Operations and Properties

Take three numbers and add them: 2 + 3 + 4. You can compute (2 + 3) + 4 = 5 + 4 = 9. Or you can compute 2 + (3 + 4) = 2 + 7 = 9. Same answer either way.

Now do the same with subtraction: 2 - 3 - 4. Compute (2 - 3) - 4 = -1 - 4 = -5. Now compute 2 - (3 - 4) = 2 - (-1) = 3.

The two ways of grouping give different answers — five units apart. With addition you could regroup freely; with subtraction you cannot. This isn't a quirk of those particular numbers. It is something about subtraction itself — a property addition has and subtraction does not.

Arithmetic looks like one thing from the outside ("the four operations"), but the four are not interchangeable. They follow different rules, and the differences are not optional. This article is the operating manual: a tour of the rules each operation does and does not obey, plus the rules that govern equations and inequalities. Once you know which rule is which, school algebra stops feeling like memorisation. Solving an equation becomes "apply rules in the right order until x is alone" — and you stop guessing.

Closure: does the answer even live in the set?

Before any of the famous rules — commutative, associative, distributive — there is one so basic it usually goes unsaid. When you write 3 + 4 = 7, three things have to be true at once. You need a set of numbers you are allowed to use. You need an operation you are allowed to apply to two of them. And the answer has to land back inside the set. If the answer escapes the set, the operation didn't really work on it — you stepped outside.

That third condition has a name.

You met this idea informally in Number Systems. Each new layer of the number system was forced into existence by an operation that escaped the previous layer. Subtraction escapes the natural numbers: 3 - 5 is not in \mathbb{N}. Division escapes the integers: 7 \div 4 is not in \mathbb{Z}. Closure is the formal name for what was happening every time. Each time the old set wasn't closed under an operation you needed, you built a bigger one.

Here is the full picture for the four arithmetic operations and the four nested number systems.

This is also why "\div 0" is undefined and not just inconvenient. There is no real number x such that 0 \cdot x = 7 — every x gives 0. So 7 / 0 has no answer to point to, and closure under division survives only after you carve out the single forbidden divisor. The reals are closed under division by anything except zero. That tiny exception is the only crack in the whole tower.

The structural rules

Closure asks whether an operation makes sense at all. The next rules ask how it behaves. There are exactly three structural rules that matter for + and \times, and almost every algebraic move you ever make is one of them.

Commutative — order doesn't matter

For addition and multiplication of real numbers,

You can swap the two inputs and get the same answer. 3 + 7 = 7 + 3 = 10. 4 \times 5 = 5 \times 4 = 20.

Subtraction and division do not commute. 5 - 3 = 2 but 3 - 5 = -2. 8 \div 2 = 4 but 2 \div 8 = 0.25. The order matters, and swapping it gives the wrong answer.

Associative — grouping doesn't matter

For addition and multiplication,

You can move the brackets around and get the same answer. This is the rule that lets you write 2 + 3 + 4 without any brackets at all — the answer is 9 no matter how you group the terms. Same for 2 \times 3 \times 4 = 24.

Subtraction and division break this rule. The opening hook of this article was the failure of subtraction: (2 - 3) - 4 = -5 but 2 - (3 - 4) = 3. Division breaks it the same way: (8 \div 4) \div 2 = 1 but 8 \div (4 \div 2) = 4. With - and \div, brackets aren't optional decoration — they change the answer.

Distributive — multiplication spreads across addition

This one connects multiplication and addition. For all real numbers,

You can multiply a into a bracketed sum by spreading it across the terms. This is the engine of expanding brackets in algebra: 3(x + 2) = 3x + 6. It is also the rule you use in reverse when you factor: 5x + 5y = 5(x + y).

Distributivity is not just a notational trick — it has a geometric meaning. If you draw a rectangle of height a and total width b + c, you can split it into two smaller rectangles of widths b and c, both with the same height a. The big rectangle's area is a(b + c). The two pieces' areas are ab and ac. Both descriptions have to give the same number, because they describe the same rectangle.

A small caution: distributivity only goes one way. Multiplication distributes over addition, but addition does not distribute over multiplication. 2 + (3 \times 4) = 2 + 12 = 14, but (2 + 3) \times (2 + 4) = 5 \times 6 = 30. Not the same. The distributive rule names a specific direction, and reading it backwards gives nonsense.

Identity and inverse

Two more rules close out the structural picture for addition and multiplication. They name special numbers.

The additive identity is 0. It is the unique number with the property that

for every a. Adding zero changes nothing. The multiplicative identity is 1, with the property

for every a. Multiplying by one changes nothing.

Each operation also has an inverse for every input. The additive inverse of a is -a, the number that adds to a to give the identity:

The multiplicative inverse of a (when a \neq 0) is 1/a, the number that multiplies with a to give the identity:

These are the rules that let you "undo" things. Subtracting is just adding the additive inverse: 5 - 3 is 5 + (-3). Dividing is just multiplying by the multiplicative inverse: 5 \div 3 is 5 \times \tfrac{1}{3}. So the four operations are really two operations — addition and multiplication — paired with the inverse-of operation that turns one into the other. The reason 0 has no multiplicative inverse is that there is no number x with 0 \cdot x = 1, and that single missing inverse is exactly what makes division by 0 undefined.

Reading expressions: BODMAS and PEMDAS

When you write an expression that mixes more than one operation, you need a rule for which operation to do first. Take 2 + 3 \times 4. Without a rule, this could be (2 + 3) \times 4 = 20 or 2 + (3 \times 4) = 14 — two valid answers, depending on which order you reach for. To make written arithmetic unambiguous, mathematicians agreed on a single convention: multiplication binds tighter than addition, so 2 + 3 \times 4 means 2 + 12 = 14, not 5 \times 4 = 20.

The convention has a name. In Indian and British schools you learn it as BODMAS:

• B — Brackets (anything in brackets is computed first)
• O — Of, or Order (powers and roots)
• DM — Division and Multiplication (tied — left-to-right)
• AS — Addition and Subtraction (tied — left-to-right)

American schools say PEMDAS (Parentheses, Exponents, Multiplication, Division, Addition, Subtraction). The two are the same rule with different names — both put brackets first, then exponents, then multiplication-and-division as a tied pair, then addition-and-subtraction as a tied pair. There is no real difference between the Indian and the American convention; only the acronym changed when the rule crossed an ocean.

Two refinements catch almost everyone the first time:

• D and M tie. Division and multiplication share a single level — when both appear, do them left-to-right in the order they are written. 24 \div 4 \times 2 is (24 \div 4) \times 2 = 6 \times 2 = 12, not 24 \div (4 \times 2) = 24 \div 8 = 3. The "M before D" reading of the acronym is the wrong reading.
• A and S tie. Addition and subtraction also share a level, also left-to-right. 10 - 3 + 2 is (10 - 3) + 2 = 7 + 2 = 9, not 10 - (3 + 2) = 5.

It is worth being precise about what kind of rule this is. BODMAS is a convention, not a theorem. Nothing in the structure of the real numbers forces multiplication to come before addition; mathematicians simply chose this rule so everyone reads the same expression the same way. The rule exists because notation about numbers needs to be unambiguous, the same way punctuation exists so sentences can be unambiguous. If you put enough brackets in by hand, you can write any expression with no ambiguity at all and BODMAS becomes unnecessary — the convention is a labour-saving agreement, not a property of arithmetic.

Here is what reading a real expression looks like, step by step.

Equality and inequality

Rules so far have been about operations on numbers. Now look at relations between numbers — when one number equals another, or when one is bigger than another. Equations and inequalities each have their own rules, and the rules for inequalities have one famous trap.

Equality

The relation = obeys three properties that look obvious until you try to do without them.

• Reflexive. a = a for every a. Every number is equal to itself.
• Symmetric. If a = b, then b = a. The relation is two-sided.
• Transitive. If a = b and b = c, then a = c. Equality chains.

These three together let you treat equations as objects you can manipulate. The single most useful rule that follows is the do-the-same-thing-to-both-sides principle: if a = b, then for any operation \star and any value c,

You can add the same number to both sides, subtract the same, multiply both by the same, or (when the value isn't zero) divide both by the same. The equality survives. This is the only move you ever make when solving an equation.

Inequality

The relations <, \le, >, \ge have most of the same flexibility, but they break in one place that catches almost everyone.

If a < b, you can:

• Add or subtract the same number on both sides. a + c < b + c. The inequality direction stays.
• Multiply or divide both sides by the same positive number. a \cdot c < b \cdot c when c > 0. Direction stays.
• Multiply or divide both sides by the same negative number — but the inequality direction flips. a \cdot c > b \cdot c when c < 0.

That last rule is the trap. If you forget to flip, you get the wrong half of the number line as your answer. The reason it happens is geometric: multiplying by a negative number reflects the number line around zero. If a was to the left of b before the reflection, it is to the right of b after. The strict ordering reverses.

The modulus of a sum and a difference

The modulus (or absolute value) of a real number a, written |a|, is its distance from zero on the number line — always positive, regardless of which side of zero a sits on. So |3| = 3 and |-3| = 3 and |0| = 0.

When you combine two numbers and then take the modulus, two clean rules show up.

The first is the triangle inequality:

The modulus of a sum is always at most the sum of the moduli. Equality happens exactly when a and b have the same sign (or one of them is zero). If they have opposite signs, they partly cancel each other out before you take the modulus, and |a + b| comes out smaller than |a| + |b|.

Try a quick numerical test. Take a = 3 and b = 5. Same sign. Then |a + b| = |8| = 8 and |a| + |b| = 3 + 5 = 8. Equal. Now take a = 3 and b = -5. Opposite signs. Then |a + b| = |-2| = 2 and |a| + |b| = 3 + 5 = 8. The sum on the left is much smaller. The cancellation showed up exactly because the signs disagreed.

You can feel this directly in the figure below. Drag the two red points anywhere on the line. The two readouts at the top show |a| + |b| and |a + b| as you move. When the points are on the same side of zero, the two values are equal. When they cross to opposite sides, |a + b| falls below |a| + |b|, and the gap between them is exactly the cancellation.

The companion rule is the reverse triangle inequality:

The modulus of a difference is always at least the difference between the moduli. The proof is short and worth seeing. Write a = (a - b) + b. Apply the triangle inequality to the right-hand side: |a| \le |a - b| + |b|, which rearranges to |a| - |b| \le |a - b|. Now run the same argument with a and b swapped to get |b| - |a| \le |b - a| = |a - b|. The two together say that |a - b| is at least as large as both |a| - |b| and |b| - |a| — which is exactly the statement that |a - b| \ge \big| |a| - |b| \big|.

These two rules will reappear constantly later: in trigonometric identities, in geometry on the plane (where they really do describe a triangle), in proofs about limits and continuity, and in any chapter of physics that involves displacement.

Two worked examples

The properties become useful the moment you start solving things. Both examples below name each property as it gets used, so you can see the structure of the moves.

Common confusions

A few traps that catch almost everyone the first time.

• "If addition is associative, subtraction must be too." False. The opening of this article showed it: (2 - 3) - 4 \neq 2 - (3 - 4). Subtraction is not associative, even though you can rewrite every subtraction as an addition. Once you do that rewrite — a - b - c becomes a + (-b) + (-c) — the additions are associative, and the brackets stop mattering. But the original expression in subtraction form genuinely needs left-to-right evaluation.

• "a + (b \times c) = (a + b) \times (a + c)." No. Distributivity goes one way only — multiplication distributes over addition, not the other way around. The left side here is a + bc and the right side is a^2 + ab + ac + bc. Different things.

• "|a + b| = |a| + |b|." Only when a and b have the same sign. If they have opposite signs, the left side is strictly smaller because of cancellation. The general rule is the inequality |a + b| \le |a| + |b|, with equality the special case.

• "To solve -2x > 6, divide both sides by -2 to get x > -3." No — you forgot to flip. The correct answer is x < -3. Any time you multiply or divide an inequality by a negative number, the direction reverses. This is the single most common mistake in school inequality problems and almost always shows up on JEE.

• "You cannot divide by zero because the calculator says error." True but unenlightening. The real reason is closure: there is no real number x such that 0 \cdot x equals anything other than 0, so 7 / 0 has no answer to point to. It is not a slow operation — it is a question with no valid answer.

• "BODMAS says multiplication comes before division." No — the M and D are tied. The expression 24 \div 4 \times 2 is read left-to-right as (24 \div 4) \times 2 = 12, not 24 \div (4 \times 2) = 3. Treating BODMAS as a strict left-to-right priority list (B then O then D then M then A then S) is the most common mis-reading of the rule.

Going deeper

These rules have a name as a package

The bundle of properties you just met — closure, associativity, identity, inverse, commutativity, plus the distributive law that ties two operations together — is the definition of an algebraic object called a field. The real numbers \mathbb{R} form a field. The rationals \mathbb{Q} also form a field. The integers \mathbb{Z} are almost a field but fail one rule: not every integer has a multiplicative inverse inside \mathbb{Z} (the inverse of 5 is 1/5, which is not an integer). \mathbb{Z} is what algebraists call a ring instead.

The point of giving the package a name is that the same properties show up far away from arithmetic. Polynomials can be added and multiplied and obey all the field rules except inverse — they form a ring. Square matrices can be added and multiplied, but multiplication is not commutative (AB generally is not BA) — they form yet another structure. Once you know which rules an object obeys and which it doesn't, you know exactly what you are allowed to do with it. Almost all of higher algebra is the study of "what survives when one of these rules is dropped."

Closure failure is the engine of mathematical extension

Every layer of the number system in Number Systems was a closure patch: subtraction broke \mathbb{N}, division broke \mathbb{Z}, the existence of \sqrt{2} broke \mathbb{Q}. The same mechanism works one level higher. The square root of a negative number breaks \mathbb{R} — there is no real x with x^2 = -1 — and the patch is to invent a new number i with i^2 = -1, which gives the complex numbers \mathbb{C}. Same pattern, one layer up. Every time mathematicians have run into an operation that escapes its set, they have built a bigger set rather than declared the operation forbidden.

The triangle inequality is geometric

On the real line, |a + b| \le |a| + |b| might look like a curiosity. It becomes vivid in two dimensions. If a and b are vectors in the plane, then |a + b| is the length of the diagonal of a parallelogram, and |a| + |b| is the length of two sides of the parallelogram laid end to end. The inequality is now literally the statement that a triangle's third side is shorter than the sum of the other two — the rule that gives the triangle inequality its name. The 1D version you met above is the same rule with the parallelogram squashed flat onto a line.

This is one of the most far-reaching inequalities in mathematics. It is the foundation of any notion of distance, in any setting — from coordinate geometry, to function spaces, to the analysis of error in numerical methods. The same six symbols pin down the meaning of "how far apart" in every one of those contexts.

Where this leads next

The properties on this page are the operating manual for everything that follows.

• Number Systems — the full picture of how each number system was built to fix a closure failure in the one inside it.
• Fractions and Decimals — how the rational number system handles arithmetic in practice, including the rules that make rationalising denominators legal.
• Exponents and Powers — a fifth operation that obeys its own laws built directly out of multiplication and the distributive idea.
• Sets — Introduction — the formal language of "the set of all numbers such that…" that this article has been using informally throughout.
• Real Numbers and Their Properties — the field axioms written out in full, with the order axioms that make < and \le behave the way they do.