Question

If you changed the length of the bit strings being used to represent integers in binary from 4 bits to 6 bits, what change would be made in the value of the largest integer you could represent? What if you were using two's complement notation?

Short answer

Switching from 4 to 6 bits increases the max unsigned integer from 15 to 63; with two's complement, it changes the range from -8 to 7, to -32 to 31.

Step-by-step solution

Understanding Bit Length Representation

Binary numbers are a way to represent integers using bits, with each bit representing a power of 2. The more bits you have, the more numbers you can represent. A binary number with n bits can represent values from 0 to \(2^n - 1\) if using unsigned integer representation.

Calculating Maximum Value with 4 Bits

Using 4 bits, the maximum integer you can represent is given by \(2^4 - 1\), which equals 15. This is because each bit represents a power of 2 (i.e., 1, 2, 4, 8), and the sum of these when all bits are set to 1 is 15.

Calculating Maximum Value with 6 Bits

Using 6 bits, the calculation for the maximum integer is \(2^6 - 1\), which equals 63. Adding two more bits allows for more combinations, increasing the possible number of represented integers.

Understanding Two's Complement Representation

Two's complement notation is used for representing signed integers. For an n-bit two's complement number, the range is from \(-2^{(n-1)}\) to \(2^{(n-1)}-1\). This allows the representation of both positive and negative numbers.

Calculating Range with 4-bit Two's Complement

With 4 bits in two's complement, the integer range is from \(-2^{3}\) to \(2^{3} - 1\), which means the range is from -8 to 7.

Calculating Range with 6-bit Two's Complement

With 6 bits in two's complement, the integer range extends to \(-2^{5}\) to \(2^{5} - 1\), which means the range is from -32 to 31.

Key concepts

Unsigned Integer Representation

In binary systems, unsigned integer representation is a method where all the bits are used to represent positive numbers. This means that you can represent values starting from zero up to a certain maximum value, determined by the number of bits.

• The more bits you have, the higher the number you can express.
• For an n-bit binary number, the largest unsigned value is given by \(2^n - 1\).
• For example, with 4 bits, the largest value is \(2^4 - 1 = 15\).

When you increase the number of bits, say from 4 to 6 bits, the maximum becomes \(2^6 - 1 = 63\). More bits mean more room for numbers!

Two's Complement

Two's complement is a popular way of representing signed integers in computing, allowing both positive and negative values.

• In this system, the most significant bit (the leftmost one) is used as the sign bit.
• If the sign bit is 0, the number is positive; if it is 1, the number is negative.
• The range for an n-bit number in two's complement is from \(-2^{(n-1)}\) to \(2^{(n-1)}-1\).

With 4 bits in two's complement, the range is from -8 to 7. Increase the bit length to 6 bits, and you extend the range to -32 to 31. This system simplifies arithmetic operations on signed numbers.

Bit Length in Binary

The bit length in binary refers to the number of bits used to represent a number. Every bit added doubles the available range of numbers.

• With more bits, you can handle larger numbers or a wider range of values.
• In binary, each bit represents a power of 2, i.e., the bit furthest to the right represents \(2^0\), the next one \(2^1\), and so forth.

Consider a "bit" as a building block; more blocks create a bigger structure. Therefore, enhancing a binary sequence from 4 to 6 bits increases the dimension of the numerical range considerably, whether you're using unsigned integers or two's complement.

Maximum Value for Bit Length

The maximum value for a given bit length depends on how many bits are available and how they are used.

• For unsigned representations, the maximum is \(2^n - 1\), where n is the number of bits.
• In two's complement, the positive maximum is \(2^{n-1} - 1\).

Increasing bit length not only extends the maximum possible value but may also affect whether you're using signed or unsigned integers. For example:

• A 4-bit unsigned integer's maximum is 15, while a 6-bit maximum is 63.
• For two's complement, a positive 4-bit maximum is 7, and a 6-bit is 31.

More bits mean more flexibility in representing larger numbers, both positive and negative, depending on the context.