Comparing Fractions: 2/3 vs 5/8 and Fibonacci Ratios

The comparison between two fractions can often be a challenge, especially when they have different denominators. In this article, we will explore the comparison between the fractions 2/3 and 5/8, and delve into the fascinating world of Fibonacci ratios.

1. Comparing 2/3 and 5/8

When comparing 2/3 and 5/8, the first step often involves finding a common denominator. The smallest common denominator for 3 and 8 is 24. To perform the conversion, we multiply the numerator and denominator of each fraction by a number that will give us 24 as the denominator:

2/3 2*8/3*8 16/24 5/8 5*3/8*3 15/24

Now that both fractions have the same denominator, the comparison can be made by looking at the numerators:

16/24 > 15/24

Therefore, 2/3 is greater than 5/8.

2. Comparing Decimals: 0.666… vs 0.625

Another approach is to convert the fractions to their decimal form and directly compare them:

2/3 0.666…

5/8 0.625

It is immediately clear that 0.666… is greater than 0.625, confirming that 2/3 is indeed larger than 5/8.

3. The Standard Comparison Method Using Common Denominators

The standard method involves finding a common denominator, which in this case is 24. The conversions are:

2/3 16/24 5/8 15/24

Again, 16/24 is greater than 15/24, which supports the conclusion that 2/3 is larger than 5/8.

4. Comparing Fibonacci Ratios

Fibonacci numbers and their ratios have a deep mathematical significance. To explore this, we will use Binet’s formula and delve into the properties of the ratio between consecutive Fibonacci numbers.

4.1 Binet’s Formula for Fibonacci Numbers

Binet’s formula for the Fibonacci sequence is given by:

$$ F_n frac{phi^n - (-phi)^{-n}}{sqrt{5}} $$

From this, we can deduce the ratio of the Fibonacci numbers:

$$ frac{F_{n-1}}{F_n} frac{phi^{n-1} - (-phi)^{-n-1}}{phi^n - (-phi)^{-n}} $$

Further simplification yields:

$$ frac{F_{n-1}}{F_n} frac{phi - (-phi)^{-n} - (-phi)^{-n-1}}{1 - (-phi)^{-n} - (-phi)^{-n}} $$

We can write this as:

$$ f(-phi^{-2n}) $$

where:

$$ f(x) frac{phi - phi^{-1} x}{1 - x} $$

It is evident that $$ f(x) $$ is an increasing function of $$ x $$ for $$ x . This can be verified by differentiating or simply by observing the behavior of the function.

Therefore, the ordering of the ratios $$ frac{F_{n-1}}{F_n} $$ is the same as the ordering of the powers $$ -phi^{-2n} $$. Since $$ -1 , we can determine the ordering based on the sign and magnitude of the powers.

- The even powers of $$ -phi^{-2} $$ are positive and decreasing.

- The odd powers of $$ -phi^{-2} $$ are negative and increasing.

Therefore, we have the following ordering of the ratios:

$$ frac{F_{n-1}}{F_n} > frac{F_{m-1}}{F_m} $$ if: $$ n > m $$ and both $$ n $$ and $$ m $$ are odd, or $$ n > m $$ and both $$ n $$ and $$ m $$ are even, or $$ n $$ is odd and $$ m $$ is even.

In all other cases where $$ n eq m $$, we have:

$$ frac{F_{n-1}}{F_n}

Note that this assumes the Fibonacci sequence is numbered starting with $$ F_0 0, F_1 1 $$.

5. Reciprocals of Fibonacci Ratios

When comparing the reciprocals of the Fibonacci ratios, such as 2/3 and 5/8, the inequalities are reversed. Therefore, if 2/3 is greater than 5/8, then 3/2 is less than 8/5.

This information provides enough to order all ratios of consecutive Fibonacci numbers. If you're curious about how to order these ratios, you can use the insights derived from Binet’s formula and the properties of the power of $$ -phi^{-2} $$.

Conclusion

Through various methods, we have demonstrated that 2/3 is greater than 5/8. Furthermore, we have explored the fascinating world of Fibonacci ratios and provided a method to order them based on their reciprocals. By leveraging mathematical formulas and properties, we can not only solve these problems but also uncover deeper mathematical truths.