Numbers start with the basics: the ones we use for counting. If you're counting apples, you use 1, 2, 3, and so on. These are called natural numbers.

But what about zero? Or what happens when you owe someone money? For that, we need a bigger set of numbers. Integers include all the natural numbers, zero, and the negative versions of the natural numbers. They represent whole things, without any fractional or decimal parts.

A number line is a great way to visualize how integers relate to each other. Zero sits in the middle. Positive numbers stretch out to the right, getting larger, while negative numbers go to the left, getting smaller.

The world isn't always made of whole things. You can eat half a pizza or run a race in 9.58 seconds. For these situations, we need numbers that live between the integers. Fractions, decimals, and percentages are three ways to talk about these parts of a whole.

A fraction shows a part of a whole, like $1/2$. A decimal uses a decimal point to show the same thing, like $0.5$. A percentage represents a part out of 100, like $50\%$. They all mean the same quantity, just written differently.

Knowing how to switch between these forms is incredibly useful. It helps you compare prices at the store or understand statistics in the news. Here are a few common conversions:

Once we have our numbers, we can use them to calculate things. The four basic tools for this are the arithmetic operations: addition ($+$), subtraction ($-$), multiplication ($\times$), and division ($\div$).

• Addition combines values. $5 + 3 = 8$.
• Subtraction finds the difference. $10 - 4 = 6$.
• Multiplication is like repeated addition. $4 \times 3$ is the same as $4+4+4$, which equals $12$.
• Division splits a value into equal parts. $20 \div 5 = 4$.

These operations work with all types of numbers, whether they're integers, fractions, or decimals. For example, $0.5 + 0.25 = 0.75$.

Some numbers are inconveniently large, like the number of stars in our galaxy (around 100,000,000,000). Others are incredibly small, like the width of a human hair (about 0.00007 meters). Writing all those zeros is tedious and can lead to mistakes.

To make things easier, we use a system called standard form, or scientific notation. This method expresses a number as a product of a number between 1 and 10 and a power of 10.

Here, $a$ is a number greater than or equal to 1 but less than 10, and $n$ is an integer. A positive $n$ means it's a large number, and a negative $n$ means it's a small number (a decimal).

Let's convert our examples:

• The number of stars, 100,000,000,000, becomes $1 \times 10^{11}$.
• The width of a hair, 0.00007 meters, becomes $7 \times 10^{-5}$ meters.

This notation is much more compact and is used all the time in science and engineering.

Let's review the main ideas.

Now, let's check your understanding.

These different types of numbers and the rules for using them form the foundation of all mathematics. From calculating your grocery bill to understanding the universe, it all starts here.