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Different Bases

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Different Bases

Study Guide

Key Definition

Numbers can be represented in different bases by using different sets of symbols and place values based on powers of the base. For example, in base 10, we use powers of $10$.

Important Notes

Base 10 uses 10 symbols: 0 to 9.
Base 3 uses 3 symbols: 0, 1, and 2.
Place values in base 3 are powers of $3$: 1, 3, 9, 27, etc.
Carrying occurs when the sum in a place value exceeds the base.
Conversion between bases requires understanding of place values.

Mathematical Notation

$+$ represents addition

$-$ represents subtraction

$\times$ or $*$ represents multiplication

$\div$ or $/$ represents division

$a_b$ denotes the number $a$ in base $b$ (subscript notation).

Remember to use proper notation when solving problems

Why It Works

Using different bases allows us to represent numbers in various ways, which is particularly useful in computing and digital systems where binary (base 2) is prevalent.

Remember

Each base system uses its own set of symbols and place values. For example, base 16 uses 16 symbols: 0-9 and A-F.

Quick Reference

Base 10:$0, 1, 2, 3, 4, 5, 6, 7, 8, 9$

Base 3 Example:$102$ in base 3 equals $11$ in base 10.

Understanding Different Bases

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Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

In base 10, each digit represents a power of $10$: ones, tens, hundreds, etc.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

What is $10_{10}$ in base 3?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

Imagine your favorite video game uses a health bar that counts in base 3 instead of base 10. If your health is displayed as $202$ in base 3, what is your health in base 10?

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

If you have a number $121$ in base 3, how would you express it in base 10?

Challenge Quiz

Single Choice Quiz

Advanced

What is $210_3$ in base 10?

Recap

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Review key concepts and takeaways