Base-10 (Decimal) Number System

Understanding the decimal number system - the foundation of everyday mathematics and the gateway to understanding other number systems in computing.

What is Base-10?

Base-10, also known as the decimal system, is the standard number system used in everyday life. It uses ten distinct digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9.

Why Base-10? Humans likely adopted base-10 because we have 10 fingers, making it natural to count in groups of ten.
Key Characteristics:
  • Base: 10
  • Digits: 0-9
  • Place Values: Powers of 10 (1, 10, 100, 1000, etc.)
  • Common Usage: Everyday counting, money, measurements

Understanding Place Value

In base-10, each position represents a power of 10. The rightmost position represents 10⁰ (ones), the next position represents 10¹ (tens), then 10² (hundreds), and so on.

Example: The number 3,547
Thousands
(10³)		 Hundreds
(10²)		 Tens
(10¹)		 Ones
(10⁰)
3 5 4 7
3 × 1000 = 3000 5 × 100 = 500 4 × 10 = 40 7 × 1 = 7

Total: 3000 + 500 + 40 + 7 = 3,547

Basic Operations

Addition with Carrying
  367
+ 485
-----
  852

Step-by-step:
7 + 5 = 12 (write 2, carry 1)
6 + 8 + 1 = 15 (write 5, carry 1)
3 + 4 + 1 = 8 (write 8)
Subtraction with Borrowing
  523
- 178
-----
  345

Step-by-step:
3 - 8 can't do, borrow from tens: 13 - 8 = 5
1 - 7 can't do, borrow from hundreds: 11 - 7 = 4
4 - 1 = 3

Base-10 in Computing

While computers use binary (base-2) internally, understanding base-10 is crucial because:

  • User Interface: Humans input and read numbers in decimal
  • Conversion: We constantly convert between decimal and binary/hexadecimal
  • IP Addresses: Often displayed in decimal format (e.g., 192.168.1.1)
  • Programming: Most programming languages accept decimal literals by default
  • Data Storage: File sizes often shown in decimal (GB, MB)
Note: Storage manufacturers often use decimal (1 KB = 1000 bytes) while operating systems use binary (1 KiB = 1024 bytes), leading to apparent "missing" space.

Practice Exercises

Exercise 1: Place Value Breakdown

Break down these numbers into their place values:

  1. 8,942
  2. 50,376
  3. 123,456
Exercise 2: Mental Math

Practice these calculations:

  1. 456 + 789
  2. 1,234 - 567
  3. 25 × 40
Exercise 3: Real-World Application

A computer has 8,589,934,592 bytes of RAM. Express this in:

  1. Billions of bytes (GB in decimal)
  2. Scientific notation
  3. Words (eight billion, five hundred...)

Next Steps

Now that you understand base-10, you're ready to explore other number systems used in computing:

Learn Base-2 (Binary)
Learn Base-16 (Hexadecimal)