Database System Concepts
Database System Concepts
7th Edition
ISBN: 9780078022159
Author: Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Publisher: McGraw-Hill Education
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Computational Systems
The computation systems can be defined as the systems that are capable of solving a problem that includes calculations either mathematical or logical, and are able to produce the result as an output. For example, a simple calculator that is given a set o…
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## Problem 2: Multiplying Two Base 2 Numbers using an Algorithm in Python or R ### Task Description: Write a program in a language of your choice (preferably Python or R) to multiply two positive integers using the following algorithm. ### Algorithm to Perform Multiplication: ```plaintext procedure multiply(a, b: positive integers) {the binary expansions of a and b are (a_n-1 a_n-2 ... a1 a0)₂ and (b_n-1 b_n-2 ... b1 b0)₂, respectively} for j := 0 to n - 1 if b_j = 1 then c_j := a shifted j places else c_j := 0 {c₀, c₁, ..., c_n-1 are the partial products} p := 0 for j := 0 to n - 1 p := p + c_j return p {p is the value of ab} ``` ### Explanation: 1. **Binary Expansion**: The numbers `a` and `b` are represented in their binary formats. 2. **Iterate through Digits of b**: - For each bit `b_j` of `b`: - If `b_j` is 1, then calculate the partial product `c_j` by shifting `a` left by `j` positions. - If `b_j` is 0, then the partial product `c_j` is 0. 3. **Sum Partial Products**: The final product `p` is obtained by summing all the partial products `c_j`. 4. **Return the Result**: The result `p` is the multiplication of `a` and `b`. ### Diagrammatic Representation: Figure 2: Multiplying two base 2 numbers. This algorithm effectively leverages the properties of binary numbers to perform the multiplication by bitwise shifts and accumulation. By following this algorithm, you can implement a program to perform binary multiplication in any programming language of your choice, such as Python or R.
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Transcribed Image Text:## Problem 2: Multiplying Two Base 2 Numbers using an Algorithm in Python or R ### Task Description: Write a program in a language of your choice (preferably Python or R) to multiply two positive integers using the following algorithm. ### Algorithm to Perform Multiplication: ```plaintext procedure multiply(a, b: positive integers) {the binary expansions of a and b are (a_n-1 a_n-2 ... a1 a0)₂ and (b_n-1 b_n-2 ... b1 b0)₂, respectively} for j := 0 to n - 1 if b_j = 1 then c_j := a shifted j places else c_j := 0 {c₀, c₁, ..., c_n-1 are the partial products} p := 0 for j := 0 to n - 1 p := p + c_j return p {p is the value of ab} ``` ### Explanation: 1. **Binary Expansion**: The numbers `a` and `b` are represented in their binary formats. 2. **Iterate through Digits of b**: - For each bit `b_j` of `b`: - If `b_j` is 1, then calculate the partial product `c_j` by shifting `a` left by `j` positions. - If `b_j` is 0, then the partial product `c_j` is 0. 3. **Sum Partial Products**: The final product `p` is obtained by summing all the partial products `c_j`. 4. **Return the Result**: The result `p` is the multiplication of `a` and `b`. ### Diagrammatic Representation: Figure 2: Multiplying two base 2 numbers. This algorithm effectively leverages the properties of binary numbers to perform the multiplication by bitwise shifts and accumulation. By following this algorithm, you can implement a program to perform binary multiplication in any programming language of your choice, such as Python or R.
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