# Math problem from book Contact: tens of billions binary = ? base 10 **URL:** \ **Category:** Factual Questions\ **Created:** [March 7, 2014, 3:01pm UTC](https://boards.straightdope.com/t/math-problem-from-book-contact-tens-of-billions-binary-base-10/683113 "2014-03-07T15:01:44Z")\ **Posts on this page:** 8\ **Page:** 1
**Author:** ![Earl\_Snake-Hips\_Tucker](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/earl_snake-hips_tucker/32/3269_2.png) [@Earl\_Snake-Hips\_Tucker](https://boards.straightdope.com/u/Earl_Snake-Hips_Tucker)\ **Post date:** [March 7, 2014, 3:01pm UTC](https://boards.straightdope.com/t/math-problem-from-book-contact-tens-of-billions-binary-base-10/683113/1 "2014-03-07T15:01:44Z")
This came up last year, and I want to rephrase the question based on my understanding: Going back over the text of the book, where they’re about to decrypt the message, the key is that it is being transmitted over and over as a number that is “tens of billions of bits long” that is the product of three prime numbers. I inferred from that that that is in binary, so. . . With that, how long in base 10 would a number that is tens of billions of binary digits be? (I figure with a number that big, there would be quite a few products of three primes that would provide a solution, so I’m not asking for that. 😃 ) ---
**Author:** ![Francis\_Vaughan](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/francis_vaughan/32/3093_2.png) [@Francis\_Vaughan](https://boards.straightdope.com/u/Francis_Vaughan)\ **Post date:** [March 7, 2014, 3:13pm UTC](https://boards.straightdope.com/t/math-problem-from-book-contact-tens-of-billions-binary-base-10/683113/2 "2014-03-07T15:13:37Z")
About ten billion decimal digits. Assuming tens, means “a few” times ten". Where “a few” means 2 or three If “tens” means 33.2, the answer is exactly 10 billion. ---
**Author:** ![Earl\_Snake-Hips\_Tucker](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/earl_snake-hips_tucker/32/3269_2.png) [@Earl\_Snake-Hips\_Tucker](https://boards.straightdope.com/u/Earl_Snake-Hips_Tucker)\ **Post date:** [March 7, 2014, 3:17pm UTC](https://boards.straightdope.com/t/math-problem-from-book-contact-tens-of-billions-binary-base-10/683113/3 "2014-03-07T15:17:58Z")
Thanks. Not a particularly dramatic reduction in digits, so it would seem. ---
**Author:** ![pulykamell](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/pulykamell/32/3166_2.png) [@pulykamell](https://boards.straightdope.com/u/pulykamell)\ **Post date:** [March 7, 2014, 3:19pm UTC](https://boards.straightdope.com/t/math-problem-from-book-contact-tens-of-billions-binary-base-10/683113/4 "2014-03-07T15:19:03Z")
Should be about 3 billion-ish digits for 10 billion decimal digits. Wolfram alpha isn’t letting me get quite as high as 2^(10 000 000 000), but at 2^billion, it gives me a number of length around 300 milliion, and at 2^(100 000 000), it gives me a length around 30 million. Basically, you could divide the number of bits by 3.3 and get a reasonable estimate of the length of your decimal number. ---
**Author:** ![JWT\_Kottekoe](https://avatars.discourse-cdn.com/v4/letter/j/3ec8ea/32.png) [@JWT\_Kottekoe](https://boards.straightdope.com/u/JWT_Kottekoe)\ **Post date:** [March 7, 2014, 3:42pm UTC](https://boards.straightdope.com/t/math-problem-from-book-contact-tens-of-billions-binary-base-10/683113/5 "2014-03-07T15:42:38Z")
… ---
**Author:** ![septimus](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/septimus/32/410_2.png) [@septimus](https://boards.straightdope.com/u/septimus)\ **Post date:** [March 7, 2014, 4:57pm UTC](https://boards.straightdope.com/t/math-problem-from-book-contact-tens-of-billions-binary-base-10/683113/6 "2014-03-07T16:57:51Z")
> [@Earl\_Snake-Hips\_Tucker](#): > > (I figure with a number that big, there would be quite a few products of three primes that would provide a solution, so I’m not asking for that. 😃 ) No, the three primes would be unique. [QUOTE=Euclid of Alexandria, Elements Book VII] If two numbers, multiplied by one another make some number, and any prime number measures the product, then it also measures one of the original numbers. [/QUOTE] ---
**Author:** ![Thudlow\_Boink](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/thudlow_boink/32/320_2.png) [@Thudlow\_Boink](https://boards.straightdope.com/u/Thudlow_Boink)\ **Post date:** [March 7, 2014, 5:42pm UTC](https://boards.straightdope.com/t/math-problem-from-book-contact-tens-of-billions-binary-base-10/683113/7 "2014-03-07T17:42:17Z")
I got the same answer as others. Here’s how: Say N is the number being transmitted. log[sub]10[/sub] N would be the number of decimal digits of N, and log[sub]2[/sub] N would be the number of binary digits. [Then](http://en.wikipedia.org/wiki/Logarithm#Change_of_base) log[sub]2[/sub] N = log[sub]10[/sub] N / log[sub]10[/sub] 2, so the number of decimal digits = the number of binary digits times log[sub]10[/sub] 2, which is about 0.301. ---
**Author:** ![Malacandra](https://avatars.discourse-cdn.com/v4/letter/m/45deac/32.png) [@Malacandra](https://boards.straightdope.com/u/Malacandra)\ **Post date:** [March 7, 2014, 5:54pm UTC](https://boards.straightdope.com/t/math-problem-from-book-contact-tens-of-billions-binary-base-10/683113/8 "2014-03-07T17:54:30Z")
Quick and dirty, 2[sup]10[/sup] ~= 10[sup]3[/sup].