Think of numbers as belonging to different families, each one a bit bigger and more inclusive than the last. We start with the simplest and build our way up.

The most basic family is the Natural Numbers. These are the counting numbers you first learned: 1, 2, 3, 4, and so on. They’re the numbers you’d use to count apples in a basket.

If we add just one more number to that family, zero, we get the Whole Numbers: 0, 1, 2, 3, and so on. The only difference is the zero.

But what about negative values, like a temperature below freezing or being in debt? For that, we need Integers. This family includes all the whole numbers, plus their negative counterparts: ..., -3, -2, -1, 0, 1, 2, 3, ...

Of course, numbers exist between the integers. Rational Numbers include any number that can be written as a ratio, or fraction, of two integers, like $1/2$, $-3/4$, or even $7$ (which is just $7/1$). As decimals, rational numbers either terminate (like $0.25$) or repeat in a predictable pattern (like $0.333...$).

Finally, there are the loners of the number world: the Irrational Numbers. These are numbers that cannot be written as a simple fraction. Their decimal representations go on forever without ever repeating. The most famous irrational number is $\pi$ (pi), which starts 3.14159... but never ends or repeats. Another common example is the square root of 2, $\sqrt{2}$.

All of these numbers—rational and irrational together—make up the set of Real Numbers. For most of the algebra you'll do, you'll be working with real numbers.

Numbers don't just exist; they interact with each other through operations like addition and multiplication. These interactions follow specific rules, or properties, that are always true. Understanding them makes calculations easier and is the key to manipulating algebraic expressions later on.

The Commutative Property says that for addition and multiplication, the order of the numbers doesn't matter. For example, $3+8$ is the same as $8+3$. Likewise, $5 \times 6$ is the same as $6 \times 5$. This is helpful for mental math, as you can rearrange numbers to make them easier to work with.

The Associative Property deals with grouping. It says that when you are only adding or only multiplying, it doesn't matter how you group the numbers. You can add $(2+3)+4$ or $2+(3+4)$ and you'll get the same result: 9.

The Distributive Property is a bit different because it links multiplication and addition. It tells us how to handle a number that is multiplying a sum in parentheses. You can "distribute" the multiplication to each number inside the parentheses.

For instance, to solve $4 \times (5 + 2)$, you could add first to get $4 \times 7 = 28$. Or, you could distribute the 4: $(4 \times 5) + (4 \times 2) = 20 + 8 = 28$. The result is identical. This property becomes extremely important in algebra when you have variables inside the parentheses.

When an expression has multiple operations, like $3 + 5 \times 2$, which do you do first? If you add first, you get $8 \times 2 = 16$. If you multiply first, you get $3 + 10 = 13$. To avoid this confusion, mathematicians agreed on a standard order of operations. A common way to remember this order is with the acronym PEMDAS.

It's crucial to remember that multiplication and division are on the same level; you do them as they appear from left to right. The same is true for addition and subtraction.

Let's try an example: $10 - 2 \times (6 - 3)^2 + 8 \text{ ÷ } 4$.

1. Parentheses: $(6-3) = 3$. The expression becomes $10 - 2 \times 3^2 + 8 \text{ ÷ } 4$.
2. Exponents: $3^2 = 9$. Now we have $10 - 2 \times 9 + 8 \text{ ÷ } 4$.
3. Multiplication and Division (left to right): First is $2 \times 9 = 18$. The expression is $10 - 18 + 8 \text{ ÷ } 4$. Next is $8 \text{ ÷ } 4 = 2$. Now we have $10 - 18 + 2$.
4. Addition and Subtraction (left to right): First is $10 - 18 = -8$. The expression is $-8 + 2$. The final answer is $-6$.

Let's check your understanding of these fundamental concepts.

Mastering these classifications, properties, and the order of operations provides the solid foundation you'll need for all of your future work in mathematics.