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The Zero Trick

Zero ($0$) is a relative newcomer to the world of mathematics. It arrived in 6th century India, but really came into its own in what is now Iraq (Baghdad) in the 8th century CE with one of the great genii of the mathematical world, Muhammad ibn Musa al-Khwarizmi. He is the most famous person that no-one has ever heard of and this stinks. What it most particularly stinks of, of course, (and par for the course) is racism and Islamophobia, but we'll let that one pass for the moment. Even people who've never heard of the man still speak his name frequently, as the word algorithm which means a "set of instructions which can be followed by a person or computer" and much used in the tech world, is an English corruption of his name (al-Khwarizmi) and every schoolchild's terror, Algebra, we get from the name of his book on the subject, which has the following catchy title.

The Compendious Book on Calculation by Completion and Balancing
(Arabic: ٱلْكِتَاب ٱلْمُخْتَصَر فِي حِسَاب ٱلْجَبْر وَٱلْمُقَابَلَة‎, al-Kitāb al-Mukhtaṣar fī Ḥisāb al-Jabr wal-Muqābalah).

However it is more commonly know by the name Al-Jabr. In western Europe, we got to hear about this quite a bit later, thanks to a chap called Leonardo (no, not that one!) Bonacci of Pisa who is himself better known by his nickname Fibonacci, which means Son of Bonacci, Bonacci being Leo's Dad. His book which confusingly is called Liber Abaci (Eng: The Book of Calculation) contains much of the material which has made school children wretched with misery for many centuries. However, it is an extraordinary work and when time permits, I shall write up some of the material in it, on this site.

But back to Zero ($0$). Zero is a number with an amazing property: If a bunch of numbers are multiplied together and the result is zero, the only way that can happen is iff (if and only if) at least one of the numbers being multiplied is, itself, Zero. Just as a little thought experiment, have a go at finding numbers which multiply together to give Zero, which don't involve using Zero itself. It can't be done; there must be a Zero in there somewhere. $9 \times 8 \times 7 \times 6 \times 5 \times 0 = 0$

Now, in this section, we are going to use this property to solve quadratic equations, so buckle up and have a look at the worked examples.

Please note that if you have not covered the Quadratic Factorisation section, it is important to work on that (follow the link) before attempting this topic.