0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

We can see that the pattern in the one's column will continue, but what of the next column to the left? In the familiar decimal system, this is called the "ten's column"; and the unique single-digit decimal series is repeated in this column, as it was in the one's column but with a slight difference, namely, each digit appears ten times before the next digit appears. That gives us ten ones concatenated with the single-digit decimal series to produce the "teens", then ten two's concatenated with the decimal series for the "twenties", and so on.

Think for a moment about the third column that would eventually appear if we were to continue, i.e. if we were to count to 100. Here again we would see the single-digit decimal series repeated over and over in the one's column, and in the ten's column we would see each numeral of the decimal series repeated ten times before the appearance of the next numeral. In the third column the series would again be repeated indefinitely, but each numeral would appear one hundred times before the next numeral appears.

So in the one's column a "1" appeared once as the decimal series was iterated; in the ten's column a "1" appeared ten times during one series' iteration, and in the "hundred's column" it would appear one hundred times. Will it then appear one thousand times in the "thousand's column", before we see a "2"? Of course, as we all know. We say that the number of times a digit will be repeated in each "place to the left" is increased by a power of ten. A "1" in the one's column appears "ten to the zeroth power" times (10^0 = 1), that is, one time; in the ten's column it appears "ten to the first power" times (10^1 = 10) or ten times; in the hundreds column, "ten to the second power" times (10^2 = 100); and so on. This pattern, though familiar, is worth noting, for we will have reason to recall it momentarily.

The ten single-digit numerals, "0" through "9", make up the symbols of our numbering system. No matter how high we count, we still use only these ten numerals in an ordered, repetitive pattern. These are Arabic numerals, the ten symbols which were adopted for numerals as Arab culture became literate centuries ago. Other numeral systems have been used by various societies, but this one has become the modern, world-wide standard, universally adopted in literate countries today. An example of a different numeral system which is familiar to Western students is the system of Roman numerals, which uses selected Roman letters to double as numerals as well as alphabetic characters: I, II, III, IV, X, L, C, etc.

Our numbering system uses ten different numerals and is known
as a decimal system (from __decem__, the Latin word for __ten
__, cf also __deka__, Greek). Why are there ten numbers
in the series? Why not six or nine? The answer is in your hands
. . . literally. Had we no thumbs, for instance, we would undoubtedly
have developed an octal numbering system instead of a decimal
one for our ordinary counting purposes. What would such a system
look like? Assuming the same historical development and the same
conventions in ordering the series, we can surmise that it would
be similar to the decimal one:

0 1 2 3 4 5 6 7 10 11 12 13 14 15 16 17 20 21 22

So where is the eight in the series above, someone might ask. It's just where you would expect it, right after the seven, just as the ten is right after the nine in the first series. But that looks like a ten, you object. Absolutely right! In base ten a "10" is a ten, in base eight a "10" is an eight. What about base five? Right, "10" is five; and so on for numbering systems with other bases. (What about base twelve, you ask? Even there, a "10" is a twelve. With bases higher than ten, alphabetic characters are used to "fill in the gaps", i.e. A is ten and B is eleven.)

Now let's take a look at base two, what is known as the binary
system. Just as there are __ten__ numerals in the __deci
__mal system, so are there __two__ numerals in the __bi
__nary system: "0" and "1". Using the principles
we have been developing, take a pencil and paper and write down
what you think would be the first three numbers in the binary
series of numbers, i.e. starting at zero, count to two. You should
have the following series:

0 1 10

0 1 10 11

0 1 10 11 100 101

Another way to think about it is to notice that the numerals "0" and "1" alternate in the one's column but that they then come in pairs in the next column. The pattern is more clear if we show the first eight numbers in the series (and include leading zeros).

000 001 010 011 100 101 110 111

We use a procedure analogous to that which we use in base ten; but in base ten it is automatic, in base two we have to do it deliberately. Recall that each column "goes up" by a factor of two, and then move through the number from right to left in the following fashion: for 10011, we have one 1, one 2, no 4's, no 8's, and one 16:

`1 0 0 1 1`

`16 + 0 + 0 + 2 + 1 = 19`

`1 1 0 0 1 1 0`

`64 + 32 + 0 + 0 + 4 + 2 + 0 = 102`

1 ? ? ? ? 16's 8's 4's 2's 1's

1 1 ? ? ? 16's 8's 4's 2's 1's

1 1 ? ? 1 16's 8's 4's 2's 1's

1 1 0 0 1 16's 8's 4's 2's 1's

(Answers are at the end of this paper.)Binary --> Decimal Decimal --> Binary a. 1010100 ________ h. 5 _____________ b. 1010 ________ i. 9 _____________ c. 110011 ________ j. 19 _____________ d. 101111 ________ k. 22 _____________ e. 10001 ________ l. 31 _____________ f. 1111 ________ m. 32 _____________ g. 10000111 ________ n. 46 _____________ o. 67 _____________

0 0 1 1 + 0 + 1 + 0 + 1 ---- ---- ---- ---- 0 1 1 10

1 0 + 1 0 0 -------- 1 1 0

1 1 + 1 0 1 -------- 1 0 0 0

1 0 1 1 + 1 1 1 ---------- 1 0 0 1 0

(Answers are at the end of this paper.)a. 1 0 1 b. 1 1 1 c. 1 0 1 0 + 1 0 + 1 0 + 1 1 0 ----- ----- ------- d. 1 0 1 0 e. 1 0 1 0 f. 1 0 1 1 + 1 0 1 0 + 1 1 1 1 + 1 0 1 --------- --------- ------- g. 1 1 1 1 0 h. 1 0 0 1 1 0 i. 1 0 1 1 1 0 + 1 1 1 0 + 1 1 0 1 + 1 0 1 --------- ----------- -----------

With decimal subtraction we borrow one of the units from the next place to the left and increase the digit in the minuend by a factor of ten (the base), i.e. add ten to it. The same is done with binary subtraction. (Obviously we have to have a value in the next place to the left from which to borrow; otherwise we're trying to subtract a smaller number from a larger. We don't know anything about negative numbers, nor will we concern ourselves with them in this introduction to binary numbers.) If we are trying to subtract one from zero, we have to borrow one unit from the place to the left and increase the minuend by a factor of two (since we're now in base two). Let's look at a simple example, subtracting one from two; and let's put in a leading zero as a reminder.

1 0 - 0 1 -----

0 10 - 0 1 ------- 1

0 10 - 0 1 ------- 0 1

1 0 0 0 - 0 1 0 1 ---------

0 10 0 0 - 0 1 0 1 ------------

0 1 10 0 - 0 1 0 1 ------------

0 1 1 10 - 0 1 0 1 -----------

0 1 1 10 - 0 1 0 1 ----------- 0 0 1 1

a. 1 0 1 b. 1 0 0 1 c. 1 0 0 0 - 1 1 - 1 0 0 - 1 0 1 ----- ------- ------- d. 1 1 1 1 1 e. 1 0 1 0 1 f. 1 1 0 0 1 1 - 1 0 1 0 1 - 1 0 1 0 - 1 1 0 0 ----------- --------- ----------- g. 1 1 1 0 1 1 1 h. 1 1 1 0 1 0 1 - 1 1 0 1 1 - 1 1 0 1 1 ------------- -------------

(Answers are at the end of this paper.)

Exercise 1 Exercise 2 Exercise 3a. 84 h. 101 a. 111 a. 10 b. 10 i. 1001 b. 1001 b. 101 c. 51 j. 10011 c. 10000 c. 11 d. 47 k. 10110 d. 10100 d. 1010 e. 17 l. 11111 e. 11001 e. 1011 f. 15 m. 100000 f. 10000 f. 100111 g. 135 n. 101110 g. 101100 g. 1011100 o. 1000011 h. 110011 h. 1011010 i. 111001

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