# Debating topics on mathematics

**Topic 1.**

The first ideas we present are simply to make people think about numbers, and in particular to encourage the use of the history archive to find birth dates and death dates before making calculations. Remember to count leap years.

How many days did Gauss live?

How many days did Euler live?

Each year in the Chinese calendar is named after one of 12 animals. The year 1985 was the Year of the Ox. How many in the 20th century were "Years of the Ox"?

**Topic 2.**

There are so many things that we accept without thought. However things usually are the way they are for a reason.

Why do we have 60 seconds in a minute, 60 minutes in an hour etc.

Why are they called seconds?

In one sense the answer might be that we still use the system set up by the Babylonians and their number system had a base of 60. But this is only a partial answer. Why did the Babylonians have 60 as a base? Nobody knows the answer to this question but it is easy to express and opinion. Perhaps it was chosen because it had lots of factors. But surely ancient civilisations do not choose the base of their number system. Base 10 must result from counting on 10 fingers. The Babylonians did not have 60 fingers! So why 60?

The word "seconds" come from the second sexagesimal place in the base 60 expansion.

**Topic 3.**

We could debate the nature of number.

What is a number?

Well it is not too hard to see what 2, 3, 4, 5, ... are. What about negative numbers.

Is -2 a number?

Suppose child A has 3 apples and child B takes 5 apples away from child A. Then child A has -2 left. Isn't this nonsense! If child A has 3 apples then B can't take 5 apples away. So is -2 a number?

Is 1 a number?

This looks more obvious. One is tempted to say "Of course it is." But the ancient Greeks did not consider 1 to be a number. Why not? Investigate.

Is the square root of 2 a number?

What is wrong with having the hypotenuse of a right angled triangle, whose shorter sides are each of one unit, not corresponding to a number? Why should there be a number corresponding to the length of every line we draw?

**Topic 4.**

Consider the number π.

Why do we use the symbol π?

Why is π such an interesting number?

If π is the ratio of the circumference of a circle to its diameter, then why is its area π times the radius squared.

Are there still things we do not know about π? In fact there are lots. Investigate.

**Topic 5.**

Building all numbers from the symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 is very clever indeed. Think about it - it really is fantastically clever!

Who invented the nine symbols that we use as numerals?

Is it a better number system than the Greek or Roman number system? Why?

Did other civilisations develop the same type of number system?

If the place value number system was a better system than that of the Romans then why was there so much resistance to using it?

Is the 0 in 2304 the same as the number 0?

**Topic 6.**

We have looked at how numbers are built from the symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.

Is 0 a number?

Does 0 obey the same rules as other numbers?

Is there a case for considering 1/0 to be a number?

**Topic 7.**

What is an equation?

What is $x$ in $2x = 3$? Is it a number?

What about $x$ in $0x = 1$? Is this an equation?

Can we solve equations without having "unknowns"?

How did the Chinese represent equations?

**Topic 8.**

Does the equation $ax = b$ always have a solution? Do quadratic, cubic and quartic equations always have solutions?

What does it mean to say that equations of degree 5 cannot be solved?

Why did the ancient Chinese not worry about solving cubic, quartic, quintic equations?

**Topic 9.**

Here are questions about a more advanced topic, namely complex numbers.

The equation $x^{2} + 1 = 0$ has no real number solution. Let $i$ be a symbol representing its solution. Is this a logical way to think?

What about $x + 1 = 0$?

Do we need to introduce negative numbers to get solutions of such equations?

Why then did people introduce negative numbers? Why did people introduce $i$?

Was it a good idea?

**Topic 10.**

Fifty years ago hard mathematical calculations were done using logarithms. Today logarithms are not used for calculating. Why?

Fifty years ago children were taught to use a slide rule. Today they are not. Why?

What will replace the pocket calculator?

Written by
J J O'Connor and E F Robertson

Last Update December 2003

Last Update December 2003