FREQUENTLY ASKED QUESTIONS

After many years teaching maths and encouraging students to ask questions, I strongly believe that there is no such thing as a silly question. When a student experiences worries, uncertainties and losses in confidence, it is always for a good reason. Below is a collection of just a few of the questions that I have been asked by students over the years.

Leap years
How can you work out which years are leap years?
Leap years occur every four years in years that are divisible by 4. However, century years (e.g. 1900, 2100) are not leap years unless they are divisible by 400 (e.g. 1600, 2000). To make your calculation simple, note that a four-digit number is divisible by 4 if its final two digits are divisible by 4. So, you can deduce that the year 2012 is a leap year because its final two digits, 12, is divisible by 4 and so 2012 is divisible by 4.

Dividing fractions
How do you divide one fraction by another?
Suppose you want to calculate 3⁄4 ÷ 5⁄8. The standard method is to turn the second fraction upside down and multiply the two fractions. This gives: 3⁄4 X 8⁄5, giving an answer, when cancelled down, of 6⁄5.

Billion
My uncle says that a billion is a million million. Is he right?
Well, he is and he isn’t! There are two meanings for the word 'billion'. The ‘long scale’ meaning of one million million, or 1 000 000 000 000, is the one your uncle probably learned at school and this was the version in use in the UK until 1974. Unfortunately this meaning of billion defined a number too large to be of much practical use and the definition was altered in 1974 to the so-called ‘short scale’ version with a value of one thousand million (1 000 000 000), thereby bringing its definition into line with the rest of the world.

Largest number
What is the largest possible number?
There is no such thing as the largest number – no matter how big a number you can think of, you can always make it bigger (try adding one!). If millions, billions and trillions aren’t big enough for you, try a googol, which is a 1 followed by one hundred zeros (in other words, 10100). Bigger again than a google is a googolplex, which is 10 raised to the power of a googol (i.e. 1 followed by a googol of zeros!)
By the way, note that Google (with a capital G and different spelling) is a computer search engine and ‘Googleplex’ is what the Google company calls their headquarters.

Pi
What is pi?

Pi is a number with a value slightly bigger than 3. The word 'pi' refers to the Greek letter that gives it its name and is written π. It is a number that crops up everywhere in maths but it essentially derives from an important property of the circle: take a circle of any size, measure its circumference (C) and measure its diameter (D) and then calculate C/D. You find that you get the same answer for each and every circle – the number 3.14159… .  Note that the actual value of pi is a decimal number whose decimals go on for ever and the pattern of its decimals never repeats. So it is never possible to state the numerical value of pi exactly. Two useful approximations are 22/7 (with a value of 3.14285…) and an even closer approximation, 355/113, which is correct to the first 5 decimal places.

Raised to the power zero
Why does raising any number to the power zero equal to 1?

Let’s take an example, based, say, on the number 5.
The question here is, why does 50 = 1?
Let’s look at a sequence of numbers, starting with 53, which is 125.
Now, divide by 5 and you get 52, which is 25.
Again, divide by 5 and you get 51, which is 5.
Notice that every time you divide by 5, the power reduces by 1 (53 - 52- 51 -…)
So, divide by 5 once more and you get the result that 50 = 1.
Now try the same thing with a different starting number, say 8, and the final step is that 80 = 1.
In general, X0 = 1, where X is any real number (i.e. either an integer or a fraction).

Two minuses
What does it mean when people say that two minuses make a plus?

This statement is open to misinterpretation. For example, here are two minuses: –3 – 5 and the answer to this calculation is -8. The important thing to remember is that the rule ‘two minuses make a plus’ only applies when the two numbers are multiplied or divided. For example, all of the following calculations give positive answers:
–2 X –3 = 6; –5 X –4 = 20; –10 ÷ –2 = 5; –14 X – 4 = 3.5.
In general, when two negative numbers are multiplied or divided, the answer is a positive number.

Binary
What is binary and what does it have to do with computers?

Before explaining binary numbers, let’s look at denary numbers. As you are probably aware, our number system is based on the number 10. What this means is that the digits contained in a denary number are drawn from ten different digits (0, 1, 2, 3, 4, 5, 6, 7, 8 and 9).
Take a particular example of a base 10 number – say, 3185:
the 5 is the number of units,
the 8 is the number of tens (101),
the 1 is the number of hundreds (102),
the 3 is the number of thousands (103),

By contrast, binary numbers are based on the number 2. The digits contained in a binary number are drawn from just two digits (0 and 1).
Take a particular example of a base 2 number – say, 1001.
Although like denary numbers binary numbers are read from left to right, let’s look at these digits from right to left.
the final 1 is the number of units,
the penultimate digit (the second zero) is the number of twos (21),
the first 0 is the number of fours (22),
the first 1 is the number of eights (23),
So, converting the binary number 1001 to denary, we get 8 + 0 + 0 + 1 = 9.

Digital circuits (such as those in computer chips) work by switching pulses of electrical current on and off - and consequently information is stored as patterns of ons and offs, represented as 1s and 0s. Numbers can be stored in binary code (eg 1001 is the number 9) and so can letters (eg 01100010 is the letter b).

Early computers worked only in binary - hence the panels of lights (on for 1, off for 0); but in modern computers binary is so far under the bonnet that most of us will never use it.

It's still evidenced in the fact that we buy memory chips in 1, 2 or 4 Gb sizes, games consoles that are 8-bit, 16-bit, 32-bit or 64-bit (which is how many 1s or 0s the chip handles in one unit of information), use a palette of 256 standard colours for the web, and have 1024Kb in a Mb, all of which are powers of 2.

Maths type
 How can I type mathematical symbols using a word processor?

There are several applications available for creating mathematical text, but here are a few tips using a standard word processing package such as Microsoft Word.
* mathematical letters like x and y should be italicised – simply select them and choose italic in the 'Formatting Palette' or otherwise.
* to avoid confusing the letter x with the multiplication sign, X , it is usual to choose this special symbol, X, for multiplication which can be found in 'Insert' - 'Symbol').
* division can be represented using the slash key, /, or alternatively it can be written as ÷, using 'Invert' and then 'Symbol'
* superscripts (x2) and subscripts (x2) are formatted using the Formatting Palette.