Nothing really matters I.
Lesson 2
Your special number
Is there a number that you specially like ? Is there a number that you don’t like at all ? Do you think some numbers have “special meanings” ?
II.
A powerful invention
Listen to the first extract from Simon Singh’s radio programme, fill in the gaps, and then answer the questions in your notebook. 1 2 3 4 5
So, where does the history of zero start ? Well there was a time when mathematicians didn’t even know about zero. The word, the symbol, the very concept of 0 hadn’t been invented ; or is that discovered ? Either way, as Ian Stewart of Warwick University points out, there came a time when mathematics couldn’t progress without 0.
7
Arithmetic works much better if you think of 0 as a number. What’s 3 take away 3 ?
8
Zero.
9
Exactly. And it’s a physical thing you can do.
6
10
So you need 0 in order to represent nothing in an equation. Questions : 1. What is this programme about ? 2. What mathematical operation is Ian Stewart talking about when he says “3 take away 3” ? 3. What is an equation ? 4. Can you think of other reasons why zero is such an important number ? 5. Simon Singh says, “Zero hadn’t been invented ; or is that discovered ?” Do you think zero was invented or discovered ? Seconde Euro
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III.
How do you relate to zero ?
In the next extract, Simon Singh interviews Adam Spencer, an Australian mathematician. When I’ve spoken to mathematicians, they tend to give numbers personalities. How do you relate to zero ?
1 2
Zero. Underrated, stubborn, at times helpful, at times very irritating, refuses to go away.
3 4
And if you really want to see the irritating side of 0 then try dividing by it.
5 6
Questions : 1. What happens if you try to divide by zero ? 2. Do you know why ?
IV.
A question of irreversibility
To find out exactly why you can’t divide by zero, Simon Singh finally interviews Charles Seife, author of “Zero, biography of a dangerous idea”. 1 2
In short, never divide by 0, because the result is chaos, logic breaks down, paradoxes proliferate. The root of all these problems is irreversibility.
4
For instance if you multiply 2 by 3 you get 6. To get back, you divide 6 by 3 and get to 2. That’s a reversible operation.
5
Right.
3
6 7
8 9
Multiplying by 0 is not a reversible operation. Because multiply 0 by 3 you get 0. But if you multiply 2 by 0 you get 0. So, if you try and work backwards there’s no obvious way to tell where you came from. Questions : 1. Can you find other examples of reversible operations ? of irreversible operations ? 2. Let a and b be two numbers, with a = b. What do you think of the following proof ? : Seconde Euro
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a=b
this is the hypothesis
a×a=a×b
multiply by a
a2 = ab a2 − b2 = ab − b2
subtract b2
(a − b)(a + b) = b(a − b)
factorise
a+b=b
divide by a − b
a+a=a
because a = b
2a = a 2a a = a a 2=1
Seconde Euro
divide by a isn’t that interesting ? !
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