Dr. Shvetha Soundararajan
shvetha@gmu.edu
Material adapted from Chapter 2 of Computer Science Illuminated, 7th Edition
NUMBERS
Natural Numbers
Zero and any number obtained by repeatedly adding one to it
Integers
A natural number, a negative number
Examples: 100, 0, 45645, 32
Examples: 249, 0, –45645, –32
Negative Numbers
Rational Numbers
A value less than 0, with a – sign
Examples: –24, –1, –45645, –32
An integer or the quotient of two integers
Examples: –249, –1, 0, 3/7, –2/5
NUMBER REPRESENTATION
There is only one set of integers, no matter what you call them.
|
5 = V = 1012 = ||||
Each base gives a different perspective on how to name these numeric values.
NUMBER REPRESENTATION
§ Decimal
|
0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,…
§ Binary
0,1,10,11,100,101,110,111,1000,1001,1010,1011,1100,1101,1110,1
111,10000,…
§ Hexadecimal
0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F,10,11,12,13,14,15,16,17,…
§ Roman
I,II,III,IV,V,VI,VII,VIII,IX,X,XI,XII,XIII,XIV,XV,XVI,XVII,XVIII,XIX,XX,XXI,...
|
| |||, ||||
| ||, ||||
| ||||,
§ Tallies: |, ||, |||, ||||, ||||, |||| |, ||||
| ||||,
| …
||||
DIFFERENT BASES
§
A base defines how many symbols we have
for numbers, and also defines the value of
each column.
§ Those symbols are used
"odometer-style" to count upwards.
DIFFERENT BASES
§ We count in decimal (base 10), thanks to our ten fingers.
→ Base 10: ten symbols:
0, 1, 2, 3, 4, 5, 6, 7, 8, 9.
§ Computers store everything in binary (base 2), thanks to the simplicity of having just
2 states.
→ Base 2: two symbols:
0, 1.
DIFFERENT BASES
§ We use hexadecimal (base 16) as a convenient shorthand for longer binary numbers.
→ Base 16: 16 symbols:
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F.
§ Base N: use any N symbols. Example:
0,1,…, (n-1).
COUNTING UP
§ We count up though the symbols of
our base, and when we run out, we
'clock over' to the next column,
starting over in the earlier columns.
§ It just happens a lot faster in binary…
§ Notice that binary clocks over into 4 new columns
exactly when hexadecimal clocks over once. Not a
coincidence!
reason: 16 = 24.
§ Think of counting in each base as using an odometer
that has a different number of symbols on its wheels.
Base 10: Base 2: Base 16: Base 8:
0
0
0
0
1
1
1
1
2
10
2
2
3
11
3
3
4
100
4
4
5
101
5
5
6
110
6
6
7
111
7
7
8
1000
8
10
9
1001
9
11
10
1010
A
12
11
1011
B
13
12
1100
C
14
13
1101
D
15
14
1110
E
16
15
1111
F
17
16 10000
10
20
17 10001
11
21
18 10010
12
22
19 10011
13
23
…
…
…
…
POSITIONAL NOTATION
How many ones are there in 642?
600 + 40 + 2 ?
Or is it
384 + 32 + 2 ?
Or maybe…
1536 + 64 + 2 ?
642 is 600 + 40 + 2 in BASE 10
The base of a number determines the number of different digit symbols (numerals) and the
values of digit positions.
POSITIONAL NOTATION
Continuing with our example…
642 in base 10 positional notation is:
6 x 102 = 6 x 100 = 600
+ 4 x 101 = 4 x 10 = 40
+ 2 x 10º = 2 x 1 = 2
= 642 in base 10
This number is in
base 10
The power indicates
the position of
the number
POSITIONAL NOTATION
R is the base
of the number
As a formula:
dn * Rn - 1 + dn - 1 * Rn - 2 + ... + d2 * R1 + d1 * R0
n is the number of
digits in the number
642 is 6 * 102 + 4 * 10 + 2 * 1
d is the digit in the
ith position
in the number
POSITIONAL NOTATION
What if 642 has the base of 13?
6 x 132 = 6 x 169 = 1014
+ 4 x 131 = 4 x 13 = 52
+ 2 x 13º = 2 x 1 = 2
= 1068 in base 10
642 in base 13 is equivalent to 1068 in base 10
CONVERTING BINARY TO DECIMAL
What is the decimal equivalent of the binary number 1101110?
1 x 26
+ 1 x 25
+ 0 x 24
+ 1 x 23
+ 1 x 22
+ 1 x 21
+ 0 x 2º
=
=
=
=
=
=
=
1 x 64
1 x 32
0 x 16
1x8
1x4
1x2
0x1
= 64
= 32
=0
=8
=4
=2
=0
= 110 in base 10
CONVERTING OCTAL TO DECIMAL
What is the decimal equivalent of the octal number 642?
6 x 82 = 6 x 64 = 384
+ 4 x 81 = 4 x 8 = 32
+ 2 x 8º = 2 x 1 = 2
= 418 in base 10
CONVERTING HEXADECIMAL TO DECIMAL
What is the decimal equivalent of the hexadecimal number DEF?
D x 162 = 13 x 256 = 3328
+ E x 161 = 14 x 16 = 224
+ F x 16º = 15 x 1
= 15
= 3567 in base 10
Remember, the digit symbols in base 16 are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F
ARITHMETIC IN BINARY
Remember that there are only 2 digit symbols in binary, 0 and 1
1 + 1 is 0 with a carry
1011111
1010111
+1 0 0 1 0 1 1
10100010
Carry Values
SUBTRACTING BINARY NUMBERS
Remember borrowing? Apply that concept here:
012
02
1010111
- 111011
0011100
CONVERTING BINARY TO OCTAL
• Mark groups of three (from right)
• Convert each group
10101011
10 101 011
2 5 3
10101011 is 253 in base 8
CONVERTING BINARY TO HEXADECIMAL
• Mark groups of four (from right)
• Convert each group
10101011
1010 1011
A
B
10101011 is AB in base 16
CONVERTING DECIMAL TO OTHER BASES
Algorithm for converting number in base 10 to other bases:
While the quotient is not zero:
Divide the decimal number by the new base
Make the remainder the next digit to the left in the answer
Replace the original decimal number with the quotient
CONVERTING DECIMAL TO BINARY
Note that the rightmost digit in a binary number tells us if it is even or odd. Thus, if we divide a number by 2, the
remainder tells us the rightmost binary digit.
We can convert a number (say, 13410) to binary by dividing by two until we reach zero. The digits in
reverse order represent the resulting number:
134 / 2 = 67
rem 0
67 / 2 = 33
rem 1
33 / 2 = 16
rem 1
16 / 2 = 8
rem 0
8/2
=4
rem 0
4/2
=2
rem 0
2/2
=1
rem 0
1/2
=0
rem 1
§
The resulting number is thus 100001102. We can check ourselves by converting back to base 10.
CONVERTING DECIMAL TO HEXADECIMAL
What is 3567 (base 10) in base 16?
222
16 3567
32
36
32
47
32
15
Answer: D
EF
13
16 222
16
62
48
14
0
16 13
0
13