Numbers and Lines

This chapter is about numbers.

Some numbers are called rational numbers. We can put these numbers on a line.

Some numbers are not like that. We call them irrational numbers.

We will learn about both kinds.

Making a Number Line

Let's draw a line.

Pick a point. Call it 0.

Pick another point. Call it 1.

The space between 0 and 1 is called one unit.

Now count to the right. Mark 2. Mark 3. Keep going!

We can also count to the left of 0. We call these numbers -1, -2, -3.

We can also find spots between the numbers.

The middle of 3 and 4 is 3.5.

We can split the line into small parts, like thirds or tenths.

This way, every number has its own spot on the line.

Measuring with Decimals

We can use decimals to show where a number is on the line.

First, find the whole number part. This is called the integer part.

Then we look closer. We split the space into ten parts.

This gives us the next number after the decimal point.

We can keep splitting to get closer and closer to the exact spot.

Some numbers, like 1/3, never stop with this splitting. Their decimals go on and on.

Making a Flat Picture (The Plane)

Now let's draw another line. Make it go up and down.

This line crosses our first line at 0.

Now we have two lines. This makes a flat picture called a plane.

We can find any point in the plane. We go right or left first. Then we go up or down.

Squares and Special Numbers

Let's draw a square. Tilt it inside a bigger square.

Look at Figure 2. The big square is made of 4 small unit squares.

The tilted square inside takes up half of that space.

So the tilted square has an area of 2.

The side of that tilted square is a special number.

We call it the square root of 2. We write it like this: √2.

A square root is a number that, when multiplied by itself, gives you the area.

More Tilted Squares

Let's try a bigger square. Make it 3 units by 3 units.

Its area is 9.

Inside, we tilt another square. The corners make triangles.

Each triangle has an area of 2.

There are 4 triangles. That's 4 times 2 = 8... wait, let's check the math from the picture: the four triangles take away 4 total.

So the tilted square's area is 9 − 4 = 5.

The side length is √5 — the square root of 5.

Perfect Squares

Some numbers have whole-number square roots.

We call these perfect squares.

4 is a perfect square. Its square root is 2.

9 is a perfect square. Its square root is 3.

25 is a perfect square. Its square root is 5.

But numbers like 2, 3, 5, 6 are not perfect squares.

Their square roots are not whole numbers.

A Big Surprise!

Long ago, a group called the Pythagoreans found something surprising.

They found that √2 is not a fraction!

It cannot be written as one whole number divided by another.

We call numbers like this irrational numbers.

The same is true for √5 and other square roots like it.

A Special Triangle Rule

When we make tilted squares, we also see right triangles.

A right triangle has one square corner.

The two short sides are called legs. We call them a and b.

The long side is called the hypotenuse. We call it c.

There is a special rule for right triangles:

a² + b² = c²

This is called the Pythagorean Theorem.

We will learn much more about it later!