The numbers you see in everyday life are "base 10" numbers. In other words,
we have 10 different number symbols (0, 1, 2, 3, 4, 5, 6, 7, 8, and 9),
that are the basis of every other number you can write.
Every number you see in base 10 is some combination of these 10
symbols.
The binary number system is the set of "base 2" numbers. In binary
numbers, every numerical value can be written using only two symbols, 0 and
1.
We as humans don't really care, since binary numbers look really strange to
us. (For example, "100" in binary is really what we think of as "4"). However,
we should care, because binary numbers don't look strange to computers.
In fact, computers actually operate in binary. We'll get to "why" later, but if
you want to know computers inside and out, you need to know how binary numbers
work.
What do I have to do to
understand them?
If you have an understanding of how base 10 numbers work, figuring out base 2
numbers isn't much of a stretch. To do this, we will need to know some basic
exponent rules.
What am I going to do in
this module?
Basically, you're going to learn how different number systems work. We'll
discuss binary, base 10, and some other systems of numbers.
Most importantly, you'll learn how to convert binary numbers into base 10
numbers, and base 10 numbers into binary numbers. In other words, you're going
to be able to translate human number symbols into computer number
symbols.
Basically, the base ten number system works like this: certain human beings
came up with ten distict number symbols (0-9), and started
counting....
0, 1, 2, 3, 4, 5, 6, 7, 8, 9....Oops!
When they got to the last symbol, they needed to figure out how to write the
number for the value that comes after 9. So, they started over at 0, and tacked
on a number to the left that helped keep track the number of times we got past
"9". So, that's where the tens place comes in...
...1, 2, 3, 4, 5, 6,7 , 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, (hey! we got to 9 again, so we have to bump
that tens place up one), 20, 21, 22......and so on,
and so on....
Once the tens place gets past 9, you have to
add another number that counts up the number of times you get to one hundred.
97, 98,
99, 100,
101, 102, ....
So, this is where you get the ones place, the tens place, the hundreds
place, the thousands place, and all the other places you tack on as your numbers
get bigger and bigger.
Take the number 1,234:
1
2
3
4
1000s
place
100s Place
10s place
1s place
You could
rewrite 1,234 as:
1*1000 + 2*100+ 3*10 + 4*1
Why would we want to?
Note that these "places" are powers of "10" . You know, 100=1, 101=10,
102=100, 103=1000, 104=10000, etc, etc.
(remember: any number other than 0 to the power
"0" equals one!)
Of course! We're in a base 10 system. All the places are powers of
10.
Question
1: Why do you think that we came up with a base ten number system? Why not
base eight, or base sixteen? What's so special about the number
ten?
_______________________________________________
II.
and number systems OTHER THAN base
10:
Okay, so what's so special about the number 10? That's for you to figure out,
but here's a hint:
Let's just say that The Simpsons might
just have a very different number system. In fact, they might have
a base 8 system. (Maybe this is why Bart has such trouble
with math...)
Yeah, the answer to why we operate in a base 10 system probably has to do
with the number of digits we have (and I'm not talking about numbers here).
Let's suppose that we operated in the world of the Simpsons,
a base eight system.
Question 2: How many numeric symbols would we have to use? What
would they be?
___________________________________________
Maybe one way to get the idea of different systems of numbers
would be to think about your car's odometer.
As the miles increase (in base 10), whenever any number rotates past "9", the
odometer "turns over". In other words, the "9" returns to "0" and the number
immediately to the left increases by one.
Now, think of an odometer that turns over after only eight
digits, 0-7.
The number on the Simpson's car odometer might look something
like this:
Simpsons' Odometer (base eight)
Our odometer (base ten)
0
0
1
1
2
2
3
3
4
4
5
5
6
6
7
7
10 <-It turns over here!
8
11
9
12
10
13
11
14
12
15
13
16
14
17
15
20 <-It turns over
again!
16
21
17
22
18
And so on, and so on....
The same pattern happens when you get bigger and bigger
numbers:
Simpsons' Odometer (base eight)
Our odometer (base ten)
...
...
74
60
75
61
76
62
77
63
100 <- Here's where the first two
numbers turn over to "0"
64
101
65
102
66
103
67
And so on, and so on.....
So, you can see that numbers in a different number system mean very different
things.Maybe those outrageous prices in the Kwikee-Mart aren't so
bad!.
After all, when Apu charges Homer $10 for a burrito, that's only $8 of our
money! (well, that's still pretty bad, I guess.)
Question
3:
Fill in the blanks! What would the Simpsons' odometer look like
here?
Simpsons' Odometer (base eight)
Our odometer (base ten)
...
...
174
204
?
205
?
206
?
207
?
208
?
209
?
210
?
211
?
212
?
213
?
214
?
215
?
216
The last section was to get you used to counting in a different system of
numbers.
Let's take a look at how base eight compares to base
ten.
Remember when we looked at the number 1,234
in base ten? We noticed that you could rewrite 1,234 as:
1*1000 + 2*100+ 3*10 +
4*1
2
4
1000s
place
100s place
10s place
1s place
In base eight, however,1,234
has an entirely different meaning.
In base eight, you have
1
2
3
4
512s place
64s place
8s place
1s place
This turns over after eight groups of eight groups of eight! This place
tells us how many groups of 512 there are.
This turns over after eight groups of eight! This place tells us how
many groups of 64 there are.
Since we turn over after "7", this place tells us how many groups of
8 we have!
This stays the same.
Or: 83=512
Or: 82=64
Or: 81=8
Or: 80=1
Important!
The only difference between base ten
numbers and any numbers of any other base is that the "places" are just powers
of different numbers.
So, if we want to translate "1, 234" (base 8) into base 10,
we could rewrite 1,234 as:
1*512 + 2*64 + 3*8 +1*1
=512 + 128 + 24 +1
=665
Here's another example:
Translate 943 (base eight) into
its base ten equivalent.
Step1:
multiply down
9
4
3
64s place
8s place
1s place
9*64=576
4*8=32
3*1=3
Step
two: add across:
576 + 32 + 3 = 611.
576+32+3=611
in base ten.
Question
4:
Using the above as an example, translate these base eight numbers into base
10 numbers!
a) 62 in base eight equals _________ in base
ten.
b) 146 in base eight equals _________ in
base ten.
c) 2405 in base eight equals _________ in
base ten.
d) 24134 in base eight equals _________ in
base ten.
!
Okay, enough of this base eight stuff. What we're really interested in is
base 2 numbers, also called binary numbers. But, if you understood how base ten
and base eight numbers are connected, binary is EASY.
Question
5:
Why do computers use binary numbers? (In other words, why do we care about
learing binary?)
To answer this, search through the "links" that you can get to via the menu
to your left.
_______________________________________
So, hopefully, you've answered the above question one way or
another and now understand the importance of knowing binary. It sort of makes
sense, right?. As smart as we sometimes think computers are, they really only
understand two things: on and off.
Okay, let's get a feel for binary numbers using a binary
odometer.
Remember, we're in base 2, which means we only have two numbers: 0 and
1.
Once an odometer number reaches "1", it next turns over to "0", and the
number immediately to its left increases by one (and if that's already "1", THAT
number turns over to "0", and the next number increases by one, and so
on...).
Binary Odometer (base eight)
Our odometer (base ten)
0
0
1
1
10<-It turns over
here!
2
11
3
100 <-It turned over
again!
4
101
5
110
6
111
7
1000
8
1001
9
1010
10
1011
11
1100
12
1101
13
1110
14
1111
15
10000
16
10001
17
10010
18
Question
6:
Take a look at the binary numbers 1, 10, 100, 1000, and 10000, along with
their base 10 equivalents.
What pattern do you notice? Why does this seem like a pretty natural pattern
for BInary numbers?
_______________________________________
_______________________________________
Question
7:
Fill in the blanks of this binary odometer:
Binary Odometer (base two)
Our odometer (base ten)
...
...
11001100
204
?
205
?
206
?
207
?
208
?
209
?
210
?
211
?
212
?
213
?
214
?
215
?
216
More Number Bases!!
Alright, this is the name of the game. We have to be able to take a
binary number, like 110111, and convert
it to a base ten number that we can understand.
It's the same process as converting a base eight number to base ten,
except that the "places" are different.
Here, 110111 means a different value in different number
systems:
Base
1
1
0
1
1
1
10
105=100000
104=10000
103=1000
102=100
101=10
100=1
8
85=32768
84=4096
83=512
82=64
81=8
80=1
2
25=32
24=16
23=8
22=4
21=2
20=1
Places for the different bases!
Of course, 110111 just means "one hundred and ten thousand, one hundred
and eleven" in base ten.
In base eight, we can use the "multiply down, add
across" method to convert it to base ten:
1*32768 + 1*4096 + 0*512 + 1*64 + 1*8 + 1*1=
36937 in base ten!
In base 2, we use the same method, except that we will
mulitply by different numbers, the powers of 2!
Step1: multiply down
1
1
0
1
1
1
32s place
16s place
8s place
4s place
2s place
1s place
1*32=32
1*16=16
0*8=0
1*4=4
1*2=2
1*1=1
Step two: add across
32 + 16 +
0 + 4
+ 2 + 1 =55 in base ten.
Hey!
Look!
Actually, the fact that we're using 1's and 0's makes the
base2/base 10 calculations really easy! Multiplying any number by
"1" is always that number, and multiplying by "0" is always
"0".
So, if we want to convert 100011 (base 2) to base ten, we know that we
have ones in the 1's place, the 2's
place, and the 32's place.
100011 = 32 + 0 + 0 + 0 + 2 + 1 =35 in base 10
Question
8:
Convert these base 2 numbers to base 10!
a) 1 is equal to ____________ in base 10.
b) 1101 is equal to ____________ in base 10.
c) 111110 is equal to ____________ in base 10.
d) 10101010101 is equal to ____________ in base 10.
Okay,the conversion from binary to base 10 isn't so bad. You just have
to keep track of the places, which are just the powers of
2!
256
128
64
32
16
8
4
2
1
As you get more places, they just keep
doubling!
Just as review, if you have a binary number like
110101100
you just match up the ones and zeroes with the corresponing power of 2,
and add!
1
1
0
1
0
1
1
0
0
256
128
64
32
16
8
4
2
1
256
128
32
8
4
256 + 128 + 32+ 8 + 4 = 428 in base 10!
Now, to convert
from base two to base 10 is kind of like reversing this process.
The idea is to first split up your base 10 number into a sum of
powers of two, (like 256 + 128 + 32+ 8
+ 4 = 428)
Let's take an example:
Suppose we want to convert the number 39 into binary.
We want to split it up into a sum of powers of two (1, 2, 4, 8, 16, 32, 64, 128, 256, etc, etc,
):
Step 1: Look for the largest power of two that
is less (or equal to) than 39:
In this case, 32
works!
So, 32 + 7 = 39
Step 2: Look at what's left over! (7 in this
case). Look for the largest power of two that is less than (or equal to)
7:
In this case,
4 works! 7= 4
+ 3, so.....
32 + 4 + 3 =
39
Step 3: Look at what's left over! (3 in this
case). Look for the largest power of two that is less than (or equal to)
3:
In this case,
2 works! 3= 2 + 1, so.....
32 + 4 + 2 + 1 =
39
They're all powers of 2! We're done with this
part.
Step 4: Reverse the process!
Unlike the problem we did at the beginning, we're going from the bottom
up this time!
?
?
?
?
?
?
?
?
?
256
128
64
32
16
8
4
2
1
32
0
0
4
2
1
** We know:
32 + 4 +
2 + 1 =
39 in base 10!
** So, we have a 32, a 4, a 2,
and a 1!
** Which means our base 2
number is: 100111*
1
0
0
1
1
1
32
16
8
4
2
1
32
4
2
1
* Don't forget the zeroes!
Actually writing a chart like the one above will help you keep the powers
of two straight.
Lets' try another example, and then you can try it. (It's actually
really easy once you get the hang of it!)
Convert 123 to
binary:
STEP 1:
123
=
64
+
59
STEP 2:
123
=
64
+
32
+
27
STEP 3:
123
=
64
+
32
+
16
+
11
STEP 4:
123
=
64
+
32
+
16
+
8
+
3
STEP 5:
123
=
64
+
32
+
16
+
8
+
2
+
1
So, 123
= 64 +
32 + 16 + 8 +
2 + 1
STEP 6:
1
1
1
1
0
1
1
64
32
16
8
4
2
1
64
32
16
8
0
2
1
So, 123 is equal to 1111011 in base 2!
Question
9: Convert these decimal numbers to binary!
Question
1: Why do you think that we came up with a base ten number system? Why not
base eight, or base sixteen? What's so special about the number
ten?
!
