Arithmetic Expressions

Why This Matters

Think about a normal day at a shop.

You buy a biscuit packet for ₹15 and a packet of dal for ₹56. You hand over ₹100. How much change do you get back?

In your head you are already doing maths. You add ₹15 and ₹56 to get the total. Then you take that total away from ₹100. You get ₹29 back.

Now look at how we might write that on paper. We could write it as 100 − 15 + 56. But wait. If we just go left to right, we get 100 − 15 = 85, then 85 + 56 = 141. That says the shopkeeper gives you more money than you paid! That is clearly wrong.

So writing maths down is not enough. We also need clear rules for how to read it. Otherwise two people can read the same line and get two different answers.

This chapter is about exactly that. We will learn what an arithmetic expression is, how to read and write one, and the rules that make sure everyone gets the same answer. These rules are the grammar of maths. Once you know them, you will never be tricked by a tricky-looking line of numbers again.

The Big Idea

An arithmetic expression is just numbers joined by the operations +, −, × and ÷. Every expression has one value — the single number it works out to. But the same line of numbers can be read in different ways, so maths uses two tools to keep things clear: brackets ( ) tell you what to do first, and the idea of terms tells you which parts stay together. Once you know these rules, every expression has exactly one correct value — and you can often compare two expressions, or regroup them, without doing all the hard calculation.

Let’s Break It Down

Before we start, let us quickly refresh the four basic operations. We will use all of them, so it helps to have them fresh.

What is an arithmetic expression?

An arithmetic expression is a phrase made of numbers joined by operation signs. Here are some:

13 + 2, 20 − 4, 12 × 5, 18 ÷ 3.

Each of these is an expression. Each one stands for a number.

The number an expression works out to is called its value. For example, the value of 13 + 2 is 15. We use the equals sign = to join an expression to its value:

13 + 2 = 15

We can read 13 + 2 out loud as “13 plus 2” or as “the sum of 13 and 2”. Figure 2.1 below puts the expression and its value side by side, so you can see the difference clearly.

Tap to zoom

One more nice thing: many different expressions can have the same value. Here are four different expressions, and every one of them has the value 12:

10 + 2, 15 − 3, 3 × 4, 24 ÷ 2.

So a single number, like 12, can be written in many ways. That freedom is useful, as we will see.

Terms of an expression

When an expression mixes operations, we need a way to see its parts. That is what terms are for.

Terms are the parts of an expression separated by + signs. You break the expression at each + sign, and each piece is one term.

Take 12 + 7. The + sign splits it into two terms: 12 and 7. Easy.

Now take 30 + 5 × 4. There is only one + sign here. So there are two terms: 30 and 5 × 4. Notice the second term is the whole thing 5 × 4, kept together. A × (or ÷) does not split a term — only a + does. Figure 2.2 shows this splitting.

Tap to zoom

But what about subtraction? Where are the + signs in 83 − 14?

Here we use the handy fact from the recap: subtracting a number is the same as adding its negative. So:

83 − 14 = 83 + (−14)

Now there is a + sign, and we can split. The two terms of 83 − 14 are 83 and −14. The minus sign sticks to the number it sits in front of.

Let us see this turn-a-minus-into-a-plus trick on a slightly longer expression.

Brackets and what they change

Go back to the shop problem from the start. We had 30 + 5 × 4 (Mallesh had 30 marbles; Arun brought 5 bags of 4). What is its value?

One student, Purna, read it left to right. He added 30 + 5 to get 35, then did 35 × 4 = 140. Another student, Mallesh, did 5 × 4 = 20 first, then 30 + 20 = 50. They got different answers — 140 and 50 — from the same line!

This is the confusion brackets are made to fix. Brackets ( ) tell you which part to do first. Whatever sits inside the brackets, you work out before anything else.

If we mean “do 5 × 4 first”, we write:

30 + (5 × 4) = 30 + 20 = 50

If instead we mean “do 30 + 5 first”, we write:

(30 + 5) × 4 = 35 × 4 = 140

Same three numbers. Same signs. The only difference is where the brackets sit — and the answer jumps from 50 to 140. Figure 2.3 puts the two versions side by side so you can see how much the brackets matter.

Tap to zoom

Brackets also rescue our shop change problem. Remember 100 − 15 + 56 went wrong and gave 141. With brackets we write what we really mean — take away the total cost:

100 − (15 + 56) = 100 − 71 = 29

Now the bracket adds 15 + 56 first, then we subtract. We get the right answer, ₹29.

The order of operations (and why the order matters)

So far we have used brackets to say what comes first. But what if there are no brackets, like in plain 30 + 5 × 4? We still need one fixed answer.

Maths solves this with a fixed order of operations. It is a short ladder of rules. You always go from the top of the ladder down:

1. Brackets first. Work out anything inside ( ).
2. Then × and ÷. Do all multiplication and division next.
3. Then + and −. Add and subtract last, working left to right.

Figure 2.4 shows this ladder. Always start at the top step and work down.

Tap to zoom

Now, why is the order set up this way — why × before +? It is not a random rule someone invented. It matches what the numbers actually mean.

Remember, × is repeated addition. In 30 + 5 × 4, the part 5 × 4 means “five fours added up”, which is one single quantity: 20. It belongs together as one term. So it makes sense to work it out first, then add it to the 30. This is exactly the “terms” idea from before: the terms are 30 and 5 × 4, you find each term’s value, then add the terms.

This is the deep reason: doing × and ÷ first is the same as finding the value of each term first, and then adding the terms. The order rule and the terms idea are really the same idea.

Let us use this on the marble problem, carefully.

Let us try one with everything in it — brackets, a ×, and a +.

Reading expressions from real situations

A great way to be sure of an expression is to picture a real story behind it. The story tells you what to do first, all by itself.

Look again at the marble story. Mallesh already had 30 marbles. Arun brought 5 bags with 4 marbles in each bag. Arun’s marbles are 5 × 4 = 20. The total is 30 + 20 = 50. The picture makes it obvious that the 5 × 4 belongs together — you would never add 30 and 5 first, because that mixes loose marbles with the number of bags. Figure 2.5 shows the story turning into its expression.

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Stories also help you compare without heavy calculation, and they help you choose between expressions. Here is a worked example using a real bill.

Notice we did not need brackets here. The terms idea already keeps 4 × 23 together, so 4 × 23 + 5 correctly means “92, then add 5”.

Comparing expressions

Just as we compare plain numbers with =, < and >, we can compare two expressions. We simply compare their values.

For example, 10 + 2 is greater than 7 + 1, because 10 + 2 = 12 and 7 + 1 = 8, and 12 is bigger than 8. So we write:

10 + 2 > 7 + 1

The clever part is that you can often compare two expressions without fully working them out. You just reason about how they differ. This saves a lot of effort. Picture a real story to feel why.

The same trick works for subtraction. Let us see one.

There is one more comparing idea worth knowing, and it feels almost magical: in an expression with only additions, you can add the terms in any order and the value never changes. For example, (−7) + 10 + (−11) gives −8 no matter which two you add first. Grown-up maths calls “swapping does not change the sum” the commutative property, and “grouping does not change the sum” the associative property. You do not need the big names — just remember that for additions, order and grouping are free.

This is why Manasa, who added a long list of numbers and then realised she forgot to include 9055, does not have to start all over again. She can simply add the missing 9055 to her earlier total. The order of adding never changes the result.

Common Mistakes

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Practice Problems

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Medium

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Summary

• An arithmetic expression is numbers joined by +, −, × or ÷. Its value is the single number it works out to, joined by an = sign (for example, 13 + 2 = 15).
• Many different expressions can share the same value (10 + 2, 15 − 3, 3 × 4 and 24 ÷ 2 are all 12).
• The terms of an expression are its parts separated by + signs. A × or ÷ stays inside one term. A subtraction is read as “add a negative”, so 83 − 14 has terms 83 and −14.
• Brackets ( ) tell you what to do first. Moving the brackets can change the value: 30 + (5 × 4) = 50 but (30 + 5) × 4 = 140.
• The order of operations is a ladder: brackets first, then × and ÷, then + and −. Doing × and ÷ first is the same as finding each term’s value, then adding the terms — so 30 + 5 × 4 = 50.
• A real-life situation often shows you the right expression and the right order, because the story tells you which parts belong together.
• You can compare two expressions just by their values, often without full calculation, by reasoning about how they differ.
• For additions only, order and grouping do not change the sum — so a forgotten number can just be added in at the end.

What’s Next

You now know how to read, write, compare and correctly evaluate expressions made of whole numbers. But what happens when the numbers are not whole — when there is something beyond the point, like 3.6 or 12.75?

In Chapter 3 — A Peek Beyond the Point, you will step past the decimal point and explore decimal numbers: what the digits after the point really mean, how to compare them, and how to add and work with them. Everything you learned here about value, terms and order will carry straight over.