People younger then I were handed an acronym to help them
process equations such as the one above, this acronym is PEMBAS and the alternate which is BODMAS. The expansion of the acronyms are provided below:
·
PEMDAS: Parentheses, Exponents, Multiplication
and Division (from left to right), Addition and Subtraction (from left to
right).
·
BODMAS stands for Brackets, Order, Division,
Multiplication, Addition, and Subtraction.
It must also be noted that MATH is not a natural language.
In a natural language one can still get their point across if the sentence is
grammatically wrong, where as with math when you get the grammar wrong you get
the wrong answer.
For the question above some people get to the answer 1,
while others get to the answer 9. The answer is 1, and not 9, below are
examples of how people get to their answer. Things will be made a bit tricky
later on to prove the point.
Following the explanations we can return to the original
equation:
Right: 6 ÷ 2(1+2) = 6 ÷ 2(3) = 6 ÷ (2×3) = 6
÷ 6 = 1
Also right: 6 ÷
2(1+2) = 6 ÷ (2+4) = 6 ÷ 6 = 1
Wrong-1: 6 ÷
2(1+2) = 6 ÷ 2 × (1+2) = 6 ÷ 2 × 3 = 3 × 3 = 9
Wrong-2: 6 ÷
2(1+2) = 6 ÷ 2 + 4 = 3 + 4 = 7
Recalling the idea that equations within brackets, as part
of a larger equation are to be resolved first, brackets can be nested. In some
cultures there was an emphasis put on the brackets e.g. {5 + [5 × (1+2)]} = 20
or in the more modern form 5 + 5(1+2) = 20.
Now for the tricky part, in the original equation we are
going to declare that (1+2) equals ‘x’ and the answer will be provided, giving
us the two possible equations 6÷2x=1 and 6÷2x=9. These equations can be
expressed as shown below.
Right
6÷2x=1 is 6 =
1 multiply both sides by x thus arriving
at 6 = x OR x= 3
(2x) 2
Wrong
While 6÷2x=9 can be expressed as 6÷2×x=9 when ignoring the
influence of the brackets and then moving from left to right it can be seen
that the equation gets altered. This new equation would result in 3×x=9 and
while dividing both sides by three still results in x =3 there is a serious problem.
The problem is that both equations can’t be right seeing as
1 ≠ 9. For those of you who do not know or maybe forgot, ≠ means ‘not equal
to’.
The Proper
Perspective
As with most situations, laziness causes confusion and
mistakes. The most accurate way of writing the original equation would be “{6 ÷
[2 × (1+2)]} =?”. Know that when it comes to brackets, or parentheses, there is
also a hierarchal order of operations where the mathematician starts with the
round brackets, and then on to the square brackets and finally the curly or
brace brackets.
Most people who enjoy maths, or those unfortunate souls who
hate math yet have to learn it, known that (a+b)2= x results in a2
+ 2ab + b2 = x. While this shortcut formula can provide the answer
for x, does anyone remember why this comes about? Just to finish off the page
it will be shown below using the bracket types provided above.
(a+b)2= x
Expanding on the above this is what we get
[(a+b) × (a+b)] = x
Multiplying each element in the first brackets with each
element in the second bracket we get
[(a×a) + (a×b) + (b×a) + (b×b)] = x
Because a×b is equal to b×a we get
{(a×a) + [(a×b) + (b×a)] + (b×b)} = x
All of which results in
a2 + 2ab + b2 = x
Of course, if a=3 and b=5 it is simply easier to add them
together to get 8 and then return the answer of 64. Mathematics is not just
about playing with numbers, while that can be fun; mathematics also provides a
methodology and framework towards successful problem solving.
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