# Base 10 and Place Value
## Plain English first
We only have 10 digit symbols: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.
With just those 10 symbols we can write any number. The trick is **place value**: the position of a digit determines how much it is worth.
The digit `3` means three. But `300` means three hundreds. And `3000` means three thousands. Same symbol, completely different value depending on where it sits.
## The column system
Each column is worth ten times the one to its right:
```text
... | Thousands | Hundreds | Tens | Ones |
```
The number **4,302** means:
```text
4 thousands + 3 hundreds + 0 tens + 2 ones
= 4000 + 300 + 0 + 2
= 4,302
```
## Standard math notation
```text
N = d₃ × 10³ + d₂ × 10² + d₁ × 10¹ + d₀ × 10⁰
Example: 4302
= 4 × 1000 + 3 × 100 + 0 × 10 + 2 × 1
= 4000 + 300 + 0 + 2
```
## Verbose Python with descriptive names
```python
def break_number_into_place_values(whole_number):
"""
Decompose a number into the value each digit contributes.
Each column is worth 10× the column to its right — that's base 10.
"""
column_names = ["ones", "tens", "hundreds", "thousands",
"ten-thousands", "hundred-thousands", "millions"]
breakdown = {}
for position, digit_character in enumerate(reversed(str(whole_number))):
digit_value = int(digit_character)
column_worth = 10 ** position
value_in_column = digit_value * column_worth
column_label = column_names[position] if position < len(column_names) else f"10^{position}"
breakdown[column_label] = {"digit": digit_value, "contributes": value_in_column}
return breakdown
# Demonstration
breakdown = break_number_into_place_values(4302)
for column, data in breakdown.items():
print(f" {column:15s}: digit={data['digit']}, contributes={data['contributes']}")
# Sum confirms: 4000 + 300 + 0 + 2 = 4302
```
## Why base 10?
Probably because humans have 10 fingers. Other bases work identically:
- **Base 2 (binary)**: computers — only digits 0 and 1
- **Base 16 (hex)**: programming — memory addresses, colors (`#FF5733`)
- **Base 60**: ancient Babylonians — why we have 60 minutes and 60 seconds
## Common mistakes
- **The digit `4` in 4,302 is worth 4,000 — not 4.**
- **Dropping placeholder zeros.** 4,302 ≠ 432.
- **Thinking "ones" means unimportant.** Everything is built in multiples of the ones column.
## See also
- [Zero — The Number That Changed Everything](/articles/6995c841-40bf-4d34-a787-c459d6084b5d)
- [What Is a Number?](/articles/8510b9a8-a3ad-4790-a928-b2360ac8679e)
- [The Number Line](/articles/257eeafa-b654-40a8-97f5-6be800ad1c8e)
- [Math Foundations — Visual Table of Contents](/articles/d404884f-54fc-4289-b3f1-baaad2bec6b2)