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binary numbers

summary

    Binary numbers are the natural internal representation for digital computers.

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binary numbers

    Binary numbers are the natural internal representation for most digital computers. For the first few decades there were some computers that were primarily decimal and many processors still include some binary coded decimal (BCD) operations.

    Most modern computers have two different internal sizes for binary integers, a standard int and a long. Many also have a short binary integer, which is typically a single byte long.

    Most modern computers allow for both signed and unsigned integers. Signed integers include zero, positive integers, and negative integers. Unsigned integers include only zero and positive integers (excluding negative integers). Unsigned integers allow for approximately double the number of integers and are commonly used for addressing hardware memory (as well as for software pointers).

    Most programming languages tie their integer choices to these underlying hardware characteristics.

C

PL/I

assembly language instructions

number systems

    Binary is a number system using only ones and zeros (or two states).

    Decimal is a number system based on ten digits (including zero).

    Hexadecimal is a number system based on sixteen digits (including zero).

    Octal is a number system based on eight digits (including zero).

    Duodecimal is a number system based on twelve digits (including zero).

binary	octal	decimal	duodecimal	hexadecimal
0	0	0	0	0
1	1	1	1	1
10	2	2	2	2
11	3	3	3	3
100	4	4	4	4
101	5	5	5	5
110	6	6	6	6
111	7	7	7	7
1000	10	8	8	8
1001	11	9	9	9
1010	12	10	A	A
1011	13	11	B	B
1100	14	12	10	C
1101	15	13	11	D
1110	16	14	12	E
1111	17	15	13	F
10000	20	16	14	10
10001	21	17	15	11
10010	22	18	16	12
10011	23	19	17	13
10100	24	20	18	14
10101	25	21	19	15
10110	26	22	1A	16
10111	27	23	1B	17
11000	30	24	20	18

integer representations

    Sign-magnitude is the simplest method for representing signed binary numbers. One bit (by universal convention, the highest order or leftmost bit) is the sign bit, indicating positive or negative, and the remaining bits are the absolute value of the binary integer. Sign-magnitude is simple for representing binary numbers, but has the drawbacks of two different zeros and much more complicates (and therefore, slower) hardware for performing addition, subtraction, and any binary integer operations other than complement (which only requires a sign bit change).

    In one’s complement representation, positive numbers are represented in the “normal” manner (same as unsigned integers with a zero sign bit), while negative numbers are represented by complementing all of the bits of the absolute value of the number. Numbers are negated by complementing all bits. Addition of two integers is peformed by treating the numbers as unsigned integers (ignoring sign bit), with a carry out of the leftmost bit position being added to the least significant bit (technically, the carry bit is always added to the least significant bit, but when it is zero, the add has no effect). The ripple effect of adding the carry bit can almost double the time to do an addition. And there are still two zeros, a positive zero (all zero bits) and a negative zero (all one bits).

    In two’s complement representation, positive numbers are represented in the “normal” manner (same as unsigned integers with a zero sign bit), while negative numbers are represented by complementing all of the bits of the absolute value of the number and adding one. Negation of a negative number in two’s complement representation is accomplished by complementing all of the bits and adding one. Addition is performed by adding the two numbers as unsigned integers and ignoring the carry. Two’s complement has the further advantage that there is only one zero (all zero bits). Two’s complement representation does result in one more negative number (all one bits) than positive numbers.

    Two’s complement is used in just about every binary computer ever made. Most processors have one more negative number than positive numbers. Some processors use the “extra” neagtive number (all one bits) as a special indicator, depicting invalid results, not a number (NaN), or other special codes.

    In unsigned representation, only positive numbers are represented. Instead of the high order bit being interpretted as the sign of the integer, the high order bit is part of the number. An unsigned number has one power of two greater range than a signed number (any representation) of the same number of bits.

bit pattern	sign-mag.	one’s comp.	two’s comp	unsigned
000	0	0	0	0
001	1	1	1	1
010	2	2	2	2
011	3	3	3	3
100	-0	-3	-4	4
101	-1	-2	-3	5
110	-2	-1	-2	6
111	-3	-0	-1	7

See also Data Representation in Assembly Language

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