1. Find an irrational number between 6 and 7.

As 6² = 36 and 7² = 49

The square root of any number between 37 and 48 can be used as the answer e.g. Ö40

Take the number half-way between two rational numbers to get another rational number i.e. 3.15. (note there are lots of others also, but this is the easiest way to find one).

Ö10 is an example of an irrational number between 3.1 and 3.2

3.  Find a fraction equivalent to 0.545454545454.....

Let x = 0.5454545454.......

and multiply by 100 (there are 2 digits '54' in each recurring bit - if you are looking at e.g. 0.356356356...., then multiply by 1,000 and so on).

100x = 54.5454545454

Subtract one from the other giving:

100x - x = 54

99x = 54

x = 54/99 = 6/11

4.  Write down one example of each of the following:

a) A rational number between 3/4 and 4/5

b) An irrational number y such that y³ is rational.

c) An irrational number less than 1.

a) Find a common denominator for both fractions - the easiest way to do this is to multiply the bottom numbers in each fraction together i.e. 4 x 5 = 20.

Now work out each as a fraction of 20:

Multiply top and bottom of 3/4 by 5 giving 15/20
Multiply top and bottom of 4/5 by 4 giving 16/20

This doesn't help so multiply each by 2:

2 x 15/20 = 30/40 and 2 x 16/20 = 32/40

Instantly giving us 31/40 as a perfect answer.

b) The cube root of 2 written as 3Ö2 is a possiblity (along with many others).

c) Think of the obvious here - forget about looking for obscure fractions - instead go into negative numbers!

e.g. -Ö5

If you insist on looking for fractions try 1/Ö2.

5. a) Which of the following numbers are irrational?

Ö4¼,    Ö6¼,    ¹/₃ + Ö3,   (¹/₃Ö3)²

b) Express the remaining rational numbers in a) in the form p/q where p and q are integers.

a) Work them out on your calculator. Any which have recurring or terminating decimals are Rational.

Hence Ö4¼ , ¹/₃ + Ö3 and (¹/₃Ö3)² are irrational.

b) Ö6¼ = Ö25/4 = Ö25 / Ö4 = 5/2

6. If    a = 2 + Ö5    and   b = 2 - Ö5

a) Calculate a - b and state whether your answer is rational or irrational.

b) Calculate the product ab and state whether your answer is rational or irrational.

a) This question is easier than it looks:

a - b = (2 + Ö5) - (2 - Ö5)

2 - 2 = 0, leaving us with Ö5 - (- Ö5)
which = Ö5 + Ö5 = 2Ö5 (Irrational)

b) (2 + Ö5) (2 - Ö5)

= 4 - 2Ö5 + 2Ö5 - (Ö5)²

The - 2Ö5 + 2Ö5 = 0 and the square of a square root = the number)

= 4 - 5 = -1 (Rational)

7. The numbers p and q are irrational and not equal.

a) If p + q is rational, what are possible values for p and q?

b) If two other numbers a and b are irrational and not equal also. Write down possible values for a and b if the product ab is rational.

a) Possible values are 1 - Ö5 and 1 + Ö5. along with loads of others. Don't just think in terms of single roots e.g. Ö5, Ö8 etc. or you'll never work out the answer - it's actually quite easy when you know how!

b) From the "Ones to remember" list, possible examples are:

Ö3 X Ö12 = Ö36 = 6
Ö5 X Ö7 = Ö35 = 5

8.
a)What does the statement "Ö8 is an irrational number" mean?

b) If n is a rational number and k x Ö8 = n, work out a non-zero value of n and a non-zero value of k which satisy this equation.

c) Write Ö8 in the form 2^y where y is a rational number.

d) If 4^t+1 = Ö8, what is the value of t?

a) There are two possible answers here:

"cannot be expressed in the form p/q where both p
and q are integers".

or

"is neither recurring nor terminating"

b) The obvious value for k = Ö8, because:

Ö8 x Ö8 = Ö8x8 = Ö64 = 8

making n = 8

c) Ö8 is the same as Ö2³ = 2^3/2 or 2^1.5.

d) 4³ = 64 and Ö64 = 8

we need to do a square root again to get Ö8

hence 4^{3/4 = Ö8 (know your indicies here...)}

3/4 = 1 - 1/4 hence t = -1/4