Complements: r's and (r-1)'s - Radix and Diminished Radix Complements

Complements are used in the digital computers in order to simplify the subtraction operation and for the logical manipulations. For each radix-r system (radix r represents base of number system) there are two types of complements.

S.N.	Complement	Description
1	Radix Complement	The radix complement is referred to as the r's complement
2	Diminished Radix Complement	The diminished radix complement is referred to as the (r-1)'s complement

Binary system complements

As the binary system has base r = 2. So the two types of complements for the binary system are 2's complement and 1's complement.

1's complement

The 1's complement of a number is found by changing all 1's to 0's and all 0's to 1's. This is called as taking complement or 1's complement. Example of 1's Complement is as follows.

2's complement

The 2's complement of binary number is obtained by adding 1 to the Least Significant Bit (LSB) of 1's complement of the number.

2's complement = 1's complement + 1

Example of 2's Complement is as follows.

 

 

Decimal system complements

 In digital logic, decimal complements are employed less frequently than their binary equivalents. The majority of contemporary digital systems rely on binary representations and complements, such as 2's complement, for effective arithmetic operations, even if they still have some historical relevance and few uses.

Below is a summary of the various decimal complements:

1. Ten's complement, or Radix complement:

•     found by deducting each number from nine.
•     Example: 999 - 375 = 624 is the ten's complement of 375.

2. Nine's Complement, or Diminished Radix Complement:

•     found by deducting each number from eight.
•     Example: 888 - 375 = 513 is the complement of 375, which equals nine.

3. The Complement of (R-1):

•     Reversed each digit from (base - 1) to find.
•     For instance, the complement of 375 in (9-1) is equal to 888 - 375 = 513 in decimal notation.

Uses:

•     used for subtraction in mechanical calculators in the past.
•     utilized in certain applications of some circuits for decimal arithmetic.
•     A setting for education in order to comprehend complements.

Restrictions:

•     more intricate computations in contrast to binary complements.
•     Need more circuits to handle overflows and carry.
•     Not effective for big values because to possible overflow of digits.

Decimal vs. Binary Complements:

•     Because binary complements—ones and twos—have more uses and are easier to implement, they are utilized more frequently in digital logic.
•     With binary complements, representing negative values and carrying out arithmetic operations are more effective.

conversion of numbers from one radix to another radix

We have learned and use the decimal numbering system simply because humans are born with ten fingers! Hence, the numeric system we is the decimal number system, but this system is not convenient for machines since the information is handled codified in the shape of ON or OFF bits.

This means, we have to learn the binary system in addition to the decimal system. We also will discuss the octal and hexadecimal systems because conversion to/from binary is easy and numbers in these systems are easier to read than binary numbers for humans. 

This way of codifying takes us to the necessity of knowing the positional methods of calculation which will allow us to express a number in any base where we need it.

A base of a number system or radix defines the range of values that a digit may have.

Converting numbers from one radix (base) to another is a fundamental concept in computer science and mathematics. Here's an overview of the different methods for conversion:

1. Integer Conversion:

a) Decimal to Another Radix:

1. Divide the number repeatedly by the target radix, noting the remainders from each division (in reverse order). These remainders become the digits in the new base.
2. If the number is negative, convert the absolute value first, then add a negative sign at the end.

Example: Convert 235 (decimal) to binary.

1. 235 / 2 = 117 R 1 -> 1 (least significant digit)
2. 117 / 2 = 58 R 1 -> 1
3. 58 / 2 = 29 R 0 -> 0
4. 29 / 2 = 14 R 1 -> 1
5. 14 / 2 = 7 R 0 -> 0
6. 7 / 2 = 3 R 1 -> 1
7. 3 / 2 = 1 R 1 -> 1
8. 1 / 2 = 0 R 1 -> 1

Therefore, 235 (decimal) = 11101011 (binary).

 

Conversion of decimal number to octal

Now, let's express the same decimal number 1341 in octal notation. 

Conversion of decimal number to hexadecimal

Let's express the same decimal number 1341 in hexadecimal notation. 

The easiest way to convert fixed point numbers to any base is to convert each part separately. We begin by separating the number into its integer and fractional part. The integer part is converted using the remainder method, by using a successive division of the number by the base until a zero is obtained. At each division, the reminder is kept and then the new number in the base r is obtained by reading the remainder from the lat remainder upwards.

b) Another Radix to Decimal:

1. Multiply each digit by its corresponding place value (power of the original radix) and sum the results.

2. Fractional Conversion:

a) Decimal to Another Radix:

1. Multiply the fractional part by the target radix repeatedly. Note the integer part of each product (these become digits in the new base).
2. For the remaining fractional part, repeat step 1 until a repeating pattern emerges or a desired precision is reached.

Example: Convert 0.625 (decimal) to binary.

1. 0.625 * 2 = 1.25 -> 1 (integer part)
2. 0.25 * 2 = 0.5 -> 0 (integer part)
3. 0.5 * 2 = 1.0 -> 1 (integer part)

Therefore, 0.625 (decimal) = 0.101 (binary) (repeating pattern).

b) Another Radix to Decimal:

1. Sum the values of each digit multiplied by its corresponding place value (a fraction of the original radix).

 Example: Convert 234.14 expressed in an octal notation to decimal.

Types of Machine Learning-Supervised and Unsupervised

There are three main types of machine learning, each with unique strengths and applications:

1. Supervised Learning: This type learns from labeled data, where each data point has a corresponding label indicating its category or value.

Case Study: Predicting credit card fraud. Historical transaction data is labeled as fraudulent or legitimate. The model learns to identify patterns in transactions that indicate fraud risk, helping banks prevent losses.

2. Unsupervised Learning: This type analyzes unlabeled data, seeking to uncover hidden patterns and structures.

Case Study: Customer segmentation. Unsupervised algorithms group customers based on their purchase history, demographics, or other factors, revealing distinct customer segments for targeted marketing campaigns.

3. Reinforcement Learning: This type learns through trial and error, interacting with an environment and receiving rewards for desired actions.

Case Study: Self-driving cars. Reinforcement learning agents train on simulated environments and real-world data, learning to navigate roads, avoid obstacles, and make safe decisions without explicit instructions.

Here are some additional examples:

• Supervised Learning:
  • Image recognition (identifying objects in photos)
  • Spam filtering (classifying emails as spam or not)
  • Sentiment analysis (understanding the emotional tone of text)
• Unsupervised Learning:
  • Anomaly detection (identifying unusual patterns in data)
  • Recommender systems (suggesting products or content users might like)
  • Market basket analysis (finding products frequently purchased together)
• Reinforcement Learning:
  • Playing games (learning strategies to win against an opponent)
  • Robot control (learning to perform tasks like walking or manipulating objects)
  • Resource management (optimizing how to allocate resources in a complex system)