Question

Multiply 8023 by 4638 using the method of al-Uqlidis?

Short answer

Question: Multiply 8023 by 4638 using the method of al-Uqlidis (lattice multiplication). Short Answer: To multiply 8023 by 4638 using lattice multiplication, follow these steps: (1) Set up a 4x4 lattice grid with the digits of 8023 written across the top columns and the digits of 4638 written down the left rows. (2) Multiply each digit of a row by each digit of a column and write the results diagonally in each corresponding square. (3) Draw diagonals across the lattice and sum the values along each diagonal. (4) Write down and combine the diagonal sums from right to left to obtain the final product. (5) The resulting product is the multiplication of 8023 by 4638.

Step-by-step solution

Set up the Lattice

Break down the number 8023 into its digits: 8, 0, 2, 3. Do the same for the number 4638: 4, 6, 3, 8. Draw a grid with 4 columns (one per each digit of the first number) and 4 rows (one per each digit of the second number). Write the digits of the first number at the top of each column and the the digits of the second one at the start of each row. You should have a grid similar to a table.

Perform Individual Multiplications

Start multiplying each number of a row by each number of a column. Each square in the lattice should have the product of the number on top of the corresponding column and the number at the start of the corresponding row. Fill the lattice squares with these results, writing two-digit numbers diagonally.

Calculate the Diagonal Sums

Draw diagonals across the lattice. For each diagonal, find the sum of the values of all squares that the diagonal crosses. Begin from the right bottom of the lattice for the first diagonal, and proceed upwards to the left. The most right diagonal will contain only one number; the second from the right will contain two numbers and so on.

Combine the Sums

Write down all the sums calculated in Step 3 starting from the smallest diagonal sum to the largest. This sequence of numbers is the result of the multiplication 8023 by 4638 using the method of al-Uqlidis.

Result Interpretation

The final answer obtained from Step 4 is the product of multiplication of 8,023 by 4,638. Each digit in the answer corresponds to a diagonal in the lattice. The very first digit from the right in the answer is the sum of the rightmost diagonal in the lattice. The second digit from the right is the sum of the second diagonal from the right in the lattice, and so on.

Key concepts

Lattice Multiplication

Lattice multiplication is a method that breaks down large multiplication problems into smaller, more manageable parts. This technique uses a grid to organize the computation, helping to ensure that the multiplication steps are thorough and easily reviewed.

Imagine you're creating a checkerboard where each square will hold a part of the multiplication process. You'll have the same number of columns and rows as there are digits in the numbers you're multiplying. Once you've got your lattice set up, each cell will be filled with a partial product, which is simply the result of multiplying a single digit from the top number with a single digit from the side number.

An essential aspect to remember is the way the products are written within the cells—diagonally. This makes it easier to add them up later. Eventually, by summing up these diagonals, starting from the one on the far right, you put together the final result, almost like piecing together a puzzle.

Mathematical Algorithms

An algorithm in mathematics is like a recipe in cooking: a step-by-step instruction to solve a particular problem. When we talk about mathematical algorithms, we're referring to well-defined procedures that take some values, or 'inputs,' and transform them through various computational steps to produce new values, or 'outputs.'

In the case of multiplication, algorithms provide systematic ways to compute products without needing to resort to memory alone. Lattice multiplication is one of several algorithms that can accomplish this task. It can be particularly helpful for students who struggle with keeping track of all the partial products in standard multiplication, since the grid provides a visual framework to organize and double-check work.

Arabic Mathematics

Arabic mathematics, which refers to mathematical development under the Islamic civilization between the 8th and 14th centuries, made significant contributions to the field. Think of it as a bridge between ancient and modern mathematics, with scholars translating Greek, Indian, and even earlier Babylonian works, and then expanding upon them with their own innovations.

Al-Uqlidisi, for example, not only contributed to arithmetic but also improved upon the decimal positional number system. The fact that he was able to devise such a method without the modern computational tools we have today is a testament to the prowess of mathematicians in the Islamic Golden Age. Their work laid the foundation for many of the maths concepts and methods we use today.

Multiplication Techniques

There are various multiplication techniques, each with its own advantages. Alongside the lattice method, which offers a systematic and visual approach, there are others like the traditional algorithm, also known as long multiplication, which is widely taught in schools. Then there's the grid method, which is similar to lattice multiplication but involves splitting numbers into tens and units before multiplying.

Additionally, mental math strategies allow for quick multiplication of simpler numbers, and methods like Egyptian multiplication or Russian peasant multiplication take a more repetitive halving and doubling approach. The choice of technique often depends on the complexity of the numbers involved and the individual's comfort with the process. Educators aim to present various methods so that students can choose the one that makes the most sense to them and ensures they can tackle multiplication with confidence.