There is a bizarre number in maths referred to simply as ‘i’. It appears to break the rules of arithmetic - but turns out to be utterly essential for applications across engineering and physics. We’re talking about the square root of -1. WHICH MAKES NO SENSE.Professor Fry waxes lyrical about the beauty and power of this so-called ‘imaginary’ number to a sceptical Dr Rutherford. Dr Michael Brooks tells the surprising story of the duelling Italian mathematicians who gave birth to this strange idea, and shares how Silicon Valley turned it into cold hard cash. It's all about oscillations, Professor Jeff O’Connell demonstrates. And finally, Dr Eleanor Knox reveals that imaginary numbers are indispensable for the most fundamental physics of all: quantum mechanics. Imaginary, impossible…but essential! Contributors: Professor Jeff O’Connell, Ohlone College California, Dr Michael Brooks, Author of 'The Maths That Made Us', and Dr Eleanor Knox, Philosopher of Physics at KCL and a Senior Visiting Fellow at the University of Pittsburgh.Producer: Ilan Goodman

Hungry for pi? Chow down on this! Pi is the ratio between a circle’s diameter and its circumference. Sounds dull – but pi turns out to have astonishing properties and crop up in places you would never expect. For a start, it goes on forever and never repeats, meaning it probably contains your name, date of birth, and the complete works of Shakespeare written in its digits. Maths comedian Matt Parker stuns Adam with his ‘pie-endulum’ experiment, in which a chicken and mushroom pie is dangled 2.45m to form a pendulum which takes *exactly* 3.14 seconds per swing. Mathematician Dr Vicky Neale explains how we can be sure that the number pi continues forever and never repeats - despite the fact we can never write down all its digits to check! She also makes the case that aliens would probably measure angles using pi because it’s a fundamental constant of the universe. NASA mission director Dr Marc Rayman drops in to explain how pi is used to navigate spacecraft around the solar system. And philosopher of physics Dr Eleanor Knox serves up some philoso-pi, revealing why some thinkers have found pi’s ubiquity so deeply mysterious. Hannah grins with delight for most of show. It’s all maths! Producer: Ilan GoodmanContributors: Matt Parker, Dr Vicky Neale, Dr Marc Rayman, Dr Eleanor Knox

Philosopher of physics Eleanor Knox and physicist Andrew Pontzen discuss the uniqueness of atoms, exploring quantum realms and the constant movement of electrons. They ponder if no two atoms are ever the same and delve into the complexity of identifying a truly unique atom. The quantum world of hydrogen reveals galaxies far away, challenging the notion of uniqueness in an infinite universe.