One of the most amazing properties of the natural numbers -- 1, 2, 3, 4, 5, 6, 7 ... -- is that they go on forever. However, you can't write down the largest number. If you think a number is the largest, you can always add 1 to it and make it larger. Hence, the sequence of natural numbers is infinite.
So how to we deal with large numbers? For example, the earth has a mass of about 5.98 sextillion metric tons. Wait a minute! What is sextillion??
It is 1,000,000,000,000,000,000,000.
And the Great Lakes contain roughly 52.92 duodecillion water molecules !! What is duodecillion?
It is 1,000,000,000,000,000,000,000,000,000,000,000,000,000,000.
How do you represent such large numbers easily? And what if we have to add or multiply them?
Archimedes, the great scientist and mathematician of antiquity, came up with a solution. Can you guess it?
Yes, you can use exponents to reduce the number of zeroes to deal with. Archimedes wrote a letter to a local monarch, in which he proposed some tools and methods to handle arbitrarily large numbers. He informed the monarch that he could use this method to count the number of grains of sand in the universe.
Archimedes was fed up with people saying you couldn’t calculate the number of grains of sand on a beach. He believed that one could, and he calculated not just how many grains of sand there were on the beach, but how many there were in the universe. The trouble Archimedes faced was the Greek number system. It was a primitive system in which letters became numbers: A = 1, B = 2, C = 3, etc. So Archimedes invented a new classification of numbers: the exponents.
Archimedes thought that to count the number of grains of sand in the universe he needed numbers up to the eighth order, i.e. (10^8)^8 = 10^64, which is equal to:
We call this great innovation that Archimedes developed the Exponential Notation. He uses exponents of powers of 10. And the idea is to express large numbers using powers of 10, by keeping track of the number of digits in the number. So using exponential notation, we would write 100 as 10 squared. And 1,000 as 10 cubed. 10,000 -- 10 to the 4th. 100,000 -- 10 to the 5th power. So a number like 4,270,000 would be written as 4.27 times 10 to the 6th, or 4.27 x 10^6.
The advantage of this exponential notation that Archimedes discovered is, that it makes it very easy to multiply arbitrarily large numbers. And that's for the following reason. If we wanted to multiply a million (10^6) times 100 million (10^8), all we have to do is add the exponents! So 6 + 8 = 14 and you get 10^14 as the result. In other words, 10^6 x 10^8 = 10^(6+8) = 10^14.
How would we apply the exponential notation to estimating the time since the Big Bang?
Astronomers have estimated that the Big Bang is about is about 13.8 billion years old. That was a number that was obtained from the recent Planck mission. We can write it in exponential notation, as 13.8 x 10^10, as 1 Billion is 10^9. We can rewrite it in the standard form as 1.38 x 10^9 years. That's the number of years in the history of the universe.
What about the number of seconds in the history of the universe? Well, we have to take that large number-- 1.38 times 10 to the 10th power, or 1.38 x 10^10 -- and multiply it by the number of seconds in every year.
Well, how many seconds are there in a year? There are 60 seconds in a minute, 60 minutes in an hour, 24 hours every day, and 365 days in a normal year. So the total number of seconds in a year is 60 times 60 times 24 times
365, which works out exactly to 31,536,000 seconds in a year. In exponential notation, we have to convert that number. And we get 3.15 times 10 to the 7th power, or 3.15 x 10^7.
So to calculate the number of seconds in the history of the universe, we take the number of years in the history of the universe --1.38 x 10^10 -- and multiply it by the number of seconds in a year -- 3.15 x 10^7.
We multiply the constants, 1.38 times 3.15, which is about 4.4. And then we add the exponents, 10 and 7 to get 17. So, approximately, the number of seconds in the history of the Big Bang as we know it is 4.4 x 10^17 seconds.
Isn't that impressive?
Image from Pixabay
The Fibonacci sequence was first observed by the Italian mathematician Leonardo Fibonacci in 1202. He was investigating how fast rabbits could breed under ideal circumstances. He made the following assumptions:
Fibonacci asked how many pairs of rabbits would be produced in one year.
Can you create the numbers yourself? Remember to count the 'pairs' of rabbits and not the individual ones. Try it.
Were you able to come up with the Fibonacci numbers? If not, here is how you would do it.
The pattern comes out to be 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233.
Fibonacci numbers are of interest to biologists and physicists because they are frequently observed in various natural objects and phenomena. For example, the branching patterns in trees and leaves are based on Fibonacci numbers.
On many plants, the number of petals is a Fibonacci number: buttercups have 5 petals; lilies and iris have 3 petals; some delphiniums have 8; corn marigolds have 13 petals; some asters have 21 whereas daisies can be found with 34, 55 or even 89 petals.
How can we create a rule (algorithm) for the fibonacci series (sequence)?
First, the terms are numbered from 0 onwards like this:
n = 0 1 2 3 4 5 6 7 8 9 10 ...
xn =0 1 1 2 3 5 8 13 21 34 55 ...
What rule can we create here? Well, if you look, x3 = x2 + x 1 (2 = 1 + 1) and x4 = x2 + x3 (3 = 1 + 2), etc.
So we can write the rule (algorithm) as: xn = x(n-1) + x(n-2).
Example: term 7 is calculated as:
x7= x(7-1) + x(7-2)
= x6 + x5
= 13 + 8
Let's write programs in Python to calculate the Fibonacci numbers.
1. With looping:
a,b = 1,1
for i in range(n-1):
a,b = b,a+b
1. With recursion:
if n==1 or n==2:
N.B: No not copy and paste the python code as identation is important.
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