USING THE FIBONACCI NUMBERS TO CONVERT KILOMETERS TO AND FROM MILES
When we say that mathematics is always available to help in your everyday life, we can demonstrate it very nicely by showing you a clever way to convert miles to and from kilometers. Most of the world uses kilometers to measure distance, while the United States still holds on to the mile to measure distance. This requires a conversion of units when one travels in a country in which the measure of distance is not the one to which we are accustomed. Such conversions can be done with specially designed calculators or by some “trick” method. That is where the famous Fibonacci numbers come in. Just to refresh your memory, the first few Fibonacci numbers are:
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144,…
Before we discuss the conversion process between these two units of measure, let's look at their origin. The mile derives its name from the Latin word for 1,000, mille, as it represented the distance that a Roman legion could march in 1,000 paces (which is 2,000 steps). One of these paces was about 5 feet, so the Roman mile was about 5,000 feet. The Romans marked off these miles with stones along the many roads they built in Europe—hence the name, “milestones”! The name statute mile (our usual measure of distance in the United States today) goes back to Queen Elizabeth I of England (1533–1603) who redefined the mile from 5,000 feet to 8 furlongs (5,280 feet) by statute in 1593. A furlong is a measure of distance within Imperial units and US customary units. Although its definition has varied historically, in modern terms it equals 660 feet, and is therefore equal to 201.168 meters. There are eight furlongs in a mile. The name “furlong” derives from the Old English words furh (“furrow”) and lang (“long”). It originally referred to the length of the furrow in one acre of a ploughed open field (a medieval communal field that was divided into strips). The term is used today for distances horses run at a race track.
The metric system dates back to 1790, when the French National Assembly (during the French Revolution) requested that the French Academy of Sciences establish a standard of measure based on the decimal system, which they did. The unit of length they called a “meter” is derived from the Greek word metron, which means “measure.” Its length was determined to be one ten-millionth (1 · 10(–7)) of the distance from the North Pole to the equator along a meridian going near Dunkirk, France, and Barcelona, Spain. Clearly, the metric system is better suited for scientific use than is the American system of measure. By an Act of Congress in 1866, it became “lawful throughout the United States of America to employ the weights and measures of the metric system in all contracts, dealings, or court proceedings.” Although it has not been used very often, there is curiously no such law establishing the use of our mile system.
To convert miles to and from kilometers, we need to see how one mile relates to the kilometer. The statute mile is exactly 1609.344 meters long. Translated into kilometers, this is 1.609344 kilometers. One the other hand, one kilometer is 0.621371192 miles long. The nature of these two numbers (reciprocals that differ by almost 1) might remind us of the golden ratio, which is approximately 1.618, and its reciprocal, which is approximately 0.618. Remember, it is the only number whose reciprocal differs from it by exactly 1. This would tell us that the Fibonacci numbers, the ratio of whose consecutive members approaches the golden ratio, might come into play here.
Let's see what length 5 miles would be in kilometers: 5 times 1.609344 = 8.04672 ≈ 8.
We could also check to see what the equivalent of 8 kilometers would be in miles: 8 times 0.621371192 = 4.970969536 ≈ 5. This allows us to conclude that approximately 5 miles is equal to 8 kilometers. Here we have two of our Fibonacci numbers.
As mentioned above, the ratio of a Fibonacci number to the one before it is approximately ϕ, which is the symbol used to note the golden ratio. Therefore, since the relationship between miles and kilometers is very close to the golden ratio, they appear to be almost in the relationship of consecutive Fibonacci numbers. Using this relationship, we would be able to approximately convert 13 kilometers to miles by replacing 13 with the previous Fibonacci number, 8. This would reveal to us that 13 kilometers is equivalent to about 8 miles. Similarly, 5 kilometers is about 3 miles, and 2 kilometers is roughly 1 mile. The higher Fibonacci numbers will give us a more accurate estimate, since the ratio of these larger consecutive Fibonacci numbers gets closer to ϕ.
Now suppose you want to convert 20 kilometers to miles. We have selected 20 because it is not a Fibonacci number. We can express 20 as a sum of Fibonacci numbers and convert each number separately and then add them. Thus, 20 kilometers = 13 kilometers + 5 kilometers + 2 kilometers. By replacing each of these Fibonacci numbers with the one lower, we have 13 replaced by 8, 5 replaced by 3, and 2 replaced by 1. This, therefore, reveals that 20 kilometers is approximately equal to 8 + 3 + 1 = 12 miles. (Of course, if we would like to have a faster and perhaps less accurate estimate, we notice that 20 is close to the Fibonacci number 21. Using that number gives us 13 miles, a reasonable estimate done more quickly.)
Representing integers as sums of Fibonacci numbers is not a trivial matter. We can see that every natural number can be expressed as the sum of other Fibonacci numbers without repeating any one of them in the sum. Let's take the first few Fibonacci numbers to demonstrate this property as shown in this table.
express larger natural numbers as the sum of Fibonacci numbers. Each time, ask yourself if you have used the fewest numbers in your sum.
To use this process to achieve the reverse, that is, to convert miles to kilometers, we write the number of miles as a sum of Fibonacci numbers and then replace each by the next larger Fibonacci number. Converting 20 miles to kilometers, therefore, gives us a sum as 20 miles = 13 miles + 5 miles + 2 miles. Now, replacing each of the Fibonacci numbers with their next larger in the sequence, we arrive at 20 miles = 21 kilometers + 8 kilometers + 3 kilometers = 32 kilometers.
To use this procedure, we are not restricted to use the Fibonacci representation of a number that uses the fewest Fibonacci numbers. You can use any combination of Fibonacci numbers whose sum is the number you are converting. For instance, 40 kilometers is 2 · 20, and we have just seen that 20 kilometers is 12 miles. Therefore, 40 kilometers is 2 · 12 = 24 miles (approximately). It should be noted that the larger the Fibonacci numbers being used, the more accurate the estimated conversion will be.
Consequently, we have another example of how some more sophisticated mathematics can be helpful in resolving a common, everyday problem.
USING THE FIBONACCI NUMBERS TO CONVERT KILOMETERS TO AND FROM MILES