One, Two, Three, Four, Five, ... Quiz

Answer: 10

The natural numbers are the ones we learn to count with - one, two, three, four five, six, seven, eight, nine, ten, etc.

Natural numbers can be added, subtracted, multiplied and divided. For example
2 + 3 = 5
7 - 4 = 3
5 x 2 = 10
6 / 3 = 2

Answer: 7

Some natural numbers can be made by multiplying smaller numbers together. For example 10 = 2 x 5, or 30 = 2 x 3 x 5. The smaller numbers are called factors of the larger number. Obviously any number can be made by multiplying itself by one, but any number greater than one which can only be made in this way is described as prime. For example 7 = 1 x 7.

The first prime numbers are 2, 3, 5, 7, 11, 13, 17, 19.

Answer: 28

A perfect number is one where the sum of all its factors, other than itself, add up to that number.

6 is the smallest perfect number. 6 = 1 x 6 = 2 x 3. So its factors are 1, 2, 3. 1 + 2 + 3 = 6

28 = 1 x 28 = 2 x 14 = 4 x 7. So its factors are 1, 2, 4, 7, 14. Adding them together results in 28.

The next three perfect numbers are 496, 8128, 33550336.

Answer: 16

A square number is the result of multiplying a number by itself. If you imagine trying to fit small square boxes inside a larger square box, it tells you how many will be needed to fill the space. When writing a square number it is usual to put the number followed by 2 as a superscript. However, that is much harder with keyboards, and it is often written with the number followed by ^2. So 2 x 2 = 2^2.

The first few square numbers are 1^2 = 1, 2^2 = 4, 3^2 = 9, 4^2 = 16, 5^2 = 25, 6^2 = 36, 7^2 = 49, 8^2 = 64, 9^2 = 81.

Answer: 27

A cube, or cubic number is the result of multiplying the number by itself, and multiplying that result by the number. In our example of boxes, we now have to imagine making as many layers of boxes as we have on each side of the bigger box. A cubic number is indicated by ^3.

The first few cubic numbers are 1^3 = 1, 2^3 = 8, 3^3 = 27, 4^3 = 64, 5^3 = 125, 6^3 = 216, 7^3 = 343, 8^3 = 512, 9^3 = 729.

Answer: -2

Numbers can get more interesting. What happens if you try to work out 5 - 7? Thought about logically, if I have only five apples, I cannot take seven apples away from them - it is impossible. If we imagine our numbers written in a line with 1 on the left and increasing to the right, we can put 0 to the left of 1, then we can imagine the line stretching on to the left, and we call those negative numbers.

The line looks like this
__-5__-4__-3__-2__-1__0__1__2__3__4__5__

Now adding and subtracting are much easier. To work out 2 + 3, we start at 2 and count 3 more spaces to the right, giving the answer 5. Going back to 5 - 7, we now start at 5 and count back 7 spaces to the left, which brings us to -2.

Answer: 3/4

Rational numbers are numbers which can be created by dividing one whole number by another one. If the numbers are both the same the answer is 1, so we will use different whole numbers. My first example (in question 1) was 6 / 3 = 2, which is quite straightforward - if we split 6 into groups of 3, we have 2 of those groups.

But what happens if we try it the other way up 3 / 6 = ? We cannot get a group of 6 out of 3 things. But we still want an answer. If we imagine sharing 3 pizzas between 6 people, the answer becomes obvious - we cut each pizza in half and then we have a piece of pizza for each person. The same thing happens with a birthday cake. If 1 cake is to be shared equally between 15 children at the party, each child should have 1/15 of the cake.

We can use other shapes. With a box of 12 chocolates arranged in 4 rows of 3, it is easy to see that taking 1/4 of the chocolates (3) out, leaves 3 /4 of the chocolates (9).

Answer: e

An irrational number is a number which is not a whole number, but cannot be created by dividing one whole number by another.

Probably the most well known irrational number is π - the Greek letter pi. It is defined as the ratio of the circumference of a circle to its diameter. No matter how large or small the circle, the result of that division is always the same. The number is very approximately 22/7, and more often written as 3.14, or even 3.14159265358979323846. But neither of those are accurate, the decimal places go on for ever.

Another example of an irrational number is the length of the diagonal of a square with a length of 1 on each side. Using Pythagoras' theorem, we know that the square of the length of the diagonal is 2, so the diagonal itself is the square root of 2 or √2, which is approximately 1.414213562373095, with the decimal places going on for ever, again.

The number I used for the question is Euler's number, commonly known as e, which is approximately 2.718281828459045, again having the decimal places go on for ever. e is defined as the limit of (1 + 1/n)^n as n approaches infinity, or as the sum of the infinite series 1/n!

While π may be better known than e, and √2 easier to understand than e, there was no danger of e not showing up correctly in the answer boxes.

Answer: 5i

An imaginary number comes from the solution to the equation a^2 = -1. The answer is not a = 1, because then a^2 would also be 1. The answer is not a = -1, because multiplying two negative numbers gives a positive answer so a^2 would again be 1. The imaginary number, designated as i, is the square root of -1, meaning that i^2 = -1. Thus the square, and all other even powers of an imaginary number are real. Odd powers of an imaginary number are also imaginary.

Answer: 3 + 4i

There is no reason why a number has to be either real or imaginary. A complex number has two parts - a real part and an imaginary part. In fact all numbers can be regarded as complex, but in many cases the imaginary part is 0i.

The usual arithmetic operations can be performed on them. All the answers should result in a standard complex number, with a real part and an imaginary part.

Addition and subtraction are quite straightforward, with operations being performed on the real and imaginary parts separately. For example
(3 + 4i) + (5 + 7i) = (3 + 5) + (4i + 7i) = 8 + 11i
(5 + 7i) - (3 + 4i) = (5 - 3) + (7i - 4i) = 2 + 3i

Multiplication is a little more complicated. Remember that i^2 = -1.
(3 + 4i) x (5 + 7i) = 3(5 + 7i) + 4i(5 + 7i) = 15 + 21i + 20i + 28i^2 = 15 + 41i - 28 = -13 + 41i

Division is even more interesting. To calculate (3 + 4i) / (5 + 7i), it is necessary to multiply both the top and the bottom of the fraction by (5 - 7i) to leave a real number as the denominator.
(3 + 4i) / (5 + 7i)
= [(3 + 4i) x (5 - 7i)] / [(5 + 7i) x (5 - 7i)]
= [15 - 21i + 20i - 28i^2] / [25 - 35i + 35i - 49i^2]
= [43 - i] / 74
= 43/74 - i/74