IRRATIONAL NUMBERS
TIP: When doing these sort of problems, remember:
a) When you convert a fraction into a decimal you end up with either a terminating decimal or a recurring decimal. Numbers which can be expressed as a fraction are called rational numbers.
1/4 = 0.25 (Terminating decimal)
1/3 = 0.33333..... (Recurring decimal)
b) Numbers
which cannot be expressed as a fraction are called irrational numbers. These
numbers go on for ever without any apparent pattern e.g. p,
Ö2, Ö7
etc.
p has been calculated by computers to millions
of decimal places and still no pattern has been found!
Square roots
of all non-squares are irrational.
Cube roots of all non-cubes are irrational.
c) The product of 2 irrational numbers can sometimes be a rational number. Remember one of the examples below:
Ö3
X Ö12 = Ö36
= 6
Ö5 X Ö7
= Ö35 = 5
d) Expressions
which include at least 1 irrational number are called surds.
e.g. Ö2, -5 ±
Ö7 etc.
Solve the following:
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1. Find an irrational number between 6 and 7. |
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2. Write down a rational number and an irrational number between 3.1 and 3.2 |
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3. Find a fraction equivalent to 0.545454545454.....
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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. |
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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.
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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.
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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.
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8. 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 2y where y is a rational number. d) If 4t+1 = Ö8, what is the value of t?
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