ࡱ> ,.+~q` bjbjqPqP .D:::trrrrd,* % """ % % % % % % %$X&h(/%-"!""""/% K\%$$$"8  %$" %$$$  t~@r1"l$ %r%0%$\)#\)$\)$P""$"""""/%/%$"""%""""* * * d* * * T\6 Rational and Irrational Numbers A rational number can be written as fraction, so not surprisingly an irrational number can not be written as a fraction. 1.5 is rational because it can be written as  EMBED Equation.DSMT4  as is 6  EMBED Equation.DSMT4 and 0.23  EMBED Equation.DSMT4  All integers are rational. An irrational number will be a decimal which goes on forever (non-terminating) without ever repeating. The main examples of irrational numbers are surds (see later) and multiples of and . We can express irrational numbers in terms of  and surds e.g. Write down an irrational number between 7.2 and 7.3  = 3.141& & & so  + 4.1 will be irrational and between 7.2 and 7.3  EMBED Equation.DSMT4  will be irrational and lie between 7.2 and 7.3 Writing a Non-Terminating Recurring Decimal as a Fraction If a decimal goes on forever (non-terminating) but repeats, then it is rational i.e. it can be written as a fraction. The obvious example is 0.333333333 which is  EMBED Equation.DSMT4  Some numbers are not quite so obvious but can still be written as fractions e.g. Write 0. EMBED Equation.DSMT4  as a fraction Let x = 0.444444.. so 10x = 4.4444444.. 10x x = 4 = 9x If 9x = 4 then x =  EMBED Equation.DSMT4  but as x = 0. EMBED Equation.DSMT4  0. EMBED Equation.DSMT4 =  EMBED Equation.DSMT4  e.g. Write 3. EMBED Equation.DSMT4  as a fraction We know that 0. EMBED Equation.DSMT4  =  EMBED Equation.DSMT4  so 3. EMBED Equation.DSMT4  must equal  EMBED Equation.DSMT4  e.g. Write  EMBED Equation.DSMT4  as a fraction  EMBED Equation.DSMT4  Let x= EMBED Equation.DSMT4  so 100x = 23.232323. 100x x = 23.23232323. 0.23232323.. = 23 = 99x If 99x = 23 then x =  EMBED Equation.DSMT4  but x = 0.23232323.. So  EMBED Equation.DSMT4  =  EMBED Equation.DSMT4  After a while you can just recognise the answers if the recurring part of the decimal is next to the decimal point  EMBED Equation.DSMT4  However you must remember the long-winded method as you may be asked to prove that, for example,  EMBED Equation.DSMT4  =  EMBED Equation.DSMT4  If the recurring part is not next to the decimal point then we have to split the recurring and non-recurring parts and use a fancy bit of division/multiplication to put the recurring part next to a decimal point. This is probably going to appear on the Higher paper. e.g. Write 0.2 EMBED Equation.DSMT4  as a fraction 0.2 EMBED Equation.DSMT4  = 0.2 + 0.0 EMBED Equation.DSMT4  Split into recurring and non-recurring parts = 0.2 + 0. EMBED Equation.DSMT4  10 Move recurring part next to decimal point   EMBED Equation.DSMT4  e.g. Write 0.41 EMBED Equation.DSMT4  as a fraction 0.41 EMBED Equation.DSMT4   EMBED Equation.DSMT4  You can check your answer by converting the fraction to a decimal by dividing and seeing if you get the starting value e.g.  EMBED Equation.DSMT4  Surds Surds are the square root of a prime number (or a multiple thereof). They are always irrational. It is more accurate to write your answer as a square root rather than as a decimal if you are dealing with surds particularly useful in situations using Pythagoras Theorem There are two laws of surds which are often used to simplify expressions: Law 1  EMBED Equation.DSMT4  (and vice versa) Law 2  EMBED Equation.DSMT4  e.g. Simplify  EMBED Equation.DSMT4   EMBED Equation.DSMT4  The trick is to look for the largest square number (the square root of which will always be an integer) which will divide into your expression. e.g. Simplify  EMBED Equation.DSMT4   EMBED Equation.DSMT4  Rationalising a Fraction If we rationalise a fraction containing a surd, we write it so that there are no surds in the denominator (bottom). There will probably be some surds left in the numerator. Rationalising does not mean get rid of all the surds. The secret to rationalising is to use equivalent fractions and multiply by one which will not change the value of our expression (we just write one in a very clever way) e.g. Rationalise  EMBED Equation.DSMT4   EMBED Equation.DSMT4  e.g. Rationalise  EMBED Equation.DSMT4   EMBED Equation.DSMT4  Higher Level If you have a more complicated denominator then you need to be aware of the difference of 2 squares i.e. a2 b2 = (a b)(a + b) e.g. If you have to rationalise  EMBED Equation.DSMT4  there is no point multiplying by  EMBED Equation.DSMT4  as when you multiply out the denominators you will get:  EMBED Equation.DSMT4  This means we will still have a surd in our denominator so it has not been rationalised The difference of 2 squares would suggest that we multiply by  EMBED Equation.DSMT4   EMBED Equation.DSMT4  There are no surds in the denominator so it has been rationalised. You may prefer to write it without expanding the numerator i.e.  EMBED Equation.DSMT4  Past Paper Questions   EMBED Equation.DSMT4    EMBED Equation.DSMT4  Write as fractions Perform the division Find common denominator and re-write using equivalent fractions Add up !#,fp     + , - .  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