Lesson 2: Understand Whole-Number Place Value

Property

Each place represents 10 times the place just to the right. A number is a sequence of digits, and its value is the sum of each digit multiplied by its place value (a power of ten). For example:

This can also be written using exponents:

Examples

• The number 5,281 in expanded form is $5 \times 1000 + 2 \times 100 + 8 \times 10 + 1$.
• The number 709 shows the importance of zero as a placeholder. It is $7 \times 100 + 0 \times 10 + 9$.
• A larger number like 1,607,300 is written as $1 \times 1000000 + 6 \times 100000 + 0 \times 10000 + 7 \times 1000 + 3 \times 100 + 0 \times 10 + 0 \times 1$.

Explanation

Place value is like a secret code where a digit's position tells you its real worth. A 7 in the tens place is 70, but in the hundreds place, it's 700! This system lets us write any number, big or small.

Property

Expanded form breaks a number into a sum of its place values. This makes it easier to compare the value of the same digit in different positions.

Examples

In the number 34,381, what is the relationship between the value of the 3 in the ten thousands place and the 3 in the hundreds place?

• Expanded Form: $3 \times 10,000 + 4 \times 1,000 + 3 \times 100 + 8 \times 10 + 1 \times 1$.
• The value of the first 3 is $30,000$ and the value of the second 3 is $300$.
• Since $30,000 = 300 \times 100$, the first 3 is 100 times the value of the second 3.

Compare the values of the digit 9 in the number 992,450.

• Expanded Form: $900,000 + 90,000 + 2,000 + 400 + 50$.
• The value of the 9 in the hundred thousands place is $900,000$. The value of the 9 in the ten thousands place is $90,000$.
• The value of the first 9 is 10 times the value of the second 9.

Explanation

Writing a number in expanded form separates each digit into its individual place value. This allows you to directly see and compare the value of each digit. By looking at the terms in the expanded form, you can determine how many times greater one digit''s value is than another''s. This method reinforces the idea that a digit''s position determines its value.