For anyone who still cares, I?m going to attempt to explain all the answers to all the questions from this thread.
The way calculations work is
B
Do the calculations in Brackets/Parentheses first (if there are any)
I
If any, do the Indices (aka Powers, Orders, Exponents - squared, square root, cubes)
DM
Multiplication or Division, calculate them next reading left to right
AS
Finally, Add or Subtract the numbers reading left to right
(Note : Lougle?s method - technically division doesn?t precede multiplication, but you do either one reading left to right, same for addition/subtraction)
The letters give rise to BIDMAS (currently preferred in UK schools c.1995), PEMDAS/PEDMAS (USA) BODMAS/BOMDAS (UK c.1960, widespread by 1980, replaced by BIDMAS as I for Indices is a much more widely used term than O for order).
The acronyms to remember them by have only been around for about 4 generations, but the priority ranking of these operations has always been, here?s why:
Addition (and its opposite, Subtraction) is the most basic operation.
When you Multiply, it?s a shorthand for repeated Addition ?
Instead of writing 4 + 4 + 4 + 4 + 4 + 4, you write 4 x 6.
The latter is a quick way of writing the former
Similarly, indices are a shorthand for repeated multiplication,
Instead of writing 4 x 4 x 4 x 4 x 4 x 4, you write 4 to the power 6
So effectively, when you are calculating, you need to unpack the Indices (the highest operation), followed by unpacking the multiplication (or it?s opposite, division), before you get to the adding and subtracting.
40 + 40 x 0 + 1
means
40 plus 40 zeroes plus 1
In its long form, it would be
40 + 0 + 0 + 0 +... (40 times) + 1
New calculators (from the last 10/15 years) will do the order of operations for you. Basic calculators, like the ones on phones, wont. The easiest way to tell whether your calculator does it is to press the square root button, then 25 and then equals. An old calculator will still say 25, a new calculator will say 5 (the right answer).
Of course the old calculator can do the calculation, but you have to type things in a more complicated way (for harder sums).
Dividing by zero
the answer is undefined for normal arithmetic rules, and that includes zero divided by zero. Check Wikipedia en.wikipedia.org/wiki/Division_by_zero under the ?in algebra? section ? for certain fields, infinity is sometimes given as the answer but as a general principle, dividing by zero does not given a specified result.
5 + 5 + 5 - 5 + 5 + 5 - 5 + 5 x 0 =
- Multiply the 5 zeros to get zero (5 x 0 = 0),
it becomes 5 + 5 + 5 - 5 + 5 + 5 - 5 + 0
- Now we?re down to Addition and Subtraction left to right,
so just keep a running total
5
- 5 = 10
+5 = 15
- 5 = 10
+5 = 15
+5 = 20
-5 = 15
+ 5 zeroes still 15
So the answer is 15. (The other methods where you group together all the additions and subtractions are equivalent, and possibly easier)
48/2(9+3)
Here is a good example of why you should use
B Brackets
I Indices
DM Division OR Multiplication, reading left to right
AS Addition OR Subtraction, reading left to right
Using the rules:
- after doing 9+3=12, you?d see
48/2x12
2)And if you do division and multiplication left to right, that makes 48/2 = 24
then 24 x 12 = 288
Division and Multiplication are the same level of operation (one is the opposite of the other) so you read them left to right.
Some mathematicians will disagree, here is one article on it thecorridortheory.blogspot.com/2011/05/ending-48293-debate.html, but if Wolfram Alpha says it's 288, it's 288.
Final note : Wolfram Alpha does say that dividing by zero gives infinity! But if you look at the type of infinity it gives, it is one which is undefined. For normal arithmetic, dividing by zero is undefined...
I should really go to bed now...