How To Evaluate Mathematical Expressions by T E Miller BSc

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During the years I was teaching mathematics, both privately and in schools, I found that many students and even teachers struggled with evaluation of expressions. The use of BODMAS (means evaluate first brackets, then Of then Division, then Multiplication, then Addition, then Subtraction) caused considerable confusion, even amongst teachers.
 Now there is one advantage of studying computer science (which I did) and that is that you have to tell the computer what to do step by step. The computer scientists therefore took pains to tell us how to evaluate mathematical expressions so we would be able to write programs that could  calculate expressions.
 Now if one uses BODMAS on 5 – 2 + 3 we might think that A (addition) comes before S (subtraction) and we might therefore say that 5 – 2 + 3 is 5 – 5 (we did the addition of 2+3)
This is NOT correct. In fact 5 – 2 + 3 is equal to 3 + 3 (we correctly did S before A)
In fact 5 – 2 + 3 = 3 + 3 = 6 (use a scientific calculator to verify that this is correct).
 In fact the correct rules are:
1) First do brackets
2) Then Exponents
3) Then multiplication and division
4) Then addition and subtraction
How do we know whether to do addition or subtraction first? The answer is that we must read from left to right. We say that addition and subtraction have equal precedence and we read from left to right.
 We also apply this to multiplication and division. They have equal precedence and we read from left to right.
 Therefore 6 divided by 3 times 5 
 is 2 times 5. Division was done before multiplication because we read from left to right when we have operators of equal precedence
 The answer is 10 (check with a scientific calculator if you like).
 A calculator that is not a scientific one can give you a wrong answer as they often do not have a correct logical way of evaluating.
 What if we have brackets within brackets? Answer: Evaluate innermost brackets first.
 Therefore 5- (2+3(3-1)) = 5 – (2+3(2)) = 5-(2+6) In this last step multiplication was correctly done before addition.

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