0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
Observe the preceding series of numbers. These are the commonplace
numbers which we use almost every day for counting. Indeed, they
are so familiar that you will be tempted not to look at them closely
or think about them seriously; but do look at them and observe
how they proceed, how they change. Think about the pattern they
follow and how the series continues beyond what is listed above.
Note that while we usually start counting at one, here we start
at zero. Why? Because the pattern is more clear. We start at
zero and "count to nine", producing the unique single-digit
decimal series which is repeated in the "one's column"
over and over again, indefinitely.
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
This is, in fact, an octal system of numbering which is used today
in computer science. Note its similarity to the familiar decimal
system. The unique single-digit series is again repeated indefinitely
in the one's column; only this time the series has eight numbers
instead of ten. The decimal system is sometimes called base
ten, and the octal system, base eight.
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
Just as a "10" is a ten in base ten and a five in base
five, in base two a "10" is a two. And two follows
one, just as expected. We had ten numbers in the unique single-digit
series in base ten, and eight numbers in the unique single-digit
series in base eight. Likewise, we have two numbers in the unique
single-digit series in base two. If the similarities continue,
we can expect this series to be repeated indefinitely in the one's
column. Indeed, with the addition of "three" the series
becomes
0
1
10
11
We see that the pattern in the one's column continues: the two
numerals that make up the unique single-digit binary series are
repeated. What about the second "column" or "place",
just to the left of the one's column? Here we see two and three
represented by what look like ten and eleven to our "decimal
eyes". Since these correspond in a way to the decimal teens,
can we expect the next parts of this series to be the "twenties"?
If so, we could expect "20" and "21" to be
the next numbers in this series; but there is no numeral "2"
in base two. We have just seen that two is represented in base
two by "10". Do we then replace the "2" in
"20" and "21" by "10" to get "100"
and "101"? Yes! And we have
0
1
10
11
100
101
Can you guess the next number in the series? One way to think
about it is to continue the analogy in which we said that "10"
and "11" correspond to the decimal teens and "100"
and "101" correspond to the decimal twenties. We could
then expect the "thirties" to follow, and we would need
a three concatenated with a zero. Since a three in binary is "11",
we would have "110" as the next number in our series,
which is correct.
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
Recall the pattern that we discovered earlier in decimal numbers
in which each column to the left goes up by a factor of ten.
Similarly, in base two each column goes up by a factor of two,
giving us (from the right) the one's column (2^0 = 1), the two's
column (2^1 = 2), the four's column (2^2 = 4), the eight's column
(2^3 = 8), and so on. Even the short series above shows the "0's"
and "1's" alternating in the one's column but coming
in pairs in the "two's column". In the "four's
column" they even seem to be in groups of four, which is,
indeed, the case. And in the eight's column they come in groups
of eight. This is one of those interesting and handy quirks often
found in numbers which has made mathematics so fascinating to
many. It should prove helpful to you while learning the binary
number system.
Binary to Decimal Conversions
Take the binary number, 10011. Does it represent the quantity
ten thousand eleven? No, of course not. We have already learned
enough to know not to look on binary numbers with our "decimal
vision". This number would represent that quantity only
in the decimal system, not in the binary system, where it represents
nineteen. But how are we to figure out that it is nineteen?
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
Now try 1100110: no 1's, one 2, one 4, no 8's, no 16's, one 32,
and one 64:
1 1 0 0 1 1 0
64 + 32 + 0 + 0 + 4 + 2 + 0 = 102
Decimal to Binary Conversions
Simple conversions of a decimal number to binary representation
require knowledge of the first eight powers of two, viz, 1, 2,
4, 8, 16, 32, 64, 128. Let us convert the decimal number twenty-five
(25) to binary. We have to ask first what is the largest power of
two that will go into our number. Scanning the list of the first
eight powers, we see that sixteen is the largest that will go
into twenty-five. This means that the number twenty-five has
one sixteen in it, so we will want a "1" in the 16's
column of our binary number. It also means that the 16's column
will be the left-most column in our binary number. So at this
point we have:
1 ? ? ? ?
16's 8's 4's 2's 1's
Next we subtract sixteen (the highest power so far) from our original
number, giving a nine. We repeat the process: what is the highest
power of two that goes into nine? Answer: eight. So twenty-five
"has an eight in it", and we put a "1" in
the 8's column:
1 1 ? ? ?
16's 8's 4's 2's 1's
Now we subtract the eight from the nine, leaving one; and since
one is the highest power of two which will go into one, we put
a "1" in the 1's column:
1 1 ? ? 1
16's 8's 4's 2's 1's
The quantity twenty-five is thus broken down into three quantities:
sixteen, eight, and one (16 + 8 + 1 = 25); and we have "1's"
in the corresponding columns of our binary representation of this
quantity. What of the 4's and the 2's columns? Surely you can
now guess that they will contain "0's":
1 1 0 0 1
16's 8's 4's 2's 1's
Do the following as an exercise to familiarize yourself with these
methods of converting between binary and decimal representation.
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 _____________
(Answers are at the end of this paper.)
Binary Arithmetic
Arithmetic operations are possible on binary numbers just as they
are on decimal numbers. In fact the procedures are quite similar
in both systems. Multiplication and division are not really difficult,
but unfamiliarity with the binary numbers causes enough difficulty
that we will introduce only addition and subtraction, which are
quite easy.
Addition
Addition is almost as easy as "one and one is two".
Indeed, there are only "zero and zero is zero", "one
and zero is one", and "one and one is two":
0 0 1 1
+ 0 + 1 + 0 + 1
---- ---- ---- ----
0 1 1 10
Notice that adding two single-digit "1's" produces a
two-digit result, which means that we have to "carry"
into the next place, just as with our familiar decimal addition.
Let's take a simple example, two plus four equals six:
1 0
+ 1 0 0
--------
1 1 0
Starting at the right column, zero plus zero equals zero, one
plus zero equals one, and zero plus one equals one. Too easy;
we didn't even have to carry. Let's try three plus five equals eight:
1 1
+ 1 0 1
--------
1 0 0 0
Starting at the right, one plus one equals two: bring down the
zero, carry the one; (now in the 2's column) one plus zero equals
one, plus the one carried equals two: bring down the zero, carry
the one; (now in the 4's column, the leading or implied) zero
plus one equals one, plus the one carried equals two: bring down
the zero, carry the one (into the 8's column). Here is a final
example to show how to handle a "three", eleven plus
seven equals eighteen:
1 0 1 1
+ 1 1 1
----------
1 0 0 1 0
Starting at the right, one plus one equals two: bring down the
zero, carry the one; (2's column) one plus one equals two, plus
one carried equals three ("11"): bring down the one,
carry the one; (4's column) zero plus one equals one, plus one
carried equals two: bring down the zero, carry the one; (8's
column) one plus zero equals one, plus one carried equals two:
bring down the zero, carry the one (into the 16's column). Now
try the following as exercises.
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
--------- ----------- -----------
(Answers are at the end of this paper.)
Subtraction
Having learned about adding binary numbers, you can well imagine
that the process of subtracting one binary number from another
is virtually the same as with decimal numbers except for conditions
under which borrowing is done. Subtracting zero from zero or
one from one leaves zero, and zero from one leaves one, as with
decimal numbers. It is only when we have a zero in the minuend
and a one in the subtrahend that we have to think in binary terms,
i.e. when we try to subtract "one from zero".
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
-----
Since we cannot subtract the one from the zero, we have to borrow
one from the next place and add two to the zero. We can now subtract
one from two and bring down the result.
0 10
- 0 1
-------
1
We must remember that when we borrow one, we have to decrease
the quantity in the column from which we borrow by one. We borrowed
one from the 2's column, which left us with a zero in the 2's
column. For completeness we say that we now subtract zero from
zero giving zero, so that our result is
0 10
- 0 1
-------
0 1
What if there is a zero instead of a one in the place to the left?
Well, what if there were a zero in the place to the left in decimal
subtraction? We would just continue to borrow from each place
to the left until we reached some non-zero quantity. The same
is done with binary subtraction. Take eight minus five:
1 0 0 0
- 0 1 0 1
---------
Since we cannot borrow one from the 2's column or the 4's column,
we have to go all the way to the 8's column, borrowing "an
eight" and adding it to the 4's:
0 10 0 0
- 0 1 0 1
------------
Now we can borrow from the 4's column, borrowing "a four"
and adding it to the 2's. Notice that we have decreased the 4's
column by one (four) and increased the 2's by two (2's):
0 1 10 0
- 0 1 0 1
------------
Then in the same way borrow "a two" from the 2's column
and add it to the 1's:
0 1 1 10
- 0 1 0 1
-----------
Now we're ready for some simple subtraction. From the right column:
one from two leaves one, zero from one leaves one, one from one
leaves zero, and zero from zero leaves zero:
0 1 1 10
- 0 1 0 1
-----------
0 0 1 1
And, indeed, our result is three, "11" in binary. You
should now be ready for some exercises in subtraction.
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 3
a. 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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