# Booth algorithm for multiplying signed numbers

• MHB
• mathmari
In summary, we are discussing the Booth algorithm for multiplying signed numbers of fixed complement representation decimal point by 2. We are considering the algebraic expressions for $N = 3$ and $N = 4$ and applying them to find the possible combinations of digits and determine the necessary operations for each iteration. Before the algorithm can begin, it needs to know the number of bits and the sign of the given numbers. For $N = 1$ and $N = 2$, we need to construct a table with 16 rows and use improved material with single registrant Product / Multiple. This procedure is repeated for $N = 3$ and $N = 4$ using the relations derived for those values.

#### mathmari

Gold Member
MHB
Hey! The algorithm of the Booth for multiplying signed numbersrs of fixed complement representation decimal point by 2 is implemented by multi-consecutive digit stain control of the multiplier, so after each test the algorithm proceeds by $N$ digits (e.g. for $3$-bit control step $N$ is equal to $2$)

a) Starting with the algebraic expression of the value of a given number and transforming it accordingly, find the relations from which the Booth algorithm for $N = 3$ and $N = 4$ is derived. So, once you find their form, apply the relations for the possible combinations of digits and determine for each of the two cases what operations must be done in each iteration, based on the possible values of the digits of the multiplier.

b) What kind of work do algorithms need before their first act, and what does that tell you about the practical implementation of Booth's algorithm for $N\geq 3$?

c) Test in principle Booth for $N = 1$ and $N = 2$ in the two digits of 16 bits $X=0111001010011001$ and $Y=1010011100101101$, for a constant-expansion multiplication unit with successive prostheses and slides that performs the $X \times Y$, where $X$ is multiplicand and $Y$ the multiplier, by constructing a table with the values of the units at each stage of the operation. The unit shall be based on improved material with the single registrant Product / Multiple, and with the minimum range as possible.

d) Repeat for the two new cases $N = 3$ and $N = 4$.

I am trying to understand how Booth algorithm works but I am facing some problems.

I found the below diagram : But what relation is it asked for at question (a) ? The relation that is also shown in the diagram just for a specific $N$ ?

For (b), I assume the algorithms need to know the number of bits in the given numbers and the sign of the numbers before they can start their work. For (c), I assume that for $N = 1$ and $N = 2$ we need to construct a table with 16 rows, where $X$ and $Y$ are the multiplicand and multiplier respectively and the columns will contain the values of the units at each stage of the operation. For (d), I assume that we need to use the same procedure as in (c), but with the relations derived from (a) for $N = 3$ and $N = 4$.

## What is the Booth algorithm for multiplying signed numbers?

The Booth algorithm is a method for multiplying two signed numbers in binary form. It is an efficient algorithm that reduces the number of partial products required for multiplication, thereby decreasing the number of steps needed to perform the multiplication.

## How does the Booth algorithm work?

The Booth algorithm works by breaking down the two signed numbers into smaller components and then using a series of shifts and additions to find the final product. It takes into account the sign of the numbers and uses a special coding scheme to reduce the number of partial products needed for multiplication.

## What are the advantages of using the Booth algorithm?

The main advantage of using the Booth algorithm is its efficiency. It requires fewer steps and operations compared to traditional multiplication methods, making it ideal for use in hardware implementations. It also reduces the number of partial products needed, which can save time and resources.

## Are there any limitations to the Booth algorithm?

While the Booth algorithm is efficient, it is not always the most accurate method for multiplying signed numbers. It can produce errors when used with certain input values and may require additional steps to ensure accuracy. Additionally, it is more complex than traditional multiplication methods and may not be suitable for all applications.

## Can the Booth algorithm be used for multiplying numbers in other bases?

Yes, the Booth algorithm can be adapted for use with numbers in other bases, such as hexadecimal or octal. However, the coding scheme and implementation may vary depending on the base being used. It is most commonly used with binary numbers, as it was designed specifically for this base.