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### Course: Algebra (all content)>Unit 11

Lesson 31: Logarithmic scale (Algebra 2 level)

# Benford's law (with Vi Hart, 1 of 2)

Vi Hart visits Khan Academy and talks about the mysteries of Benford's Law with Sal. Created by Sal Khan.

## Want to join the conversation?

• What brought Benford to his discover of this pattern? Was it just curiousity? Did Benford apply this law to any use in science, finance, or engineering?
• Actually, it was Simon Newcomb who noticed the wear on the book and originally stated the law. Though he published in 1881, the paper didn't have much impact. The law was rediscovered by physicist Frank Benford, who published in 1936. This time people took notice, and his name is now attached to the phenomenon.
• Just two minutes ago I went to a site with the first 100000 digits of pi, and jotted down the number of times my browser said it found "1" on the page, then "2" on the page, etc etc. Expecting Benford's law, this made my results much more astounding. The number of times each digit appeared in comparison to all the others was stunningly regular, as each tally was between 9800 and 10200. I know it's kinda not related to this video, but why is this happening? What is going on?
• Pi is suspected to be a normal number. A normal number is a number that has a equal frequency for all it's digits, so five would be as common as six. However we do not know if Pi is actually a normal number.
• Would this work in other number systems like octal, hexadecimal, binary. base 7 etc.?
• Yes, it works in any number system. However, in binary it's rather trivial - Benford's law predicts, accurately, that a random number will surely begin with 1.
• Montenegro IS a country! It is on the Adriatic in Eastern Europe. But he was thinking of Monaco, which is the little country on the eastern coast of France.
• Yes,it is Monaco, which has the feat of the world's most densely populated country in the world,with 36,000 residents in about one sq. mile.

Montenegro is a country,but it was made in 2006, and was part of Serbia and Montenegro,so it may have not yet have become a country when Sal and Vi made the video.
Good catch!
• Who is Vi hart?
• She has an amazing series on "Math for Fun and Glory". Search it up on khan. Well worth your time!
• In a more general sense, isn't this because every time you cross a numerical "threshold" the numbers all start with 1? as we go from 1's to 10's to 100's to 1000's, wherever these numbers stop there will be more numbers in the earlier part of that group than the later ones. (so for example, if you just took some random sample of a lot of numbers from 1 to 60, obviously more numbers would start with 1, 2, 3, 4, 5, or 6. if you did it with 1 to 3000, a lot would start with 1, 2, or 3)
• I am thinking it is probably a property of exponential functions. Most of the examples given are exponential in nature. If you look at an exponential function it is constantly getting stepper. Hence within any given range the smaller numbers will take a larger amount of the x axis. So, if you pick a random assortment of numbers from the x axis and then find the exponential value, for any given magnitude, it is more likely to be from the lower end (ie closest to 0).
• What is the distribution of Benford's law in base two? How would one go about finding that?
• In base two the first significant digit is always one.
• About the odds and evens thing, doesn't that make the most sense? because there are only 4 evens besides 0 and 5 odds.