Basic Operations

What “Basic Operations” Means in Arithmetic

In arithmetic, “basic operations” are the four fundamental ways we combine numbers:

• Addition
• Subtraction
• Multiplication
• Division

Each of these will have its own chapter where you work with them in detail. Here, the goal is to understand how they fit together as a system, how they relate to one another, and some common ideas that are shared across all four.

You should think of this chapter as a map of the four operations, not a step‑by‑step how‑to for each one.

The Four Operations as “Ways of Combining Numbers”

Each basic operation answers a slightly different kind of question about numbers:

• Addition: “How many altogether?”
• Subtraction: “How much more?” or “What is the difference?”
• Multiplication: “How many in total if we have repeated groups?”
• Division: “How many in each group?” or “How many groups can we make?”

A useful way to see the structure is:

• Addition and subtraction are closely related (they “undo” each other).
• Multiplication and division are closely related (they “undo” each other).
• Multiplication is repeated addition.
• Division is “reverse multiplication.”

You will study each pair in more detail later, but the relationships are important from the start.

Inverse Operations: Undoing a Calculation

Two operations are called inverses if one undoes the effect of the other.

The two inverse pairs among the basic operations are:

• Addition and subtraction
• Multiplication and division

In symbols:

• If you add a number and then subtract the same number, you get back where you started:
  $$a + b - b = a$$
• If you multiply by a nonzero number and then divide by the same number, you get back where you started:
  $$\frac{a \times b}{b} = a \quad \text{for } b \neq 0$$

Thinking in terms of “doing” and “undoing” is important later, especially when you solve equations in algebra.

Order of Operations (PEMDAS/BODMAS)

When a calculation involves more than one operation, you need an agreed‑upon order so everybody gets the same answer. This is called the order of operations.

A common English mnemonic is PEMDAS:

• P: Parentheses
• E: Exponents
• M: Multiplication
• D: Division
• A: Addition
• S: Subtraction

You may also see BODMAS (Brackets, Orders, Division, Multiplication, Addition, Subtraction). The idea is the same.

Within this arithmetic course, the key point is how the four basic operations are grouped:

• Multiplication and division are done before addition and subtraction.
• Multiplication and division are treated at the same level (work left to right).
• Addition and subtraction are treated at the same level (work left to right).

So, ignoring parentheses and exponents for the moment, the rule is:

1. Do all multiplications and divisions from left to right.
2. Then do all additions and subtractions from left to right.

This convention keeps calculations unambiguous and is used everywhere in mathematics.

Properties Shared by the Basic Operations

Each operation has its own detailed properties, but some broad ideas appear repeatedly. At this stage, you only need to recognize them and know which operations they apply to. Detailed proofs and extensive practice belong in the specific operation chapters.

Commutative Property (Order of Numbers)

An operation is commutative if changing the order of the numbers does not change the result.

• Addition is commutative: $a + b = b + a$.
• Multiplication is commutative: $a \times b = b \times a$.
• Subtraction is not commutative in general.
• Division is not commutative in general.

So, for basic operations:

• You can swap numbers freely in addition and multiplication.
• You cannot simply swap numbers in subtraction or division without changing the value.

Associative Property (Grouping of Numbers)

An operation is associative if changing how the numbers are grouped (with parentheses) does not change the result, as long as the order of the numbers stays the same.

• Addition is associative: $(a + b) + c = a + (b + c)$.
• Multiplication is associative: $(a \times b) \times c = a \times (b \times c)$.
• Subtraction is not associative in general.
• Division is not associative in general.

This means:

• For a string of additions or multiplications, you may group them however is convenient.
• For a string of subtractions or divisions, you must be careful with grouping; parentheses matter.

Distributive Property (Linking Addition and Multiplication)

The distributive property tells you how multiplication interacts with addition and subtraction:

• Over addition:
  $$a \times (b + c) = a \times b + a \times c$$
• Over subtraction:
  $$a \times (b - c) = a \times b - a \times c$$

This property connects two of the basic operations and is one of the most important patterns you will use repeatedly in algebra.

Identity and Zero Ideas in Basic Operations

Each operation has special numbers that behave in simple, predictable ways.

Additive Identity and Zero for Addition/Subtraction

The additive identity is the number that does not change another number when added.

• The additive identity is $0$:
  $$a + 0 = a \quad\text{and}\quad 0 + a = a$$

In terms of subtraction:

• Subtracting $0$ does nothing: $a - 0 = a$.

Zero also appears as the result when you add a number and its “opposite”:

• $$a + (-a) = 0$$

The idea of identity and opposites will be revisited in the chapters on integers and equations.

Multiplicative Identity and Zero for Multiplication/Division

The multiplicative identity is the number that does not change another number when multiplied.

• The multiplicative identity is $1$:
  $$a \times 1 = a \quad\text{and}\quad 1 \times a = a$$

Zero interacts with multiplication in a special way:

• Any number times $0$ is $0$:
  $$a \times 0 = 0 \quad\text{and}\quad 0 \times a = 0$$

Division has two special warnings:

• Division by $1$ leaves a number unchanged: $\dfrac{a}{1} = a$.
• Division by $0$ is not defined in standard arithmetic; you cannot divide by zero.

Division by zero is not allowed for any of the number systems you will meet at this level.

Estimation and Mental Strategies Across Operations

Even before going into detailed techniques for each operation, it is useful to see that some mental strategies work across all four:

• Rounding: Replace numbers by nearby “easy” numbers (like multiples of $10$ or $100$) to estimate.
• Breaking numbers apart (using place value or the distributive property) to make calculations easier.
• Checking reasonableness: Use a simpler approximate calculation to see if your exact result makes sense.

These are not separate operations, but habits that help you use the basic operations more effectively in everyday contexts.

How the Basic Operations Build Later Topics

The four basic operations are the foundation for almost everything else in this course:

• Fractions, decimals, and percentages rely on the same operations, just on different kinds of numbers.
• Algebra uses the basic operations, but often with variables instead of only concrete numbers.
• Geometry formulas, probability calculations, and statistics all are built out of these operations.

Understanding how the operations relate (inverse pairs, properties, and order of operations) prepares you to move beyond simple “button pushing” into more flexible and confident problem solving.

In the next chapters, you will study addition, subtraction, multiplication, and division individually, learning concrete methods and practicing each operation in detail.