CISC 1115
Introduction to Programming Using Java
Numeral Systems
Numeral Systems
Hash Marks
- Primitive and simple
- Hard to keep track of 'how many'
- 'Hash marks in groups of five'
- Different numeral system
- How about arithmetic?
- No real zero
Roman Numeral System
- Decimal (not positional)
- No zero
- What about arithmetic?
Positional Numeral Systems
- Primitive version
- hands and fingers
- one finger, two fingers, …, nine fingers, one hand, one hand and one finger, …, nine hands and nine fingers, one hand of hands, one hand of hands and one finger, …
- What about an eight-fingered alien?
- Numerals vs value
- Numerical value of symbol is dependent upon its position in the representation
- 1232 - 1,000 + 200 + 30 + 2
- The two 2's use the same 'numeral', but represent different values.
- Numeral value also depends on how many numerals (fingers) in system
- Total value of the number is the sum of all the position's values\
- Iterate (cycle through) all the symbols in a given position; when run out of symbols...
- … add another position, placing first symbol there, and restart the cycle in the
original position
- Compact notation
- Simplifies arithmetic operations
The Decimal System
- Probably arose because we have ten fingers
- Ten symbols (one per finger): 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
- After counting to nine (9), we run out of symbols-- go to next position
The Hexadecimal System
- Sixteen symbols: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F
- After counting to fifteen (F), we run out of symbols-- go to next position
The Binary System
- Two symbols: 0, 1
- After counting to one (1), we run out of symbols-- go to next position
The Octal System
- Eight symbols: 0, 1, 2, 3, 4, 5, 6, 7
- After counting to seven (7), we run out of symbols-- go to next position
A comparison of the Numerals of the Different Bases
| Decimal | Hex | Octal | Binary | Light Switch
|
| 0 | 0 | 0 | 00000 | off off off off off
|
| 1 | 1 | 1 | 00001 | off off off off on
|
| 2 | 2 | 2 | 00010 | off off off on off
|
| 3 | 3 | 3 | 00011 | off off off on on
|
| 4 | 4 | 4 | 00100 | off off on off off
|
| 5 | 5 | 5 | 00101 | off off on off on
|
| 6 | 6 | 6 | 00110 | off off on on off
|
| 7 | 7 | 7 | 00111 | off off on on on
|
| 8 | 8 | 10 | 01000 | off on off off off
|
| 9 | 9 | 11 | 01001 | off on off off on
|
| 10 | A | 12 | 01010 | off on off on off
|
| 11 | B | 13 | 01011 | off on off on on
|
| 12 | C | 14 | 01100 | off on on off off
|
| 13 | D | 15 | 01101 | off on on off on
|
| 14 | E | 16 | 01110 | off on on on off
|
| 15 | F | 17 | 01111 | off on on on on
|
| 16 | 10 | 20 | 10000 | on off off off off
|
| 17 | 11 | 21 | 10001 | on off off off on
|
| 18 | 12 | 22 | 10010 | on off off on off
|
Indicating the Base/Radix of a Number
We indicate the radix or base we are dealing with using a subscript
(written in decimal):
112 = 310
118 = 910
1116 = 1710
Converting TO Decimal
Hexadecimal to Decimal
Each position reflects a power of 16
- First position (rightmost) is units (1's = 160)
- Next position is the "sixteenss" posiiton (parallel to the "tens" position in decimal) (16's = 161)
- Next position is "two-hundred-fifty-sixes" position (parallel to the "hundreds" position in decimal) (256's = 162)
- …
To convert:
- Multiply the (decimal equivalent of the) digit in each position by the power of 16 represented by that position
- Add the values
1AC16 =
C * 1 = 12 * 1 = 12
+ A * 16 = + 10 * 16 = + 160
+ 1 * 256 = + 1 * 256 = + 256
----
428
Binary to Decimal
Each position reflects a power of 2
- First position (rightmost) is the units position (1's = 20)
- Next position is the "twos" posiiton (parallel to the "tens" position in decimal) (2's = 21)
- Next position is "fours" position (parallel to the "hundreds" position in decimal) (4's = 22)
- …
To convert:
- Multiply the (decimal equivalent of the) digit in each position by the power of 2 represented by that position
- Add the values
1001012 =
1 * 1 = 1
0 * 2 = 0
1 * 4 = 4
0 * 8 = 0
0 * 16 = 0
1 * 32 = 32
---
37
Any Base to Decimal
Each position reflects a power of the base/radix (e.g. 2 for binary, 8 for octal, 10 for decimal, 16 for hexadecimal)
- First position (rightmost) is units (radix0)
- Next position is "radix to the power of 1" position (radix1)
- Next position is "radix to the [pwer of 2" position (radix2)
- …
To convert:
- Multiply the (decimal equivalent of the) digit in each position by the power of the radix represented by that position
- Add the values
Converting FROM Decimal
Decimal to Hexadecimal
To convert:
- Repeatedly divide by 16
- Hold on to the remainder at each step (represented in hexadecimal)
- Continue until the division produces a quotient of 0
- The hexadecimal result consists of:
- The first (topmost) remainder is the rightmost digit
- The second remainder is the next (i.e., the tens) digit
- etc.
428 / 16 R 12 = C
26 / 16 R 10 = A
1 / 16 R 1 = 1
0
1AC
Decimal to Binary
To convert:
- Repeatedly divide by 2
- Hold on to the remainder at each step (represented in hexadecimal)
- Continue until the division produces a quotient of 0
- The binary result consists of:
- The first (topmost) remainder is the rightmost digit
- The second remainder is the next (i.e., the tens) digit
- etc.
37 / 2 R 1
18 / 2 R 0
9 / 2 R 1
4 / 2 R 0
2 / 2 R 0
1 / 2 R 1
0
100101
Decimal to Any Base
To convert:
- Repeatedly divide by the radix
- Hold on to the remainder at each step (represented in the radix's digits)
- Continue until the division produces a quotient of 0
- The result consists of:
- The first (topomost) remainder is the rightmost digit
- The second remainder is the next (i.e., the tens) digit
- etc.
Converting Between Arbitrary Bases
In general, one can always convert between any two bases b1 and b2 by:
- Converting from b1 to decimal
- Converting from decimal to b2
Converting Between Hexadecimal and Binary
You will have frequently need to convert between hexadecimal and binary and there is fortunately a
simple shortcut:
- The fact that both hexadecimal and binary are powers of 2 (hex = 24; binary = 21)
make the conversion quite straightforward.
- Notice the following table:
| Hex | Binary
|
| 0 | 0000
|
| 1 | 0001
|
| 2 | 0010
|
| 3 | 0011
|
| 4 | 0100
|
| 5 | 0101
|
| 6 | 0110
|
| 7 | 0111
|
| 8 | 1000
|
| 9 | 1001
|
| A | 1010
|
| B | 1011
|
| C | 1100
|
| D | 1101
|
| E | 1110
|
| F | 1111
|
- Exactly four digits of binary represent the same numbers as exactly one digit of hex
- When we have to go to a second hex position, we also have to go to a fifth binary position.
- Each digit of hex there is represented by a corresponding 4-bit binary value.
Hex to Binary
To convert:
- Translate each hex digit into its 4-bit binary equivalent.
- The result is the binary equivalent of the hex number
1AC16 = 0001 1010 1100 = 0001101011002
Binary to Hex
To convert:
- Translate each 4-bit binary value into its single hex digit equivalent.
- The result is the binary equivalent of the hex number
- The only extra concern here is to make sure you begin at the right (the number of binary bits may not be exactly divisible by 4).
1101011002 = 1 A C = 1AC16
Binary Arithmetic
Addition of Binary numbers is similar to that of decimal:
- Starting at the rightmost position, add the two digits:
| 0 + 0 = 0
|
| 0 + 1 = 1
|
| 1 + 0 = 1
|
| 1 + 1 = 0 ; carry 1 to the next position
|
| 1 + 1 + 1 = 1 ; carry 1 to the next position
|
111
+ 110
----
1101