
VI Binary Math As mentioned previously in this chapter, a complex system such as a computer is really a collection of simpler devices. In the everyday human world we work with numbers. We work with numbers that the mathematicians call base 10 math because most of us have ten digits (five on each hand). Think for a moment about how we count. Starting with zero, we move to one, two, three, etc. After reaching the number nine, we have exhausted the possible amount of numbers if we included the value zero. That means we have used 10 different values, so base 10. What are we really saying when we say 10? Think of the numbers arranged in columns. The first column when using 2 numbers will be the unit column; the second column will be the numbers column. Breaking this value down means we have zero value in the unit column, and, we have a single 10 in the tens column. If the value was 70, that is the same as saying there is a grand total, of 7 tens in the tens column. If the value was 88, it is the same expression as 8 tens and 8 units. In other words, we moved over to the left one column. After 99 the next number is 100. Again, we have moved over one column, leaving no value in the tens column and no value in the units column. When you look at a computer closely, it really is not very intelligent. In fact, it is very dumb; it can only count starting from zero, up to one. That's it. Only two possible values exist at the base level of a computer. Zero and one or off and on. Zero represents off and one represents on. So what happens if you need to count to a number bigger than one? You borrow a trick from base 10, and move over a column. Now the second column can represent a value of two. Now with 2+1, we can count up to three. Need to count higher than 3? Add a column. Make that column the value of four. Now we can count 4+2+1 or a grand total, of 7. Need to see 8? Great. Add a column; call it the value of 8. Now you have 8+4+2+1 or 15 in base 10. By now, you can probably guess that to go to 16, you add a column for the value of sixteen. And the next column would be 32, and, so on. If you go to 8 columns the column most left becomes 128 with the seventh being 64. The sixth column is the value of 32, with the fifth column holding the value of 16. The fourth column becomes the value of 8, the third column being a placeholder for the value of 4 with the second column having the value of 2, and the first column the value of 1. If there is a zero in any column, it is off, and it is not counted. Notice that the zero has to be there, or the values in the column would all be skewed and the math wouldn't come out right. If all the columns are on, the maximum value is equal to 128+64+32+16+8+4+2+1 or 255, as shown in Table 3.
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