BIDMAS

BIDMAS - order of operations
(document yet to be checked by Mathagony Aunt)

Mathematics, unlike English, is not always read from left to right. The different operations have to be carried out in a particular order. This is where the mnemonic BIDMAS comes in:

B rackets
I ndices
D ivision
M ultiplication
A ddition
S ubtraction

To find the solution to a mathematical expression, the operators have to be worked through in order - the BIDMAS order. Having said that, there is a very IMPORTANT EXCEPTION, which I will show a little later. Also be careful to watch for negative values, such as 3×-2. Here the operator is ×, the first number is 3 and the other number is -2.

In general, BIDMAS gives the correct order of tackling an expression. First, look for Brackets. Work out any expression in the brackets. For example, given the expression "3(4-2)", we have to work out the 4-2, which is inside the brackets, before tackling the multiplication by three, which is outside the brackets. The 4-2 gives an answer of 2, and we can replace this in the backets: 3(2). Now we can replace the brackets. As the "3(2)" means "3×(2)", we now get 3×2, which is 6.

Indices tell us how many times to multiply a number by itself. Many of use are familiar with SQUARE numbers, that is numbers that are multiplied by themselves, such as 3×3, which is often written as 3². The superscript 2 tells us to multiply three by three. If the superscript 2 was a five then we would multiply five threes together: 3×3×3×3×3 (the answer is 243). However, in this progam we cannot show superscript very easily, so we use the ^ symbol (found above the 6 key). We would write these two examples as 3^2 and 3^5. We have to work out indices before after brackets and before multiplication.

Take this expression: 3(18-4^2)^4. Rather nasty to start with, but one step at a time.

First look inside the brackets: 18-4^2. Work this out. Although the subtraction is before the indices the indices have to be worked out first, so work out 4^2. 4^2 =16. Replace into the sum to get 18-16, which is 2. There are no more operations in that set of brackets, so replace the brackets. The expression now becomes 3×2^4.

Again, do the indices first, to get 2^4, which is 16. Now replace the 2^4 with 16 and do the final multiplication. We get 3×16=48.

Had we done things differently we may have done 18-4 to get 14, squared the 14, to get 196. Multiplied 196 by 3 to get 588 and then raise 588 to the power of 4, to get 1.19...×10^11.Horrible number!

Now we work through any divisions and multiplications. Divisions have to be done first. Look at this example 12÷4÷2×4. If the divisions are carried out first (from left to right) and replacing the answer,we get 12÷4=3, then 3÷2=1.5, then 1.5×4=6. Doing the multiplication first gives 2×4=8, and the expression would become 12÷4÷8, giving 12÷4 = 3, then 3÷8=0.375 - a completely different answer!

THE VERY IMPORTANT EXCEPTION

There is an exception to BIDMAS, and it is found right at the end when only additions and subtractions remain. When there are ONLY, I repeat, ONLY additions and subtractions remaining then you can work left to right. I'll show you why.

Suppose I get an expression 4+6-4+8. If I do the additions first I will replace 4+6 with 10 and 4+8 with 12, to get the expression 10-12. The answer to which is -2. Now working from left to right, I start off with 4 units and add to that 6 units. This gives me ten units. I now take away 4 units to leave me with 4 units. Finally I add another 8 units. This gives me 14 units in all.

This program allows you to enter an equation and see what the result is. Use your keyboard to enter letters (variables) and numbers. There are buttons at the top to enter brackets etc.

I have yet to get a check everything since the programmer has updated the program to work with exponents. The symbol ^ indicate raise to the power, so 2^6 means 2 to the power 6 or 2×2×2×2×2×2 or 64.