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NASA, Aliens and Binary Codes from the Star

The Drake Equation was intended by astronomer Frank Drake to spark a dialogue about the odds of intelligent life on other planets. He was one of the founding members of SETI – the Search for Extra Terrestrial Intelligence – which has spent the past 50 years scanning the stars looking for signals that could be messages from other civilisations.

In the following video, Carl Sagan explains about the Drake Equation:

The Drake equation is:
drake

where:

N = the number of civilizations in our galaxy with which communication might be possible (i.e. which are on our current past light cone);
R* = the average number of star formation per year in our galaxy
fp = the fraction of those stars that have planets
ne = the average number of planets that can potentially support life per star that has planets
fl = the fraction of planets that could support life that actually develop life at some point
fi = the fraction of planets with life that actually go on to develop intelligent life (civilizations)
fc = the fraction of civilizations that develop a technology that releases detectable signs of their existence into space
L = the length of time for which such civilizations release detectable signals into space

The desire to encode and decode messages is a very important branch of mathematics – with direct application to all digital communications – from mobile phones to TVs and the internet.

All data content can be encoded using binary strings. A very simple code could be to have 1 signify “black” and 0 to signify “white” – and then this could then be used to send a picture. Data strings can be sent which are the product of 2 primes – so that the recipient can know the dimensions of the rectangle in which to fill in the colours.

If this sounds complicated, an example from the excellent Maths Illuminated handout on codes:

code3

If this mystery message was received from space, how could we interpret it? Well, we would start by noticing that it is 77 digits long – which is the product of 2 prime numbers, 7 and 11. Prime numbers are universal and so we would expect any advanced civilisation to know about their properties. This gives us either a 7×11 or 11×7 rectangular grid to fill in. By trying both possibilities we see that an 11×7 grid gives the message below.

code2

More examples can be downloaded from the Maths Illuminated section on Primes (go to the facilitator pdf).

A puzzle to try:

“If the following message was received from outer space, what would we conjecture that the aliens sending it looked like?”

0011000 0011000 1111111 1011001 0011001 0111100 0100100 0100100 0100100 1100110

Hint: also 77 digits long.

This is an excellent example of the universality of mathematics in communicating across all languages and indeed species. Prime strings and binary represent an excellent means of communicating data that all advanced civilisations would easily understand.

Answer in white text below (highlight to read)

Arrange the code into a rectangular array – ie a 11 rows by 7 columns rectangle. The first 7 numbers represent the 7 boxes in the first row etc. A 0 represents white and 1 represents black. Filling in the boxes and we end up with an alien with 2 arms and 2 legs – though with one arm longer than the other!
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Cracking Codes Lesson – a double period lesson on using and breaking codes

Cracking ISBN and Credit Card Codes– the mathematics behind ISBN codes and credit card codes.

Essential resources for IB students:

1) Revision Village

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Revision Village has been put together to help IB students with topic revision both for during the course and for the end of Year 12 school exams and Year 13 final exams.  I would strongly recommend students use this as a resource during the course (not just for final revision in Y13!) There are specific resources for HL and SL students for both Analysis and Applications.

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There is a comprehensive Questionbank takes you to a breakdown of each main subject area (e.g. Algebra, Calculus etc) and then provides a large bank of graded questions.  What I like about this is that you are given a difficulty rating, as well as a mark scheme and also a worked video tutorial.  Really useful!

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The Practice Exams section takes you to a large number of ready made quizzes, exams and predicted papers.   These all have worked solutions and allow you to focus on specific topics or start general revision.  This also has some excellent challenging questions for those students aiming for 6s and 7s.

Each course also has a dedicated video tutorial section which provides 5-15 minute tutorial videos on every single syllabus part – handily sorted into topic categories.

2) Exploration Guides and Paper 3 Resources

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I’ve put together four comprehensive pdf guides to help students prepare for their exploration coursework and Paper 3 investigations. The exploration guides talk through the marking criteria, common student mistakes, excellent ideas for explorations, technology advice, modeling methods and a variety of statistical techniques with detailed explanations. I’ve also made 17 full investigation questions which are also excellent starting points for explorations.  The Exploration Guides can be downloaded here and the Paper 3 Questions can be downloaded here.

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All content on this site has been written by Andrew Chambers (MSc. Mathematics, IB Mathematics Examiner).

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IB Maths Exploration Guide

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IB HL Paper 3 Practice Questions (120 page pdf)

Eight P3 investigation questions and fully typed mark scheme (around 240 marks)

Available to download here

Modelling Guide for Explorations

A 50 page pdf guide full of advice to help with modelling explorations – focusing in on non-calculator methods in order to show good understanding.

Available to download here.

Statistics Guide

A 55 page pdf guide full of advice to help with modelling explorations – focusing in on non-calculator methods in order to show good understanding.

Available to download here.

IB Revision Notes

Full revision notes for SL Analysis (60 pages), HL Analysis (112 pages) and SL Applications (53 pages).  Beautifully written by an experienced IB Mathematics teacher.

Available to download here.

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