Question

This problem is from Brahmagupta's work on congruences. Given that the sun makes 30 revolutions through the ecliptic in 10,960 days, how many days have elapsed (since the sun was at a given starting point) if the sun has made an integral number of revolutions plus \(8080 / 10,960\) of a revolution, that is, "when the remainder of solar revolutions is 8080 ." If \(y\) is the number of days sought and \(x\) is the number of revolutions, then, because 30 revolutions take 10,960 days, \(x\) revolutions take \((1096 / 3) x\) days. Therefore, \(y=(x+808 / 1096)(1096 / 3)\), or \(1096 x+808=3 y\). Thus, solve \(N \equiv 808(\) mod 1096) and \(N \equiv 0(\bmod 3)\).

Short answer

Answer: 3000 days.

Step-by-step solution

Find the number of days for 1 revolution

Calculate the number of days it takes for 1 solar revolution by dividing 10,960 by 30: $$\frac{10,960}{30}= 365.33$$

Rewrite the given equation in terms of x and y

The given equation is \(y = (x + \frac{808}{1096})\frac{1096}{3}\), so we can rewrite it as: $$y = \frac{1096}{3}(x + \frac{808}{1096})$$

Rewrite the equation to get N

We can rewrite the equation as \(1096x + 808 = 3y\) to get N by setting N = 3y: $$N = 1096x + 808$$

Write down the congruence equations for N

The problem asks to solve for N such that: $$N \equiv 808 \pmod{1096}$$ $$N \equiv 0 \pmod{3}$$

Solve the first congruence equation for N

Solve the first congruence equation: $$N \equiv 808 \pmod{1096}$$ Since 808 is already less than 1096, we have: $$N = 808 + 1096k$$ where k is an integer.

Substitute the expression for N into the second congruence equation

Substitute the expression for N into the second congruence equation: $$808 + 1096k \equiv 0 \pmod{3}$$

Find the value of k that satisfies this congruence equation

To find the value of k that satisfies the congruence equation, we should check for divisibility by 3: $$808 + 1096k \equiv 0 \pmod{3}$$ $$808 \equiv 2 \pmod{3}$$ Since 1096 is divisible by 3, we have: $$1096k \equiv -2 \pmod{3}$$ So, we need k to be an integer such that: $$1096k \equiv -2 \pmod{3}$$ We can check the values of k, such as k=1 or k=2, and we find that k=2 satisfies this congruence equation.

Find the value of N for this k value

Now that we have the value of k, we can find the value of N: $$N = 808 + 1096(2)$$ $$N = 808 + 2192$$ $$N = 3000$$ The number of days that have elapsed when the sun has made an integral number of revolutions plus \(8080/10,960\) of a revolution is 3000 days.

Key concepts

Mathematical Congruences

Mathematical congruences are a fundamental concept in number theory, a branch of mathematics that deals with the properties and relationships of integers. Essentially, a congruence relation is a way to express that two numbers have the same remainder when divided by a given number, called the modulus.

For example, if we say that 15 and 35 are congruent modulo 10, written as \(15 \text{mod } 10 \text{and } 35 \text{mod } 10\), it means that when both 15 and 35 are divided by 10, they both leave a remainder of 5. This concept is represented by the notation \(15 \equiv 35 (\bmod 10)\). Similarly, the original exercise asked to solve \(N \equiv 808(\bmod 1096)\) and \(N \equiv 0(\bmod 3)\), which means we need to find a number N that, when divided by 1096, leaves a remainder of 808, and when divided by 3, leaves no remainder – that is, it is divisible by 3.

Understanding congruences is important for various applications, including solving Diophantine equations, cryptography, and even calculating astronomical events, as we see in the historical context of Brahmagupta's work.

Solar Revolutions Calculation

Calculating solar revolutions is a task that involves astronomy and mathematics, particularly modular arithmetic. A solar revolution is the time it takes for the sun to return to the same position in the sky, relative to the stars. It's essentially one year, measured not by the calendar, but by the stars. Brahmagupta, an Indian mathematician and astronomer, worked on such calculations using congruences.

In the given exercise, we are dealing with a part of a solar revolution expressed as a fraction \(8080/10,960\). The problem takes an historical approach to find the number of days elapsed given the sun has made an integral number of revolutions plus a fraction of a revolution. Using the relationship between days and revolutions, we leveraged the concept of mathematical congruences to find the exact number of days. This sort of calculation was critical in the past for creating accurate calendars and predicting celestial events, and it still has its importance in today's astronomy.

Modular Arithmetic

Modular arithmetic, often referred to as 'clock arithmetic', is a system of arithmetic for integers, where numbers 'wrap around' after they reach a certain value, the modulus. This provides a means of handling calculations for congruences and has a wide range of applications from computer science to cryptography to understanding patterns within the integers.

In modular arithmetic, we are often interested in finding a number that leaves a specific remainder when divided by the modulus. In the provided solution, the modulus are 1096 and 3, and we are interested in finding a value of N that is congruent to 808 modulo 1096, and to 0 modulo 3. The process involves equations and sometimes a bit of trial and error, as we saw when finding the appropriate value of k, which helps us solve for N.

This kind of problem solving is at the heart of modular arithmetic and continues to be useful in modern mathematics and technology, showing us that the work of ancient mathematicians like Brahmagupta remains relevant today.