Problem 20 - A History of Mathematics: An Introduction

Question

In the Babylonian system, multiply 25 by 1,04 and 18 by 1,21 . Divide 50 by 18 and 1,21 by 32 (using reciprocals). Use our standard multiplication algorithm modified for base \(60 .\)

Step-by-Step Solution

Answer

Question: Perform the following multiplications and divisions using the standard multiplication algorithm modified for base 60 (Babylonian system): 1. Multiply 25 by 1,04 2. Multiply 18 by 1,21 3. Divide 50 by 18 4. Divide 1,21 by 32 Answer: 1. 26\,{40 \over 60} 2. 24\,{18 \over 60} 3. 2\,{46 \over 60} 4. {1 \over 32}\,{7 \over 640}

Step 1: Multiply 25 by 1,04

First, write the numbers you want to multiply in base 60 format. For 25, it's just 25 ; for 1,04, it's 1 (integer part) and \({4 \over 60}\) (fractional part). Multiply the two numbers as you would in base 10, but carry over the products that are at least 60. $ 25 \\ \times \;1,04 \\ \_\_\_\_\_ $ Multiply the integer part: $ 25 \\ \times \;1 \\ = 25 $ Multiply the fractional part: $ 25 \\ \times \;{4 \over 60} \\ = {100 \over 60} $ Since 100 is greater than 60, we can simplify the fraction: $ = 1\,{40 \over 60} $ Add the products together to get the final answer in base 60 format: $ 25 + 1\,{40 \over 60} \\ = 26\,{40 \over 60} $

Step 2: Multiply 18 by 1,21

Again, write the numbers in base 60 format: 18 by 1,\({21 \over 60}\). Multiply: $ 18 \\ \times \;1,21 \\ \_\_\_\_\_ $ Multiply the integer part: $ 18 \\ \times \;1 \\ = 18 $ Multiply the fractional part: $ 18 \\ \times \;{21 \over 60} \\ = {378 \over 60} $ Because 378 is greater than 60, simplify the fraction: $ = 6\,{18 \over 60} $ Add the products together to obtain the final answer in base 60 format: $ 18 + 6\,{18 \over 60} \\ = 24\,{18 \over 60} $

Step 3: Divide 50 by 18

Divide 50 by 18 using the reciprocal. Since dividing by 18 is the same as multiplying by \({1 \over 18}\), multiply 50 by \({1 \over 18}\): $ 50 \\ \times \;{1 \over 18} \\ = 2\,{46 \over 60} $

Step 4: Divide 1,21 by 32

Divide 1,\({21 \over 60}\) by 32 using the reciprocal. Since dividing by 32 is the same as multiplying by \({1 \over 32}\), multiply 1,\({21 \over 60}\) by \({1 \over 32}\): $ 1\,{21 \over 60} \\ \times \;{1 \over 32} \\ = {1 \over 32} + {21 \over 1920} $ Simplify the fraction \({21 \over 1920}\): $ = {1 \over 32} + {7 \over 640} $ The final answers for each operation are: 1. 26\,{40 \over 60} 2. 24\,{18 \over 60} 3. 2\,{46 \over 60} 4. {1 \over 32}\,{7 \over 640}

Key Concepts

Base 60 Calculations

The base 60 system, also known as the sexagesimal system, was used in ancient Babylonian mathematics and is still used today for measuring time, angles, and geographic coordinates. Unlike our familiar base 10 or decimal system, which uses powers of 10, the base 60 system uses powers of 60.

For example, when multiplying 25 by 1,04 in the Babylonian system, you convert 1,04 to 1 plus 4/60, since the comma represents the separation point similar to the decimal point in base 10. Then you perform multiplication as in the decimal system, but with special attention to when the product exceeds 60, in which case the excess becomes the next higher unit's value. This step ensures that each digit in the sexagesimal system remains between 0 and 59.

Mathematical Operations in Different Bases

Understanding mathematical operations in bases other than 10 requires a recognition of positional value in these different systems. In base 60, for example, the place values from right to left are 1 (60^0), 60 (60^1), 3600 (60^2), and so on.

To perform multiplication in base 60, as seen with 18 by 1,21, convert the fractional part 21/60 and then multiply each part separately. Regular multiplication algorithms apply, but carrying over occurs at multiples of 60 instead of 10. This adaptation is necessary because the value of each place increases by a factor of 60, not 10. Similarly, the operation must be carefully managed to keep digits legitimate for the base you're working in.

Reciprocal Method for Division

The reciprocal method for division is a technique used extensively in Babylonian mathematics and can be useful when operating in bases other than the familiar base 10. In the absence of a direct division operation, the reciprocal or multiplicative inverse (1/n) is used. For instance, to divide 50 by 18, we multiply 50 by the reciprocal of 18. The result is then converted back into base 60.

This method transforms a division problem into a multiplication problem, which can be simpler to manage using the base's multiplication rules. Reciprocal multiplication is especially potent in systems like base 60, where direct division may become cumbersome due to the large number of possible remainders in the base's digit set.