Integer Encoding (数字编码)

Abstract

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• Integer encoding is the process of representing decimal integers in the form of zeros and ones, supporting both positive and negative integer
• There are 3 integer encoding variants; Sign-and-Magnitude (原码), 1’s Complement (反码) and 2’s Complement (补码), computer uses the 2’s Complement to store integer, because it solves the flaws of Sign-and-Magnitude (原码) and 1’s Complement (反码)
• Below is a 4-bits integer encoding under the 3 variants

Decimal	Signed Magnitude	One’s Complement	Two’s Complement
$+7$	$0111$	$0111$	$0111$
$+6$	$0110$	$0110$	$0110$
$+5$	$0101$	$0101$	$0101$
$+4$	$0100$	$0100$	$0100$
$+3$	$0011$	$0011$	$0011$
$+2$	$0010$	$0010$	$0010$
$+1$	$0001$	$0001$	$0001$
$+0$	$0000$	$0000$	$0000$
$-0$	$1000$	$1111$	$0000$
$-1$	$1001$	$1110$	$1111$
$-2$	$1010$	$1101$	$1110$
$-3$	$1011$	$1100$	$1101$
$-4$	$1100$	$1011$	$1100$
$-5$	$1101$	$1010$	$1011$
$-6$	$1110$	$1001$	$1010$
$-7$	$1111$	$1000$	$1001$
$-8$	-	-	$1000$

Important

For the three integer encoding variants, the encodings for positive integer are exactly the same.

Important

$1010_2$ is a positive binary number, $-(1010_2)$ is a negative binary number. All the bits are Magnitude Bit, no Sign Bit.

$1010_{sm}$ is a negative binary number encoded using Sign-and-Magnitude (原码).

$1010_{1s}$ is a negative binary number encoded using 1’s Complement (反码).

$1010_{2s}$ is a negative binary number encoded using 2’s Complement (补码).

Examples

The 8-bit 2’s complement representation of $1010_2$ is $00001010_{2s}$.

The 8-bit 2’s complement representation of $-(11111111_2)$ is $00000001_{2s}$.

Sign Bit

• $0$ denotes positive
• $1$ denotes negative
• Usually the first bit from left, the most significant bit (MSB)

Magnitude Bit

• Represent the magnitude (absolute value) of the integer in standard binary representation

Sign-and-Magnitude (原码)

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Positive Decimal	Signed Magnitude	Negative Decimal	Signed Magnitude
$+0$	$0000$	$-0$	$1000$
$+1$	$0001$	$-1$	$1001$
$+2$	$0010$	$-2$	$1010$
$+3$	$0011$	$-3$	$1011$
$+4$	$0100$	$-4$	$1100$
$+5$	$0101$	$-5$	$1101$
$+6$	$0110$	$-6$	$1110$
$+7$	$0111$	$-7$	$1111$

• Sign-and-magnitude is a method for representing both positive and negative integers using a Sign Bit and magnitude bits. The most significant bit (MSB) is reserved as the sign bit

Unable to performance subtraction with Adder

The Adder is unable to perform subtraction on sign-and-magnitude encoded integers, an explicit set of Logic Gates for substation is needed. This can be avoided using 1’s Complement (反码) or 2’s Complement (补码).

1’s Complement (反码)

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Decimal	Signed Magnitude	One’s Complement
$+0$	$0000$	$0000$
$-0$	$1000$	$1111$
$-1$	$1001$	$1110$
$-2$	$1010$	$1101$
$-3$	$1011$	$1100$
$-4$	$1100$	$1011$
$-5$	$1101$	$1010$
$-6$	$1110$	$1001$
$-7$	$1111$	$1000$

• Basically same as the Sign-and-Magnitude (原码), but for negative decimal integers, all magnitude bits are inverted

Reuse adder for subtraction

We can reuse Adder Logic Gates to perform subtraction without the need for dedicated subtraction logic gates.

Obtaining 1s-complement

We can obtain the two’s complement, also known as the negative representation of a positive number, using the formula: $-x=2^n-x-1$.

Mathematically wrong definition of 0

We have +0 (when all bits are 0) & -0(when all bits are 1) under this encoding, this can be solved with 2’s Complement (补码).

Trick to convert from 1's complement to decimal value

The decimal value equals the negative decimal value of Sign Bit + the positive sum of the magnitude bits.

The weight of the sign bit is $-2^{n-1}-1$, where $n$ is the total number of bits the binary number has.

1s Complement in Addition and Subtraction

• For addition, we can simply perform standard binary addition
• For subtraction, we need to find the 1’s complement of the number we want to subtract, and then perform standard binary addition

What if there is a carry-out from the most significant bit (MSB)?

Add 1 to the result.

2’s Complement (补码)

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Decimal	One’s Complement	Two’s Complement
$+0$	$0000$	$0000$
$-0$	$1111$	$0000$
$-1$	$1110$	$1111$
$-2$	$1101$	$1110$
$-3$	$1100$	$1101$
$-4$	$1011$	$1100$
$-5$	$1010$	$1011$
$-6$	$1001$	$1010$
$-7$	$1000$	$1001$
$-8$	-	$1000$

• This builds on top of 1’s Complement (反码), the only difference being to add $1$ to the 1’s complement (反码) integer encoding for negative numbers

Mathematically correct definition of Zero

The integer encoding for both $+0$ and $-0$ is $0000$, thus eliminating two different integer encodings for $0$.

Obtaining 2s-complement

We can obtain the two’s complement, also known as the negative representation of a positive number, using the formula: $-x=2^n-x$.

Trick to convert from 2's complement to decimal value

The decimal value equals the negative decimal value of Sign Bit + the positive sum of the magnitude bits.

The weight of the sign bit is $-2^{n-1}$, where $n$ is the total number of bits the binary number has.

Trick to obtain 2's complement

Start from the rightmost bit and move left. Write down each bit as it is until you encounter the first ‘1’. From that point onwards, invert all the remaining bits.

Complement on fraction

Given 0101.01, the 2s complement is 1010.11.

2s Complement in Addition and Subtraction

• For addition, we can simply perform standard binary addition
• For subtraction, we need to find the 2’s complement of the number we want to subtract, and then perform standard binary addition

What if there is a carry-out from the most significant bit (MSB)?

Simply ignore it.

A more hardware friendly way of detecting integer overflow

Comparing the carry-in and carry-out of the most significant bit. Integer overflow occurs if they are different.

Integer Overflow

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• Occurs when a value is outside the range that the datatype can hold

How to check for integer vverflow

If the Sign Bit of the two operand is the same, and the sign bit of the result is different. Then we have an Integer (整数) overflow.

How can the sum of two positive integer result in a negative number, and how the sum of two negative integer result in a positive number?

The sum of a negative and positive integer guaranteed no integer overflow.

Excess Representation

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• Another way to represent positive and negative numbers with binary
• A fixed bias (offset, excess) is subtracted from the represented value to get the actual value
• The range of value it can represent is $-B$ to $(2^n-1)-B$ where B is the bias

Excess 3 with 4 bits

The bias is 3 (in binary: 0011). The number represented by binary 0000 would be $0-3=-3$, 0001 would represent −2, and so on.

Important

To get an evenly distributed number of positive and negative numbers, the excess we use for an n-bit number is $2^{n-1}$.

Given an 8-bit number in excess-100 representation. How many negative values are there in this number representation?

$100$

Reference

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• Sign-and-Magnitude Addition and Subtraction
• Hello-Algo