Intro to Maths for CS: Fractions: Numerical and Algebraic

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1 Intro to Maths for CS: Fractions: Numerical and Algebraic Joshua Knowles School of Computer Science, University of Birmingham Term 1, (Slides by John Barnden)

2 Textbook Parts Programme F.1, section on Fractions, Ratios and Percentages and Programme F.2, section on Algebraic Fractions NB: Study the stuff on ratios and percentages by yourselves

3 Notes about Adding and Subtracting Fractions What s the value of ? We need to get each fraction to have the same denominator (and still a whole number), so that we re dealing with basic units of equal size. It s a bit like adding together amounts in different currencies pounds, euros, dollars. Any common denominator that is a whole number will do for the purposes of adding and subtracting fractions, as long as the numerators stay as whole numbers, because we know how to add and subtract them. We can change a fraction without changing the proportion that it signifies by mutiplying or dividing top and bottom by the same amount anything we like, whole number or not, other than 0 or 1. (WHY IS THIS? Think it through.) So we could change to 5, say, if we wanted. (Just an example it isn t right for our particular question above.) and 85 5 are equivalent fractions.

4 Adding and Subtracting Fractions, contd. For our present purpose, dividing is not a plausible way to proceed, because it might make a denominator or numerator be a non-whole number. And multiplying by something that isn t a whole number has the same problem. So we multiply the top and bottom of each fraction by a (positive) whole number. The multipliers for each fraction will usually be different. We need a way of multiplying 18, 27 and 5 to get the same value. That value is some common multiple of 18, 27 and 5. The most obvious way: Just multiply 18, 27 and 5 to get the common multiple 240. To get from 18 to 240 we multiply it by 27 and 5, i.e. by 5. To get from 27 to 240 we multiply it by 18 and 5, i.e. by 90. To get from 5 to 240 we multiply it by 18 and 27, i.e. by 486.

5 Adding and Subtracting Fractions, contd. We were dealing with This now becomes which evaluates to BUT This can be simplified by dividing top and bottom by 9, to get 270 AND NOTE THAT all the new numerators (1755, 150 and 972) can be divided by 9. This is not a coincidence!

6 Adding and Subtracting Fractions, contd. The reason is that the common denominator was bigger than it need have been (in fact 9 times bigger). This was because we didn t take the least common multiple (LCM) of 18, 27 and 5 as the common denominator. Their LCM is only 270. We were dealing with This now becomes which evaluates directly to 270. We got rid of the extra factor of 9 everywhere, because 240 is BUT there is a DOWNSIDE: to find the multiplier for each original fraction, we need to DIVIDE its denominator into the common denominator we need to divide 18, 27 and 5 into 270 (unless we re clever to remember information about how we derived the LCM).

7 Adding and Subtracting Fractions, contd. So we have a TRADEOFF between (i) dealing with larger numbers throughout, and having extra simplifciation work on the result fraction at the end, and but having overall a simpler process no LCM-finding or denominator-division (ii) having smaller numbers throughout and avoidance of unnecessary final simplification work, but with the added need for LCM-finding and denominator-division.

8 Multiplying and Dividing Fractions: Caution/Reminder Multiplying top and bottom of a fraction by some number k is NOT the same as multiplying (or dividing) the whole fraction by k!!!! It just multiplies it by 1! Multiplication examples: = (multiply numerator and denominator, but by different things) 18 5= = (only multiply numerator by 5) Division examples: = = = = = 90 In division, we can take the reciprocal of the divisor (i.e. put it upside down) and then multiply.

9 Fractions with Strange Bits Numerator and denominator don t have to be positive integers: /5 2 φ tan 20 π 2 + tan 20 5 π 2 When a numerator or denominator is itself a fraction and/or negative, it may be beneficial to multiply top and bottom to get rid of the fraction(s) and/or negativity there. 2.8 /5 changing to 14 = 14 Other possibly beneficial sorts of manipulation: 5 2 changing to (multiplying top and bottom by 2) 1 (1+ 5)/2 to to 2(1 5) (1+ 5)(1 5) (this includes using (a + b)(a b) =a 2 b 2 ) = 2(1 5) 4 to 1 5 2

10 Adding/Subtracting Fractions with Strange Bits Idea is the same as with ordinary fractions: we find a convenient common denominator. Convenient means in part that each denominator divides the common denominator without giving a fraction as the result. We no longer have a clear notion of lowest common denominator. It s just that LCMs are convenient in the whole-number case. 7 π π = 7 4π2 +(5 ) 2 12π 2 If we had used 12π 2 2 as the common denominator, then the second numerator would have got multiplied by 2/π, introducinga fraction into a term within the top part of the answer, rather than being multiplied just by 2.

11 Algebraic Fractions Algebraic fractions are ones where the numerator or denominator contains a variable rather than just numbers. But they work on exactly the same principles as numerical fractions. 5(a+b) c d2 4x = (c d2 ) 20x(a+b) 5(a+b) c d2 4x = 5(a+b) 4x 12x = c d 2 5(a+b)(c d 2 ) 5(a+b) + c d2 10x = 6x +(a+b)(c d2 ) 10x(a+b) a a+b + b a b 1 2 = 2a(a b)+2b(a+b) (a2 b 2 ) 2(a 2 b 2 ) = a2 +b 2 2(a 2 b 2 )

What is a Fraction? Fractions. One Way To Remember Numerator = North / 16. Example. What Fraction is Shaded? 9/16/16. Fraction = Part of a Whole

What is a Fraction? Fractions. One Way To Remember Numerator = North / 16. Example. What Fraction is Shaded? 9/16/16. Fraction = Part of a Whole // Fractions Pages What is a Fraction? Fraction Part of a Whole Top Number? Bottom Number? Page Numerator tells how many parts you have Denominator tells how many parts are in the whole Note: the fraction