Our entire number system is built on a simple idea: the position of a digit determines its value. We use ten digits (0, 1, 2, 3, 4, 5, 6, 7, 8, 9), and we group things in sets of ten. This is called the base-ten system.

Think about the number 472. It's not just a 4, a 7, and a 2 thrown together. The '2' is in the ones place, so it just means 2. The '7' is in the tens place, so it stands for 7 tens, or 70. And the '4' is in the hundreds place, meaning it represents 4 hundreds, or 400. So, 472 is really $400 + 70 + 2$. This concept is known as place value.

As you move to the left, each place value is ten times bigger than the one to its right. The tens place is ten times the ones place, the hundreds place is ten times the tens place, and so on.

What about numbers that aren't whole? The decimal point is our guide to parts of a whole. Everything to the right of the decimal point represents a fraction of one.

Just as place values get bigger by ten times when moving left, they get ten times smaller when moving right. The first spot to the right of the decimal is the tenths place ($1/10$). The next is the hundredths place ($1/100$), followed by the thousandths place ($1/1000$).

So, a number like 24.153 is read as "twenty-four and one hundred fifty-three thousandths." It breaks down like this: 2 tens, 4 ones, 1 tenth, 5 hundredths, and 3 thousandths.

Writing numbers in words helps connect the digits to their value. A number like 9.04 is "nine and four hundredths." We say "and" for the decimal point. The name of the final decimal place (in this case, hundredths) tells us the size of the fractional part.

Comparing numbers is a fundamental skill. To find out which of two numbers is larger, we just need a system. Start by comparing the digits from left to right, in the largest place value.

Let's compare 5,482 and 5,399.

1. Start with the thousands place. Both numbers have a 5. It's a tie, so we move to the next place.
2. Look at the hundreds place. The first number has a 4, and the second has a 3. Since 4 is greater than 3, we can stop.

5,482 is greater than 5,399. We write this as $5,482 > 5,399$.

Let's try one with decimals: compare 14.35 and 14.351.

1. Tens place: Both have a 1.
2. Ones place: Both have a 4.
3. Tenths place: Both have a 3.
4. Hundredths place: Both have a 5.
5. Thousandths place: The first number has no digit here (or a 0), while the second has a 1.

Since 1 is greater than 0, 14.351 is greater than 14.35. We write this as $14.35 < 14.351$.

Sometimes we don't need exact numbers. If a store has 1,987 books, you might just say they have "about 2,000." This is called rounding. It makes numbers easier to work with and talk about.

The rule is simple. First, find the digit in the place you are rounding to. Then, look at the digit directly to its right.

• If the digit to the right is 5 or more, you round up (add one to the rounding digit).
• If the digit to the right is 4 or less, you round down (the rounding digit stays the same).

After rounding, all digits to the right of the rounding place become zeros.

Let's round 6,821 to the nearest hundred.

1. The digit in the hundreds place is 8.
2. The digit to its right is 2.
3. Since 2 is less than 4, we round down. The 8 stays the same.
4. The digits to the right of the 8 become zeros.

So, 6,821 rounded to the nearest hundred is 6,800.

This works for decimals, too. To round 9.472 to the nearest tenth:

1. The digit in the tenths place is 4.
2. The digit to its right is 7.
3. Since 7 is 5 or more, we round up. The 4 becomes a 5.
4. Digits to the right of the new 5 are dropped.

So, 9.472 rounded to the nearest tenth is 9.5.

Now, let's test your understanding of place value.

Understanding place value is the bedrock of arithmetic. It gives you the power to handle numbers of any size with confidence.