Arithmetic Expressions

A phrase like 13 + 2, 20 - 4, or 12 × 5 is called an arithmetic expression. Every arithmetic expression has a single value — the number it works out to. The expression 13 + 2 has the value 15, and we write this using the equals sign: 13 + 2 = 15. The expression itself (the phrase) and its value (the number) are two different things, connected by '='.

Because every expression boils down to a single value, you can compare two expressions exactly the way you compare two numbers — using '<', '>' or '='. For example, compare 13 + 9 and 14 + 6. Working each one out, 13 + 9 = 22 and 14 + 6 = 20. Since 22 is bigger than 20, we can write 13 + 9 > 14 + 6. You don't always need to fully calculate both sides, either — sometimes reasoning about how much bigger or smaller one side is compared to the other gets you the answer faster, especially when the numbers involved are close to each other.

This idea of expressions having values, and being comparable, is the foundation for everything else in this chapter: once you can reliably find the value of any expression — however many operations it contains — comparing, simplifying and rewriting expressions all become much easier.

When an expression has more than one operation and no brackets, we need a consistent rule for how to read it — this is where the idea of terms comes in. The terms of an expression are the parts separated by a '+' sign. In 12 + 7, the terms are 12 and 7. But what about an expression with a minus sign, like 83 - 14? Since subtracting a number has the same effect as adding its negative (its inverse), we can rewrite 83 - 14 as 83 + (-14). Checking this: 83 - 14 = 69, and 83 + (-14) = 69 as well — they match. So the terms of 83 - 14 are 83 and -14.

Once every subtraction in an expression is rewritten this way as an addition, something powerful becomes true: the terms can be added in any order (this is called the commutative property of addition) and grouped in any way (the associative property of addition) without changing the final value. For example, 423 + 419 gives exactly the same value as 419 + 423 — swapping two terms never changes a sum.

This also means you can rebalance an equation by adjusting both sides together. Suppose you want to fill 423 + ___ = 419 + ___ with two different numbers. Since 419 is 4 less than 423, the blank on the right must be 4 more than the blank on the left to keep both sides equal. For instance, 423 + 1 = 424 and 419 + 5 = 424 — both sides land on the same value because the right side's blank compensated for 419 being smaller to begin with.

Many students know the order-of-operations rule as BODMAS or VBODMAS (Vinculum, Brackets, Of, Division, Multiplication, Addition, Subtraction) — do brackets first, then multiplication/division, then addition/subtraction. The 'terms' approach in this chapter is the same rule explained more carefully: an expression without brackets is first split into terms (at the '+' signs, after converting any '-' into a negative term), each term is evaluated on its own using its multiplications and divisions, and only after that are the resulting terms added together.

Take 30 + 5 × 4. This has two terms: 30, and 5 × 4. Evaluate the term 5 × 4 first, getting 20. Then add: 30 + 20 = 50. This is why 30 + 5 × 4 = 50 and not 140 (which you'd wrongly get by adding 30 + 5 = 35 first, then multiplying by 4) — multiplication inside a term always happens before the terms are added.

A slightly longer example: 7 × 4 + 9 × 6 has two terms, 7 × 4 and 9 × 6. Evaluating each separately: 7 × 4 = 28, and 9 × 6 = 54. Adding the terms: 28 + 54 = 82. Even expressions with several multiplications and additions mixed together become manageable once you spot the terms first and evaluate each one completely before doing any adding.

Brackets tell us exactly which part of an expression to evaluate first, which avoids any ambiguity. But once you understand terms, you can often remove brackets altogether and get an equivalent, bracket-free expression. The rule depends entirely on the sign sitting right before the bracket.

If a '+' sign comes before the bracket, the terms inside keep their original signs once the bracket disappears. For example, 28 + (35 - 10) can be rewritten as 28 + 35 - 10, with no sign changes — both work out to 53 (check: 28 + 25 = 53, and 28 + 35 - 10 = 53). This makes sense because '+' just means 'add whatever is inside,' regardless of how that inside amount was built up.

But if a '-' sign comes before the bracket, every term inside flips its sign when the bracket is removed. For example, 28 - (37 + 12) becomes 28 - 37 - 12, with both terms inside switching from positive to subtracted. Checking: 28 - (37 + 12) = 28 - 49 = -21, and 28 - 37 - 12 = -21 as well — they match. This sign-flip happens because subtracting a whole bracketed amount means taking away every part of what's inside it, not just the first part.

Consider 14 × 10 and 6 × 10 added together, as 14 × 10 - 6 × 10. Rather than computing each multiplication separately (140 and 60) and then subtracting (140 - 60 = 80), notice that both terms share the common factor 10. We can factor it out: 14 × 10 - 6 × 10 = (14 - 6) x 10 = 8 × 10 = 80 — the same answer, reached with a single, easier multiplication. This shortcut is called the distributive property: a number multiplied by a sum (or difference) equals the sum (or difference) of that number multiplied by each part separately.

The distributive property is especially handy for multiplying by a number that's just above or below a round number. To find 15 × 104, rewrite 104 as (100 + 4): 15 × 104 = 15 x (100 + 4) = 15 × 100 + 15 × 4 = 1,500 + 60 = 1,560. This avoids multiplying 15 by 104 directly, which is far more error-prone to do in your head.

The same idea works in reverse, to simplify an expression rather than just to multiply quickly. If you already know an expression like 83 - 37 - 12, you can check whether a different-looking expression equals it by rewriting both using terms. For instance, 84 - 38 - 12 also equals 34 — same as 83 - 37 - 12 — because 84 is one more than 83, but 38 is also one more than 37, so the two changes cancel out and leave the value unchanged.