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Decimal to Unsigned Binary Conversion

Converting from decimal to unsigned binary is a little more complicated, but it still isn't too difficult. Once again, there is a well-defined process. To begin with, it is helpful to remember the powers of 2 that correspond to each bit position in the binary numbering system. These were presented in Figure 4 for the powers of 20 up to 27. What we need to do is separate the decimal value into its power of 2 components. The easiest way to begin is to find the largest power of 2

that is less than or equal to our decimal value. For example if we were converting 7510 to binary, the largest power of 2 less than or equal to 7510 is 26 = 64.

The next step is to place a 1 in the location corresponding to that power of 2 to indicate that this power of 2 is a component of our original decimal value. Next, subtract this first power of 2 from the original decimal value. In our example, that would give us 7510 6410 = 1110. If the result is not equal to zero, go back to the first step where we found the largest power of 2 less than or equal to the new decimal value. In the case of our example, we would be looking for the largest power of 2 less than or equal to 1110 which would be 23 = 8. When the result of the subtraction reaches zero, and it eventually will, then the conversion is complete. Simply place 0's in the bit positions that do not contain 1's. Figure 6 illustrates this process using a flowchart.

If you get all of the way to bit position zero and still have a non-zero result, then one of two things has happened. Either there was an error in one of your subtractions or you did not start off with a large enough number of bits. Remember that a fixed number of bits, n, can only represent an integer value up to 2n 1. For example, if you are trying to convert 31210 to unsigned binary, eight bits will not be enough because the highest value eight bits can represent is 28 1 = 25510. Nine bits, however, will work because its maximum unsigned value is 29 1 = 51110.

Figure 1.6Decimal to Unsigned Binary Conversion Flow Chart

Example1Convert the decimal value 13310 to an 8 bit unsigned binary number.

Solution Since 13310 is less than 28 1 = 255, 8 bits will be sufficient for this conversion. Using Figure 4, we see that the largest power of 2 less than or equal to 13310 is 27 = 128. Therefore, we place a 1 in bit position 7 and subtract 128 from 133.

 

             

Bit position

 

 

133 128 = 5

Our new decimal value is 5. Since this is a non-zero value, our next step is to find the largest power of 2 less than or equal to 5. That would be 22 = 4. So we place a 1 in the bit position 2 and subtract 4 from 5.

 

           

Bit position

 

 

5 4 = 1

Our new decimal value is 1, so find the largest power of 2 less than or equal to 1. That would be 20 = 1. So we place a 1 in the bit position 0 and subtract 1 from 1.



 

         

Bit position

 

 

1 1 = 0

Since the result of our last subtraction is 0, the conversion is complete. Place zeros in the empty bit positions.

Bit position

 

And the result is:

13310 = 100001012

Example 2 Convert the decimal number 99 to its binary equivalent:

Divide 99 by 2. The quotient is 49 with a remainder of 1; indicate the 1 on the right.

Divide 49 by 2(the quotient from the previous division). The quotient is 24 with a remainder of 1, indicated on the right.

Divide 24 by 2. The quotient is 12 with a remainder of 0, as indicated .

Divide 12 by 2. The quotient is 6 with a remainder of 0, as indicated.

Divide 6 by 2. The quotient is 3 with a remainder of 0, as indicated.

Divide 3 by 2. The quotient is 1 with a remainder of 1, as indicated.

Divide 1 by 2. The quotient is 0 with a remainder of 1, as indicated. Since the quotient is 0, stop here.

The base 2 number is the numeric remainder reading from the last division to the first

Binary Addition

Consider the following binary addition problems and note where it is necessary to carry the 1:


Date: 2015-12-17; view: 717


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