Before diving into signed and unsigned representations, it's essential to grasp the basics of binary numbers. Binary is a base-2 numeral system, meaning it only uses two digits: 0 and 1. Each digit in a binary number is called a bit. In a 4-bit binary number, there are four positions, each representing a power of 2, starting from the right (which is (2^0)).
Example of a 4-bit binary number:
1 0 1 1
Calculating its decimal equivalent:
[
1 \times 2^3 + 0 \times 2^2 + 1 \times 2^1 + 1 \times 2^0 = 8 + 0 + 2 + 1 = 11
]
So, 1011 in binary is 11 in decimal.
Definition: Unsigned binary numbers represent only non-negative integers. All bits are used to denote the magnitude of the number.
Range for 4-bit Unsigned Binary:
- Minimum Value:
0000which is0in decimal. - Maximum Value:
1111which is (2^4 - 1 = 15) in decimal.
Representation:
0000 = 0
0001 = 1
0010 = 2
...
1111 = 15
Key Points:
- No sign bit; all bits contribute to the magnitude.
- Simple and straightforward for non-negative numbers.
Signed binary numbers can represent both positive and negative integers. There are different methods to represent signed numbers, but the most common are:
- Sign-Magnitude Representation
- One's Complement Representation
- Two's Complement Representation
We'll explore each method using 4-bit binary numbers.
Definition:
In sign-magnitude, the most significant bit (MSB) represents the sign of the number (0 for positive, 1 for negative), and the remaining bits represent the magnitude.
Range for 4-bit Sign-Magnitude:
- Minimum Value:
1111which is-7in decimal. - Maximum Value:
0111which is+7in decimal.
Representation:
0000 = +0
0001 = +1
0010 = +2
...
0111 = +7
1000 = -0
1001 = -1
1010 = -2
...
1111 = -7
Issues:
- Two representations for zero (
+0and-0), which can complicate computations. - Arithmetic operations are more complex due to the separate handling of the sign bit.
Definition: In one's complement, negative numbers are represented by inverting all the bits of the positive number (including the sign bit).
Range for 4-bit One's Complement:
- Minimum Value:
1000which is-7in decimal. - Maximum Value:
0111which is+7in decimal.
Representation:
0000 = +0
0001 = +1
0010 = +2
...
0111 = +7
1000 = -7
1001 = -6
1010 = -5
...
1111 = -0
Issues:
- Still has two representations for zero (
+0and-0). - End-around carry is needed for addition, making arithmetic operations slightly more complex.
Definition:
Two's complement is the most common method for representing signed numbers. A negative number is represented by inverting all the bits of the positive number and then adding 1 to the least significant bit (LSB).
Range for 4-bit Two's Complement:
- Minimum Value:
1000which is-8in decimal. - Maximum Value:
0111which is+7in decimal.
Representation:
0000 = 0
0001 = +1
0010 = +2
...
0111 = +7
1000 = -8
1001 = -7
1010 = -6
...
1111 = -1
Advantages:
- Only one representation for zero.
- Simplifies arithmetic operations; no need for special handling of the sign bit.
- Wider range compared to sign-magnitude and one's complement.
Example Calculation:
To find the two's complement of 5 (0101):
- Invert the bits:
0101→1010 - Add
1:1010+1=1011which represents-5.
| Representation | Minimum Value | Maximum Value |
|---|---|---|
| Unsigned | 0 | 15 |
| Sign-Magnitude | -7 | +7 |
| One's Complement | -7 | +7 |
| Two's Complement | -8 | +7 |
- Unsigned Binary: Ideal for scenarios where only non-negative numbers are needed, such as counting or indexing.
- Signed Binary (Two's Complement): Preferred in most computing systems for its efficiency in handling both positive and negative numbers, especially in arithmetic operations.
-
Confusing Sign-Magnitude with Two's Complement:
- It's crucial to distinguish between these methods as they handle negative numbers differently.
- Two's complement avoids the issue of dual zero representations and simplifies subtraction by turning it into addition.
-
Misinterpreting the Range:
- Always verify the range based on the number of bits and the representation method.
- For example, in 4-bit two's complement, the range is from
-8to+7, not-7to+7.
-
Overflow in Arithmetic Operations:
- When performing arithmetic with signed numbers, be cautious of overflow, which occurs when the result exceeds the representable range.
- For instance, adding
7(0111) and1(0001) in 4-bit two's complement results in8, which is1000and represents-8, leading to an overflow.
In summary, understanding signed and unsigned binary numbers is fundamental in computer science and digital electronics. Here's a concise breakdown using 4-bit numbers:
-
Unsigned Binary:
- Range:
0to15 - Example:
1011represents11
- Range:
-
Signed Binary:
- Sign-Magnitude:
- Range:
-7to+7 - Example:
1011represents-3
- Range:
- One's Complement:
- Range:
-7to+7 - Example:
1011represents-4
- Range:
- Two's Complement:
- Range:
-8to+7 - Example:
1011represents-5
- Range:
- Sign-Magnitude:
Key Takeaways:
- Unsigned is for non-negative numbers.
- Signed methods allow representation of negative numbers, with Two's Complement being the most efficient and widely used due to its simplicity in arithmetic operations and single zero representation.
By mastering these concepts, you'll have a solid foundation for understanding how computers represent and manipulate numerical data.
Two's complement is a method used in computing to represent signed integers. It is widely used because it simplifies the design of arithmetic circuits and allows for easy addition and subtraction of both positive and negative numbers.
Let's go through the process of representing -8 in 4-bit two's complement step by step.
First, find the binary representation of the positive counterpart of the number you want to represent. In this case, the positive number is 8.
However, in 4-bit binary, the maximum positive number we can represent is 7 (0111). The number 8 requires 5 bits to be represented (10000). This indicates that 8 is outside the range of 4-bit two's complement representation.
For an n-bit two's complement system:
- The range of representable numbers is from (-2^{(n-1)}) to (2^{(n-1)} - 1).
For 4 bits:
- Minimum value: (-2^{(4-1)} = -8)
- Maximum value: (2^{(4-1)} - 1 = 7)
So, the range is from -8 to 7.
Since -8 is the minimum value in 4-bit two's complement, its representation is straightforward.
-
Start with the binary representation of the positive number:
8in binary is1000(but as noted, this is actually-8in two's complement).
-
Invert all the bits (find the one's complement):
- Inverting
1000gives0111.
- Inverting
-
Add 1 to the least significant bit (LSB) to get the two's complement:
0111+1=1000.
So, 1000 represents -8 in 4-bit two's complement.
To verify that 1000 indeed represents -8, let's convert it back to decimal.
-
Identify the sign bit:
- The most significant bit (MSB) is
1, indicating a negative number.
- The most significant bit (MSB) is
-
Find the two's complement to get the positive counterpart:
- Invert the bits:
1000→0111. - Add
1:0111+1=1000.
- Invert the bits:
-
Convert the positive binary number to decimal:
1000in binary is8in decimal.
Since the original number was negative, 1000 represents -8.
- 4-bit Two's Complement Range:
-8to7 - Representation of -8:
1000
-
Misunderstanding the Range:
- It's crucial to recognize that in 4-bit two's complement,
-8is the smallest number and is represented as1000. - Attempting to represent
+8in 4 bits is not possible because it exceeds the maximum positive value (7).
- It's crucial to recognize that in 4-bit two's complement,
-
Incorrect Inversion:
- When finding the two's complement, ensure you invert all bits correctly and add
1to the LSB. - Skipping the addition of
1leads to the one's complement, which is not the correct representation for two's complement.
- When finding the two's complement, ensure you invert all bits correctly and add
-
Sign Bit Confusion:
- Always check the MSB to determine if the number is positive (
0) or negative (1). - In
1000, the MSB is1, confirming it's a negative number.
- Always check the MSB to determine if the number is positive (
In 4-bit two's complement representation:
- The binary number
1000represents-8. - This is because:
- The MSB
1indicates a negative number. - Applying the two's complement process to
1000yields8, confirming that1000stands for-8.
- The MSB
Understanding this representation is fundamental for working with signed integers in computing, ensuring accurate arithmetic operations and data interpretation.
Numbers in computers are represented using fixed-point or floating-point notation, depending on precision and range requirements.
Fixed-point representation stores numbers with a fixed number of digits before and after the decimal (or binary) point.
Consider an 8-bit representation with 4 integer bits and 4 fractional bits (Q4.4 format):
| Binary | Decimal Equivalent |
|---|---|
| 1010.1100 | -5.25 (in 2's complement) |
| 0011.1010 | 3.625 |
- Pros: Faster operations, simpler hardware.
- Cons: Limited range and precision.
Floating-point representation allows a wide range of values by using a mantissa (significand), exponent, and sign bit.
[ N = \text{Sign} \times \text{Mantissa} \times \text{Base}^{\text{Exponent}} ]
| Sign (1-bit) | Exponent (8-bit) | Mantissa (23-bit) |
|---|---|---|
| 0 (positive) | 10000011 (Exponent = 131, i.e., 4) | 01000000000000000000000 (1.5 in mantissa) |
For +12.375, we convert it to binary:
- Binary:
1100.011 - Normalized Form:
1.100011 × 2³ - IEEE 754 Encoding:
- Sign Bit:
0(positive) - Exponent:
3 + 127 = 130→10000010 - Mantissa:
10001100000000000000000
- Sign Bit:
- Final Representation:
0 10000010 10001100000000000000000
Normalization ensures the mantissa has only one non-zero digit before the decimal.
0.0001011 × 2⁴→ Not normalized1.011 × 2⁰→ Normalized
This allows for more efficient storage and calculations.
A: Floating-point allows a wider range of values and greater precision, crucial for large-scale calculations.
A: 01111111011111111111111111111111, which is approximately 3.4 × 10³⁸.
A: When the exponent is all zeros, the number is stored in a denormalized form, which helps represent very small values.