Expanded Notation — Definition, Formula & Examples
Expanded notation is a way of writing a number that shows the value of each digit based on its place. For example, 345 is written as 300 + 40 + 5.
Expanded notation represents a number as the sum of products of each digit and its corresponding power of ten, making the place value of every digit explicit.
How It Works
Start with the leftmost digit and multiply it by its place value. Move right, multiplying each digit by its place value. Then write the results as a sum. Zeros can be skipped since they contribute nothing to the total. For instance, in 4,072 you would write 4,000 + 0 + 70 + 2, which simplifies to 4,000 + 70 + 2.
Worked Example
Problem: Write 5,638 in expanded notation.
Step 1: Identify each digit and its place value: 5 is in the thousands place, 6 is in the hundreds place, 3 is in the tens place, and 8 is in the ones place.
Step 2: Multiply each digit by its place value.
Step 3: Write the number as the sum of these values.
Answer:
Why It Matters
Understanding expanded notation builds a strong foundation for multi-digit addition, subtraction, and multiplication, where you often work with individual place values. It also prepares students for scientific notation, which is used in science and engineering to handle very large or very small numbers.
Common Mistakes
Mistake: Assigning the wrong place value to a digit, such as writing the 6 in 2,635 as 6,000 instead of 600.
Correction: Count place positions from right to left: ones, tens, hundreds, thousands. The 6 in 2,635 sits in the hundreds place, so its value is 600.