Floating Point Arithmetic
|
|
| Binary numbers can also have decimal points, and to show you how, we will once again begin with decimal numbers. For decimal numbers with decimal points, the standard way to represent the digits to the right of the decimal point is to continue the powers of ten in descending order starting with -1 where 10-1=1/10th = 0.1. |
|
| That means that the number 6.5342 has 5 increments of 10-1 (tenths), 3 increments of 10-2 (hundredths), 4 increments of 10-3 (thousandths), and 2 increments of 10-4 (ten-thousandths). |
|
| The table below shows this graphically. |
|
|
|
|
| Therefore, our example has the decimal value 6*1 + 5*0.1 + 3*0.01 + 4*0.001 + 2*0.0001 = 6.5342. |
|
| Binary representation of real numbers works the same way except that each position represents a power of two, not a power of ten. To convert 10.01101 to decimal for example, use descending negative powers of two to the right of the decimal point. |
|
| Exponent |
2 |
1 |
0 |
-1 |
-2 |
-3 |
-4 |
-5 |
| Position Value |
4 |
2 |
1 |
0.5 |
0.25 |
0.125 |
0.0625 |
0.03125 |
| Sample Values |
0 |
1 |
0 |
0 |
1 |
1 |
0 |
1 |
|
|
| Therefore, our example has the decimal value 0*4 + 1*2 + 0*1+0*0.5 + 1*0.25 + 1*0.125 + 0*0.0625 + 1*0.03125 = 2.40625. This means that the method of conversion is the same for real numbers as it is for integer values; we’ve simply added positions representing negative powers of two. |
|
| Computers, however, use a form of binary more like scientific notation to represent floating-point or real numbers. For example, with scientific notation we can represent the large value 342,370,000 as 3.4237 x 108. |
|
| This representation consists of a decimal component or mantissa of 3.4237 with an exponent of 8. Both the mantissa and the exponent are signed values allowing for negative numbers and for negative exponents respectively. |
|
| Binary works the same way using 1’s and 0’s for the digits of the mantissa and exponent and using 2 as the multiplier that moves the decimal point left or right. |
|
| For example, the binary number 100101101.010110 would be represented as:
1.00101101010110 * 28 |
