Code_ The Hidden Language of Computer Hardware and Software - Charles Petzold [21]
The number system we use today is known as the Hindu-Arabic or Indo-Arabic. It's of Indian origin but was brought to Europe by Arab mathematicians. Of particular renown is the Persian mathematician Muhammed ibn-Musa al-Khwarizmi (from whose name we have derived the word algorithm) who wrote a book on algebra around A.D. 825 that used the Hindu system of counting. A Latin translation dates from A.D. 1120 and was influential in hastening the transition throughout Europe from Roman numerals to our present Hindu-Arabic system.
The Hindu-Arabic number system was different from previous number systems in three ways:
The Hindu-Arabic number system is said to be positional, which means that a particular digit represents a different quantity depending on where it is found in the number. Where digits appear in a number is just as significant (actually, more significant) than what the digits actually are. Both 100 and 1,000,000 have only a single 1 in them, yet we all know that a million is much larger than a hundred.
Virtually all early number systems have something that the Hindu-Arabic system does not have, and that's a special symbol for the number ten. In our number system, there's no special symbol for ten.
On the other hand, virtually all of the early number systems are missing something that the Hindu-Arabic system has, and which turns out to be much more important than a symbol for ten. And that's the zero.
Yes, the zero. The lowly zero is without a doubt one of the most important inventions in the history of numbers and mathematics. It supports positional notation because it allows differentiation of 25 from 205 and 250. The zero also eases many mathematical operations that are awkward in nonpositional systems, particularly multiplication and division.
The whole structure of Hindu-Arabic numbers is revealed in the way we pronounce them. Take 4825, for instance. We say "four thousand, eight hundred, twenty-five." That means
four thousands
eight hundreds
two tens and
five.
Or we can write the components like this:
4825 = 4000 + 800 + 20 + 5
Or breaking it down even further, we can write the number this way:
4825 = 4 x 1000 +
8 x 100 +
2 x 10 +
5 x 1
Or, using powers of ten, the number can be rewritten like this:
4825 = 4 x 103 +
8 x 102 +
2 x 101 +
5 x 100
Remember that any number to the 0 power equals 1.
Each position in a multidigit number has a particular meaning, as shown in the following diagram. The seven boxes shown here let us represent any number from 0 through 9,999,999:
Each position corresponds to a power of ten. We don't need a special symbol for ten because we set the 1 in a different position and we use the 0 as a placeholder.
What's also really nice is that fractional quantities shown as digits to the right of a decimal point follow this same pattern. The number 42,705.684 is
4 x 10,000 +
2 x 1000 +
7 x 100 +
0 x 10 +
5 x 1 +
6 ÷ 10 +
8 ÷ 100 +
4 ÷ 1000
This number can also be written without any division, like this:
4 x 10,000 +
2 x 1000 +
7 x 100 +
0 x 10 +
5 x 1 +
6 x 0.1 +
8 x 0.01 +
4 x 0.001
Or, using powers of ten, the number is
4 x 104 +
2 x 103 +
7 x 102 +
0 x 101 +
5 x 100 +
6 x 10-1 +
8 x 10-2 +
4 x 10-3
Notice how the exponents go down to zero and then become negative numbers.
We know that 3 plus 4 equals 7. Similarly, 30 plus 40 equals 70, 300 plus 400 equals 700, and 3000 plus 4000 equals 7000. This is the beauty of the Hindu-Arabic system. When you add decimal numbers of any length, you follow a procedure that breaks down the problem into steps. Each step involves nothing more complicated than adding pairs of single-digit numbers. That's why someone a long time ago forced you to memorize an addition table:
+
0
1
2
3
4
5
6
7
8
9
0
0
1
2
3
4
5
6
7
8
9
1
1
2
3
4
5
6
7
8
9
10
2
2
3
4
5
6
7
8
9
10
11
3
3
4
5
6
7
8
9
10
11
12
4
4
5
6
7
8
9
10
11
12
13
5
5
6
7