Why brackets are needed

Some expressions can be read in more than one way. Language has the same problem:

• Shalini sat next to a friend with toys. (The friend has the toys.)
• Shalini sat next to a friend, with toys. (Shalini has the toys.)

One comma changes the meaning.

Mallesh brought 30 marbles. Arun brought 5 bags with 4 in each. Mallesh wrote 30 + 5 × 4. Purna, who didn't know the story, added 30 + 5 = 35 first, then × 4, and got 140. Mallesh multiplied 5 × 4 = 20 first, then added 30, and got 50. Mallesh is right, but Purna's mistake is understandable: the expression alone doesn't say what to do first.

In maths, brackets do the job of the comma. 30 + (5 × 4) says: work out 5 × 4 first. The rule: work out what is inside the brackets first, then the rest. 30 + (5 × 4) = 30 + 20 = 50.

Brackets can save you from nonsense. Irfan buys biscuits for ₹15 and toor dal for ₹56, and pays with ₹100. His change is 100 minus the whole bill:

• 100 − (15 + 56) = 100 − 71 = ₹29. Correct.
• 100 − 15 + 56, worked left to right, gives ₹141. The shopkeeper would hand back more than Irfan paid!

So a minus sign in front of a bracket takes away everything inside the bracket, not just the first number.