Question

Write \(13.395\) and \(22.8642\) in Stevin's notation. Use his rules to multiply the two numbers together and to divide the second by the first.

Short answer

In conclusion, when we convert the given numbers 13.395 and 22.8642 into Stevin's notation, we get 13, \frac{395}{1000} and 22, \frac{8642}{10000}. Multiplying these two numbers using Stevin's rules, we obtain the product of 286, \frac{12434689}{1000000}. Dividing the second number by the first number using Stevin's rules, we obtain the quotient of 1, \frac{219402}{130000}.

Step-by-step solution

Convert the numbers into Stevin's notation

To convert the numbers 13.395 and 22.8642 into Stevin's notation, we need to write them as whole numbers followed by a separator (like a comma), and then write out the decimals with the fractional part as a numerator, and the power of 10 they represent as the denominator. As follows: 13.395 in Stevin's notation: 13, \frac{395}{1000} 22.8642 in Stevin's notation: 22, \frac{8642}{10000}

Multiply the numbers using Stevin's rules

To multiply the numbers in Stevin's notation, we will follow these steps: 1. Multiply the whole parts of the numbers: 13 × 22 = 286 2. Multiply the fractional parts of the numbers: \frac{395}{1000} × \frac{8642}{10000} = \frac{3410890}{10000000} 3. Multiply the whole part of the first number by the fractional part of the second number, and vice versa, and add the products: 13 × \frac{8642}{10000} + 22 × \frac{395}{1000} = \frac{112346}{1000} + \frac{8690}{1000} = \frac{121036}{1000} 4. Add all the results: 286, \frac{3410890}{10000000} + \frac{121036}{1000} = 286, \frac{3410890 + 121036000}{10000000} = 286, \frac{124346890}{10000000} So, the product of 13.395 and 22.8642 in Stevin's notation is 286, \frac{12434689}{1000000}

Divide the second number by the first using Stevin's rules

To divide the second number by the first in Stevin's notation, we will follow these steps: 1. Divide the whole parts of the numbers: 22 ÷ 13 = 1 2. Find the remainder of the whole parts: 22 - (13 × 1) = 9 (This remainder will be used with the first fractional part in the next step) 3. Divide the fractional parts: \frac{8642 - (395 × 1)}{10000} = \frac{8247}{10000} 4. Divide the remainder of the whole parts by the whole part of the divisor and combine it with step 3 result: \frac{9}{13} + \frac{8247}{10000} = \frac{9}{13} + \frac{8247}{10000}\cdot\frac{13}{13} = \frac{9}{13} + \frac{107211}{130000} = \frac{219402}{130000} The quotient in Stevin's notation when dividing 22.8642 by 13.395 is 1, \frac{219402}{130000}.

Key concepts

Fractional Parts

When working with numbers like 13.395 and 22.8642, we need to express them in a notation that separates their whole parts from their fractional parts. This is where Stevin's notation comes in handy. The key idea is to write the decimal numbers as two separate parts: one for the whole number and another for the fractional component:

• Take the number 13.395; the whole part is 13 and the fractional part is 0.395.
• For 0.395, we multiply the digits by the smallest power of 10 that will eliminate the decimal points, turning it into a fraction: \(\frac{395}{1000}\).

So in Stevin's notation, 13.395 becomes 13, \(\frac{395}{1000}\).

We do this similarly for 22.8642: its whole number is 22 and its fractional component is 0.8642, which becomes \(\frac{8642}{10000}\). Thus, in Stevin's notation, it reads as 22, \(\frac{8642}{10000}\).
This process helps clearly separate the integral and fractional components, making it easier to perform operations like multiplication and division.

Multiplication Rules

Multiplying numbers in Stevin's notation involves several systematic steps. Let's look into the multiplication rules to grasp the process better:

• First, multiply the whole (integer) parts of the numbers. For example, multiplying 13 and 22 will give us 286.
• Next, handle the fractional parts by multiplying them as fractions: \(\frac{395}{1000}\) and \(\frac{8642}{10000}\). This yields \(\frac{3410890}{10000000}\).
• Then, cross-multiply the whole part of one number with the fractional part of the other and add the results. So you calculate \(13 \times \frac{8642}{10000}\) plus \(22 \times \frac{395}{1000}\), which equals \(\frac{121036}{1000}\).
• Finally, sum up all these results: 286, \(\frac{3410890}{10000000} + \frac{121036}{1000}\), combining the fractions to get them with a common denominator and finalize the results.

The result, when combined and simplified, shows the product in Stevin's notation as 286, \(\frac{12434689}{1000000}\). This step-by-step approach allows for clearer and more structured multiplication of decimal numbers.

Division Rules

Dividing with Stevin's notation follows a unique but consistent method, keeping the whole and fractional parts distinct. Here's how it works:

• Start by dividing the whole numbers directly. For example, dividing 22 by 13 gives 1 as the main integer part of the quotient.
• Then, find the remainder from this division (22 - 13), which is 9, and prepare it for use in conjunction with the fractional part.
• Proceed to divide the fractional part, adjusted by any remainders if necessary: \(\frac{8642 - (395 \times 1)}{10000} = \frac{8247}{10000}\).
• Combine the adjusted fractional part with the fraction of the remainder over the divisor. Calculate \(\frac{9}{13} + \frac{8247}{10000} \cdot \frac{13}{13}\), which simplifies to \(\frac{219402}{130000}\), including converting fractions to a common denominator when needed.

In Stevin's notation, the quotient is obtained as 1, \(\frac{219402}{130000}\). This process helps to perform division, preserving the meaningful separation of whole and fractional parts throughout the calculations.