MARCUS DU SAUTOY:'At the heart of the mathematical world lie numbers 'They give us the power to describe, 'measure and count everything in the universe.
MARCUS DU SAUTOY:'But numbers aren't always what they seem.
MARCUS DU SAUTOY:'Take for example negative numbers.'
MARCUS DU SAUTOY:It's impossible to trade anything. Stocks, shares, currency, even fish without negative numbers.
MARCUS DU SAUTOY:Most of us are comfortable with them, even though we may not like it, we understand what it means to have a negative bank balance.
MARCUS DU SAUTOY:But when you start to think about it, there's something deeply strange about negative numbers because they don't seem to correspond to anything real at all.
MARCUS DU SAUTOY:'The deeper we look into the mathematical world of numbers, 'the weirder it becomes.'
MARCUS DU SAUTOY:It's easy to imagine one fish or two fish or no fish at all. It's much harder to what minus one fish looks like.
MARCUS DU SAUTOY:Negative numbers are so odd that if I have minus one fish and you give me a fish, then all you can be certain of is that I've got no fish at all.
MARCUS DU SAUTOY:'The strange and powerful thing about numbers 'is that they can exist in the mathematical world 'whether or not they seem to make sense in the real world.
MARCUS DU SAUTOY:'And some numbers are so strange 'they seem to defy even the laws of mathematics.'
MARCUS DU SAUTOY:Now this is one of the most basic facts of mathematics a positive number multiplied by another positive number is a positive number. So for example one times one is one.
MARCUS DU SAUTOY:A negative number multiplied by another negative number also gives a positive number.
MARCUS DU SAUTOY:'Whenever the signs are the same, the product is always positive.'
MARCUS DU SAUTOY:However in the code there's a special number which breaks this rule. When I multiply it by itself, it gives the answer minus one.
MARCUS DU SAUTOY:This isn't a number I can calculate, I can't show you this number. Nevertheless, we've given this number a name. It's called "i", and it's part of a whole class of numbers called imaginary numbers.
MARCUS DU SAUTOY:'Introducing these strange new numbers into mathematics 'required an act of the imagination.
MARCUS DU SAUTOY:'But despite their strange properties, 'they turn out to have some very practical applications.
MARCUS DU SAUTOY:'Air traffic control relies on radar to track planes accurately 'during their passage through the air.
MARCUS DU SAUTOY:'Complex computation is required to decode these signals, 'and distinguish moving objects like planes 'from the stationary background.
MARCUS DU SAUTOY:'And at the heart of that analysis, lies "i". 'The number that cannot exist.'
MARCUS DU SAUTOY:I mean you could do these calculations with ordinary numbers, but they're so cumbersome, by the time you've actually done the calculation, the plane's moved to somewhere else.
AIR TRAFFIC CONTROLLER:Altitude 6,000 on a squawk of 7, 7, 1, 5.
MARCUS DU SAUTOY:Using imaginary numbers makes the calculation so much simpler, that you can track the planes in real time. In fact without them, radar would be next to useless for air traffic control.
MARCUS DU SAUTOY:'Although "i" is an imaginary number, 'I trust my life to it every time I get in an aeroplane.
MARCUS DU SAUTOY:'As strange as it may seem, 'that's because even the so-called imaginary bits of mathematics 'can be used to explain, control and accurately predict the real world.'
Starting with the logical roots of arithmetic with negative numbers, Marcus du Sautoy explains how mathematicians created imaginary numbers by ‘imagining’ the square root of -1.
Though imaginary in nature, he then explores how this abstract mathematical idea has become vital to air traffic control systems.
This clip is from the series The Code.
Teacher Notes
Use as an enrichment and extension clip during a series of lessons on negative numbers or powers and roots.
Once the concept has been established, students can have a go at working with complex numbers as a way of practising surds.
Curriculum Notes
These clips will be relevant for teaching Maths at KS4 and GCSE in England, Wales and Northern Ireland and National 4/5 or Higher in Scotland.
The topics discussed will support OCR, Edexcel, AQA, WJEC in England and Wales, CCEA in Northern Ireland and SQA in Scotland.
