n storage positions,
• 2ⁿ possible bit patterns.

For example, a 4-bit register can contain:

2⁴ = 16 distinct bit patterns

from 0000 through 1111.

Registers and General Memory Compared

Binary Word

A binary word is a group of bits treated as a single unit of information.

The meaning of a word depends on context. The same pattern could represent:

• an unsigned number,
• a signed number,
• a coded decimal digit or character,
• part of an instruction,
• a set of control flags.

This connects binary storage with the representations developed in 1.2 Binary Numbers, 1.6 Signed Binary Numbers, and 1.7 Decimal Codes. Storage elements preserve bits; the selected coding convention determines what those bits mean.

1.9 Binary Logic

Definition

Binary logic is a system of logical operations in which variables have only two possible states, normally represented by 0 and 1.

A binary logic circuit receives one or more 0/1 inputs, applies a rule, and produces a 0/1 output. The rule may ask whether all inputs are 1, whether at least one is 1, whether an input should be inverted, or whether two inputs differ.

Detailed Explanation

Binary logic provides the decision-making foundation of digital circuits. Arithmetic circuits, control units, comparators, memory control, communication circuits, and processors are built from combinations of logical operations.

Binary logical values can be interpreted as:

Logic Variable

A logic variable is a variable that can take only two logical values: 0 or 1.

Examples:

A = 0 or 1

B = 0 or 1

A logic function produces an output based on one or more such inputs.

Truth Table

A truth table lists every possible input combination and the output produced by a logical operation.

For n independent binary inputs, the number of possible input combinations is:

2ⁿ

Thus:

• 1 input → 2 combinations.
• 2 inputs → 4 combinations.
• 3 inputs → 8 combinations.
• 4 inputs → 16 combinations.

NOT Operation

The NOT operation, also called logical inversion, produces the opposite binary value.

If A = 0, NOT A = 1.

If A = 1, NOT A = 0.

AND Operation

The AND operation produces 1 only when all of its required inputs are 1.

For two inputs A and B:

Y = A AND B

A useful interpretation is a permission rule requiring two conditions. If both conditions are true, the output becomes true.

OR Operation

The OR operation produces 1 when at least one input is 1.

Y = A OR B

Exclusive-OR Operation

The Exclusive-OR (XOR) operation produces 1 when its two inputs are different and 0 when they are equal.

This is the operation used in the Gray-code conversion method described in 1.7 Decimal Codes.

Basic Operations Compared

Combining Logic Operations

More complex binary decisions are formed by connecting basic logic operations.

Consider:

Y = (A AND B) OR C

The calculation follows this sequence:

A and B evaluated by AND → result combined with C using OR → final Y

Binary Logic and Digital Hardware

The operations described mathematically are implemented physically using logic gates. A logic gate is a digital circuit designed to perform a logical operation on binary inputs.

The relationship is:

Binary inputs → logic gate operation → binary output

Large digital systems consist of many gates connected to perform:

• arithmetic,
• comparison,
• data selection,
• code conversion,
• memory control,
• decision making.

Exam Questions

1. Convert the following as indicated:
   a) (F4B.B)16 to Decimal
   b) (110.625)10 to Binary and Octal (2083)
2. Write short notes on:
   a) Gray Code (2083)
3. Perform the following conversion
   a) (A57)₁₆ to binary
   b) (110001)₂ to octal (2.5+2.5) (2081)
4. Differentiate between BCD and Gray code. Convert (10)₁₀ to BCD. (3+2) (2081)
5. Perform the following operations:
   a. (011101)2-(110011)2 using 2's complement
   b. (89344)10-(98654)10 using 9's complement (2080)
6. Differentiate analog computer with digital computer. (2078)
7. How binary addition is done? Show binary addition of (11001) with (11011). (2078)

Whole Chapter Summary

Digital Systems

Digital systems work with discrete states, usually binary 0 and 1, while analog systems work with continuously varying quantities. Digital representation supports reliable logic processing, convenient storage, copying, and programmable computation.

Binary Numbers

Binary is the base-2 positional system used directly by digital hardware. Every bit position represents a power of 2, and binary arithmetic follows base-2 carry and borrow rules such as 1 + 1 = 10₂.

Number Base Conversion

Conversion changes the representation of a value without changing the value itself. Positional expansion converts numbers to decimal, repeated division converts decimal integers, repeated multiplication converts decimal fractions, and power-of-two grouping handles binary-octal and binary-hexadecimal conversion.

Octal and Hexadecimal Numbers

Octal is base 8 and represents three binary bits per digit. Hexadecimal is base 16 and represents four binary bits per digit, making both systems compact forms of long binary strings.

Complements

Complements allow subtraction to be performed through addition. Binary systems use 1's and 2's complements, while decimal complement arithmetic uses 9's and 10's complements.

Signed Binary Numbers

Signed representations encode positive and negative values using sign-magnitude, 1's complement, or 2's complement. Two's complement provides one zero representation and an n-bit range from −2ⁿ⁻¹ to 2ⁿ⁻¹−1.

Decimal Codes

Decimal codes translate decimal information into binary patterns. 8421 BCD represents each decimal digit separately, 2421 and 84−2−1 are weighted self-complementing codes, Excess-3 adds three to each digit, and Gray code provides one-bit transitions between adjacent words.

Binary Storage and Registers

A binary storage element holds one bit, and a register groups several storage elements to hold a binary word. An n-bit register has n bit positions and can store 2ⁿ distinct binary patterns.

Binary Logic

Binary logic processes 0/1 values using operations such as NOT, AND, OR, and XOR. Truth tables describe every input-output combination, while logic gates implement these rules as digital circuits.

Key Formulas and Points Sheet

Last-Minute Revision Sheet