Partition & rename numbers (expanded notation)
Theory
Partitioning splits a number into the value of each of its digits, and expanded notation writes the number as the sum of those values. Numbers can also be renamed using place value, such as writing 7 million as 7000 thousands.
Expanded notation writes a number as a sum, giving each digit its place value. For \(372\,405\) that is \(300\,000 + 70\,000 + 2\,000 + 400 + 5\).
Partitioning means splitting a number into parts. The standard partition uses place value, but a number can also be split in non-standard ways, such as \(163\,480 = 150\,000 + 13\,480\).
Renaming writes the same number in different units. Because \(1000\) thousands make \(1\) million, \(7\) million is the same as \(7000\) thousands.
Each column is worth ten times the one to its right:
| Column | Place value |
|---|---|
| Hundred thousands | \(100\,000\) |
| Ten thousands | \(10\,000\) |
| Thousands | \(1\,000\) |
| Hundreds | \(100\) |
| Tens | \(10\) |
| Ones | \(1\) |
How to write a number in expanded notation
- Name the place value of each digit, from left to right.
- Multiply each digit by its place value to get its part.
- Add the parts with \(+\) signs, leaving out any digit that is \(0\).
Give each digit its place value.
\(300\,000 + 70\,000 + 2\,000 + 400 + 5\)
The thousands digit is \(6\).
| \(6\) | \(\rightarrow\) | \(6\,000\) |
The missing part is \(6\,000\).
\(1000\) thousands make \(1\) million.
| \(7 \times 1000\) | \(=\) | \(7000\) |
So \(7\) million \(=\) \(7000\) thousands.
The \(7\) is in the thousands place, so it is \(7\,000\), not \(70\,000\).
\(400\,000 + 7\,000 + 500\)
Common pitfalls
Frequently asked questions
What is expanded notation?
It writes a number as the sum of the value of each digit, such as \(372\,405 = 300\,000 + 70\,000 + 2\,000 + 400 + 5\).
How do you partition a number?
Split it into parts. The standard way gives each digit its place value; non-standard ways such as \(150\,000 + 13\,480\) for \(163\,480\) are also fine.
What does it mean to rename a number?
Write the same number in different units. Because \(1000\) thousands make \(1\) million, \(7\) million renames as \(7000\) thousands.
What happens to a zero digit?
A zero adds nothing, so it is left out of the sum but still holds its place. \(900\,407 = 900\,000 + 400 + 7\).
How many thousands are in a million?
There are \(1000\) thousands in one million, so \(7\) million is \(7000\) thousands.