Signed Numbers

Why is having a sign bit not suited for digital computing?
Having a sign bit is not suited for digital computing because it complicates the circuitry needed to handle numbers appropriately.

1's Complement

What is a complementary system?
A complementary system is a number system where negative numbers are represented as a subtraction of its absolute value from another specific value.

What is 1's complement?
1's complement is a method to represent negative numbers in binary by flipping each bit.

How do you get the specific value that is used to find the 1's complement of a number?
To get the specific value that is used to find the 1's complement of a number, calculate , where is the number of bits in the binary number.

What is the equation for the 1's complement of a binary number?
The equation for the 1's complement of a binary number is , where is the number of bits and is the value of the binary number.

From Gemini

The "specific value" is often the maximum value that can be represented by the given number of bits. The equation for finding that value is . For example, the maximum value that can be represented by four bits is 15 because . What you do to find the 1's complement of a binary number is subtract the absolute value of the binary number from that specific value.

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Add/Sub in 1's complement

Which bit indicates that a binary number is negative in a 1's complement system?
In a 1's complement system, the bit that indicates a binary number is negative is the MSB.

What is something you have to consider when performing 1's complement arithmetic?
Something you have to consider when performing 1's complement arithmetic is end-around carry.

What is end-around carry?
End-around carry is when the final carry of an arithmetic operation, which would otherwise be in the MSB position, needs to be added to the LSB of the final binary number.

2's Complement

What is 2's complement?
2's complement is another number system like 1's complement which consists of the 1's complement of a number plus 1.

What are the steps to finding the 2's complement of a binary number?
The steps to finding the 2's complement of a binary number are:

  1. Go through the digits of the binary number from right to left.
  2. If the number is a 0, include it.
  3. If the number is a 1, and it's the first one that you encounter, include it.
  4. Flip every subsequent bit after that first 1.

What is the equation for the 2's complement of a binary number?
The equation for the 2's complement of a binary number is , where is the number of bits and is the value of the binary number.

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Add/sub in 2's complement

Do you have to consider end-around carry when performing 2's complement arithmetic?
No, you don't have to consider end-around carry when performing 2's complement arithmetic.

What do you do with the final carry in 2's complement arithmetic?
With the final carry in 2's complement arithmetic, you ignore it.

Using complements for Add/Sub

When do you convert a number to 1's or 2's complement if you're going to perform an arithmetic operation it?
If you're going to perform an arithmetic operation on a number, you convert it to 1's or 2's complement beforehand.

Binary Points: Fixed Point

What are the powers of two that each bit position represents in a binary fixed point number?
In a binary fixed point number, the powers of two that each bit position represents is:

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Floating Point

What are the three parts of a floating point number?
The three parts of a floating point number are:

  1. A Mantissa.
  2. An exponent.
  3. A sign.

How many bits does the Mantissa of a full-precision floating point number have?
The Mantissa of a full-precision floating point number has 23 bits.

How many bits does the exponent of a full-precision floating point number have?
The exponent of a full-precision floating point number has 8 bits.

How many bits does the Mantissa of a double-precision floating point number have?
The Mantissa of a double-precision floating point number has 52 bits.

How many bits does the exponent of a double-precision floating point number have?
The exponent of a double-precision floating point number has 11 bits.

What floating point precision do 32-bit systems use?
The floating point precision that 32-bit systems use is full-precision.

What floating point precision do 64-bit systems use?
The floating point precision that 64-bit systems use is double-precision.

What are three reasons you should use floating point numbers?
Three reasons you should use floating point numbers are that:

  1. They have a a significantly higher range of values.
  2. They're easy to truncate.
  3. They're convenient for multiplications and handling overflows.

Array Multipliers

Example of performing multiplication on two binary numbers using an array multiplier

Multiplying Two Numbers With An Array Multiplier.png