Arithmetic

(the study of numbers and the basic operations of addition, subtraction, multiplication, and division)

What is arithmetic?

Arithmetic is the part of math that deals with numbers and four basic actions we do with them: adding, subtracting, multiplying, and dividing. Almost everything else in math builds on top of arithmetic.

Numbers and place value

The numbers we use for counting — 1, 2, 3, and so on — are called natural numbers. If we also include zero and negative numbers, we get the integers: …, −2, −1, 0, 1, 2, …

We write numbers using place value: where a digit sits tells us how much it is worth. Take the number 352. The 2 is worth just 2. The 5 is worth 50, because it sits one place to the left. The 3 is worth 300, because it sits two places to the left. Each step to the left multiplies the value by 10. That is why 352 and 235 are different numbers, even though they use the exact same three digits — 3, 5, and 2 — just arranged differently.

The four basic operations

Addition puts two amounts together to make a total. Subtraction tells you what is left after taking one amount away from another. Multiplication is a shortcut for adding the same number over and over: 4 × 3 means "add 4 together, 3 times" (4 + 4 + 4 = 12). Division splits an amount into equal groups, and it undoes multiplication: 12 ÷ 4 = 3, because 4 × 3 gives 12 back.

Three rules that always work — and why

Three simple rules describe how addition and multiplication behave. We don't just have to trust these rules — we can actually see why each one is true.

Rule 1 — order doesn't matter (commutativity): 3 + 5 gives the same answer as 5 + 3. The same goes for multiplication: 4 × 3 is the same as 3 × 4. Here is why, for multiplication: imagine 4 rows of 3 dots each, like a small grid. If you count row by row, you get 4 groups of 3. If you instead count column by column, you get 3 groups of 4. Either way, you are counting the exact same dots — so the two totals have to match.

Rule 2 — grouping doesn't matter (associativity): when adding three numbers, it doesn't matter which two you add first. (2 + 3) + 4 gives the same answer as 2 + (3 + 4) — both equal 9. Picture joining three pieces of rope end to end: the total length is the same no matter which two pieces you tie together first.

Rule 3 — multiplying a sum (distributivity): 2 × (3 + 4) gives the same answer as (2 × 3) + (2 × 4) — both equal 14. Imagine you buy 2 bags of fruit, and each bag has 3 apples and 4 oranges. You can count the total pieces of fruit two ways: "2 bags' worth of 7 pieces each" or "all the apples, plus all the oranges." Either way you get the same number, because it's the same fruit.

A real proof: odd plus odd is always even

Let's prove something, not just check a few examples. A number is even if you can split it into two equal whole groups (like 6 = 3 + 3). A number is odd if you can't (like 7). Here is a proof that adding any two odd numbers always gives an even number — for every possible pair, not just the ones we happen to try.

Every odd number is "some even number, plus 1." For example, 7 is 6 (even) plus 1. In general, we can write any odd number as 2m + 1, where m just stands for some whole number — 2m is always even, since it's 2 times something, and adding 1 makes it odd.

Now take two odd numbers and add them. Write the first as 2m + 1 and the second as 2n + 1 (m and n can be any whole numbers, not necessarily the same as each other). Adding them together:

(2m + 1) + (2n + 1) = 2m + 2n + 2

We can pull a 2 out of every term on the right: 2m + 2n + 2 = 2 × (m + n + 1).

The result is 2 multiplied by a whole number — and any number that is 2 times a whole number is, by definition, even. Since m and n could be any whole numbers at all, this works every single time, for every possible pair of odd numbers. That's what makes it a proof, rather than just something we noticed worked in a few examples.

Prime numbers

A prime number is a whole number bigger than 1 that can only be evenly divided by 1 and itself. 7 is prime, because nothing between 2 and 6 divides evenly into it. The first few primes are 2, 3, 5, 7, 11, and 13.

Numbers bigger than 1 that are not prime are called composite, and every composite number can be broken down into primes multiplied together — for example, 12 = 2 × 2 × 3. This breakdown always exists, and it is always the same set of primes no matter how you go about finding it.

Why arithmetic matters

Every price you pay, every recipe you scale up or down, and every date on a calendar relies on arithmetic. It is also the direct foundation for algebra — the next step, where letters stand in for numbers we don't know yet.