They are presented with examples below in the first part of the article. In the second part this article explains the very basic math behind each of them.
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Knowing it may help you remember any of the algorithms if you suddenly forget them. I strongly suggest you take a notepad and a pen and perform the operations along with me to better remember the math.
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Here are the four algorithms with examples that you can find on the web. Converting decimal integer to binary To convert integer to how to create a binary, start with the integer in question and divide it by 2 keeping notice of the quotient and the remainder.
Continue dividing the quotient by 2 until you get a quotient of zero. Then just write out the remainders in the reverse order. Here is an example of such conversion using the integer So, 12 in decimal system is represented as in binary.
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Converting decimal fraction to binary To convert fraction to binary, start with the fraction in question and multiply it by 2 keeping notice of the resulting integer and fractional part. Continue multiplying by 2 until you get a resulting fractional part equal to zero.
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Then just write out the integer parts from the results of each multiplication. Here is an example of such conversion using the fraction 0.
So, 0. Converting binary integer to decimal To convert binary integer to decimal, start from the left.
Take your current total, multiply it by two and add the current digit. Continue until there are no more digits left.
The simple math behind decimal-binary conversion algorithms
Here is an example of such conversion using the fraction Converting fraction integer to decimal To convert binary fraction to decimal, start from the right with the total of 0. Take your current total, add the current digit and divide the result by 2. There you have 4 simple algorithms that will allow you to convert binary numbers to decimal and back. Base-q expansion of a number The key to understanding why those algorithms work is a base-q expansion of a number.
An integer number in any numeric system can be represented in the following form: where, N is integer x is the digit 0 through 9 for base system, 0 and 1 for base-2 system q is the base value 10 for base system, 2 for base-2 system Throughout this article this form is referred to as base q expansion of the number N, or simply base q expansion.
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The first thing we need to notice here is all summands except for the last one will be even numbers, because they all are multiples of two. Here we have number 12 which is even, so x0 is zero.
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Since all summands from x1 to xN are multiples of two, we can factor out 2 to single out x1. We can write out the polynomial inside the parenthesis as a separate statement: Here, applying the same logic from above we can see that x1 is equal to 0. We can continue factoring 2 until the quotient is zero.
So the number 12 in binary using the algorithm described above is represented as Remember that we started with the idea to show why the algorithm that involves diving by 2 works. We can also put the calculations for those four steps into one representation like this Make sure you understand how we arrive at that representation as we will need it when exploring how the algorithm of converting from binary to decimal works.
The first thing to notice here is that negative powers of 2 give us fractions with the denominator of 2 with positive powers. Just like we did with integers, we can also put the calculations for those three steps into one representation like this: Again, it is important that you fully grasp this representation as we will need it when exploring binary to decimal conversion.
Why not all fractions can be finitely represented in binary The fact that some fractions represented finitely in decimal system cannot be represented finitely in binary system comes as surprise to many developers. But this is exactly the confusion that lies in the root of the seemingly weird outcome of adding 0.
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So what determines whether a fraction can be finitely represented in a numeric system? Well, for a number to be finitely how to create a binary the denominator in a fraction should be a power of the system base.
DEL To convert the number for C, 67, into a binary number: Remember how binary numbers are read bottom to top, first position and Default Value to top position and Default Value, with each of the eight character positions assigned a unique number value?
Here we also will use the base-q expansion form of the number. The base-q expansion form of the number is the key here as well.
Click the File option from the File Operations toolbar. It will also ask for selection of the ASN. The Add button next to the ASN.