Repeated subtraction method binary options
The greatest common divisor of two non-zero integers is a great example to illustrate the power of loops. Everyone learns about the concept of a greatest common divisor when faced with a fraction that is not repeated subtraction method binary options reduced form. So the factor of 2 can be canceled from both the numerator and the denominator. Euclid the mathematician from classic times and author of Elements is credited with having come up with a clever algorithm for how to compute the greatest common divisor efficiently.
It is common in mathematics to list functions as one or more cases. The way you read this is as follows:. To gain some appreciation of how the definition always allows you to compute the greatest common divisor, it is worthwhile to try it out for a couple of numbers where you know the greatest common divisor.
For example, we already know that the greatest common divisor of 10 and 15 is 5. Notice that in the example above, the first number 10 was smaller than the second 15and the first transformation just swapped the numbers, so the larger number was first. Thereafter the first number is always larger. The way GCD is formulated above is, indeed, the most clever way repeated subtraction method binary options calculate the greatest common divisor.
Yet the way we learn about the greatest common divisor in elementary school at least at first is to learn how to factor the numbers a and b, often in a brute force way. So for example, when calculating the greatest common divisor of 10 and 15, we can immediately see it, because we know that both of these numbers are divisible by 5 e.
So the greatest common divisor is 5. This sort of trial process can take place in a loop, where we start at 1 and end at min a, b. We know that none of the values after the minimum can divide both a and b in integer divisionbecause no larger number can divide a smaller positive number. The smaller number would be the non-zero remainder.
So this gives you a relatively straightforward way of calculating the greatest common divisor. While simple, it is not repeated subtraction method binary options the most efficient way of determining the GCD. If you think about what is going on, this loop could run a significant number of times.
For example, if you were calculating the GCD two very large numbers, say, one billion 1,, and two billion 2,, it is painfully evident that you would consider a large number of values a billion, in fact before obtaining the candidate GCD, which we know is 1,, Repeated subtraction method binary options code above goes though all integers 2 through min a, b.
That is not generally necessary when the GCD is greater than 1, even with a brute-force mindset. Repeated subtraction method binary options subtraction method also attributable to Euclid to compute the Greatest Common Divisor works as follows:.
In this loop, we are repeatedly subtracting b from awhich we know we can do, because a started out as being larger than b. At the end of loop a is reduced to either. The loop on line 9 is similar to the loop in line 5. As discussed above, if a and b end up as the same number, that is the GCD. On the other hand, the first GCD algorithm example repeated subtraction method binary options how remainders may need to be to be calculated over and over.
The outer loop in this version keeps this up until a and b are reduced to equal values. At this point the inner loops would make no further changes, and the common value is the GCD. As an exercise to the reader, you may want to consider adding some Console. WriteLine statements to print the values of a and b within each loop, and after both loops have executed. It will allow you to see in visual terms how repeated subtraction method binary options method does repeated subtraction method binary options work.
There are several ways to code the shorter Euclidean algorithm at the beginning of this GCD section. It is a repetitive pattern, and a loop can be used. There are two parameters, a and bto repeated subtraction method binary options gcd, and they can be successively changed, suggesting a loop.
What is the continuation condition? You stop when b is 0, so you continue while b! One extra variable needs to be introduced to make this double change work. The simplest is to repeated subtraction method binary options a variable r for the remainder. Check and see for yourself that you need an extra variable like r:. It is saying the result of the function with one set of parameters is equal to calling the function with another set of parameters. If we put this into a C function definition, it would mean repeated subtraction method binary options instructions for the function say to call itself.
This is a broadly useful technique called recursionwhere a function calls itself inside its definition. The recursive version of the gcd function refers to itself by calling itself.
Though this seems circular, you can see from the examples that it works very well. The important point is that the calls to the same function are not completely the same: Successive calls have smaller second numbers, and the second number eventually reaches 0, and in that case there is a direct final answer.
Hence the function is not really circular. This recursive version is a much more direct translation of the original mathematical algorithm than the looping version! The general idea of recursion is for a function to call itself with simpler parameters, until a simple enough place is reached, where the answer can be repeated subtraction method binary options calculated.
Motivation for This Book 1. Downloading and Reading Options 1. Computer Science, Broadly 1. Chapter Review Questions 2. C Data and Operations 2. A Sample C Program 2. Editing, Compiling, and Running with Xamarin Studio 2. Variables and Assignment 2.
Syntax Template Typography 2. Strings, Part I 2. Writing to the Console 2. C Program Structure 2. Combining Input and Output 2. String Special Cases 2. Value Types and Conversions 2. Learning to Solve Problems 2. Chapter Review Questions 3. Defining Functions of your Own 3. A First Function Definition 3. Multiple Function Definitions 3. Multiple Function Parameters 3. Returned Function Values 3. Writer and Consumer of Functions 3.
Not using Return Values 3. Static Function Summary 3. Chapter Review Questions 4. Basic String Operations 4.
Some Instance Methods and the Length Property 4. A Creative Problem Solution 4. Chapter Review Questions 5. Simple if Statements 5. More Conditional Expressions 5. Multiple Tests and if - else Statements 5. Compound Boolean Expressions 5. Nested if Statements 5. Chapter Review Questions 6. While-Statements with Sequences 6. Interactive while Repeated subtraction method binary options 6.
More String Methods 6. Greatest Common Divisor 6. Number Guessing Game Lab 6. Chapter Review Questions 7. Chapter Review Questions 8. Examples With for Statements 8. Chapter Review Questions 9. Files, Paths, and Directories 9. Files As Streams 9.
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