Binary Numbers
Lesson notes
Overview
Every bit is a power of two: 1011₂ = 8 + 2 + 1 = 11. This topic is rated beginner and is number 32 of 117 in Computer Science. Core ideas and first intuitions. No prior knowledge needed.
What you are seeing
The panel above shows an interactive animation of the idea. Every slider changes one quantity so you can see its effect on the outcome. The animation is drawn live from the model, so everything you see responds to the sliders. Press Space to pause, use Restart to replay from the start, and change the speed to slow down quick events.
In depth
The binary system represents numbers with only two digits, 0 and 1, where each position is a power of two: 1011₂ = 8 + 2 + 1 = 11. Computers use it since transistors have two reliable states. Convert from decimal by repeatedly dividing by two and recording remainders. Eight bits make a byte with 256 values, and n bits can represent 2ⁿ different values. Hexadecimal is a shorthand where each digit stands for four bits.
Reading the relationship
The expression "Σ bᵢ·2ⁱ" is a compact summary of the idea. It uses the symbols Σ, bᵢ; name what each one stands for before you start. Notice that it contains a sum, so many small contributions are added together. Then use the sliders to see how changing one quantity changes the others.
A guided exploration
1. Run the animation once with the default settings and describe in words what happens. 2. Change one slider at a time and note which quantity responds most. 3. Push each slider to its extreme and decide whether the result still makes sense.
Try this
Start by only moving one slider at a time. Say aloud what you expect, then check.
Common pitfalls
Test small inputs by hand and then the largest ones. Off-by-one errors and forgetting how cost grows with input size are the most common traps.
How it connects
It builds on “Full Adder”, the topic before it in Computer Science. Once you are comfortable, continue with “Bitwise Operations”.
Why it matters
Computing ideas decide how fast, how safely and how well software can solve problems.
Where you might use it
- Revising “Binary Numbers” before a class, exam or certification
- Choosing the right algorithm or data structure when writing software or apps
- Debugging programs and judging how fast or how safely code will run at scale
- Understanding how search engines, games, phones and AI systems make decisions
Frequently asked questions
What is Binary Numbers?
Every bit is a power of two: 1011₂ = 8 + 2 + 1 = 11. It is a beginner topic in Computer Science.
How do I use the Binary Numbers simulation?
The panel at the top shows an interactive animation of the idea. Every slider changes one quantity so you can see its effect on the outcome. Move the sliders one at a time, then press Space to pause the animation and look closer.
What does the formula mean?
The expression Σ bᵢ·2ⁱ packs the idea into symbols. Name each symbol first, then change one quantity with the sliders and see which others respond.
Who is this topic for?
Core ideas and first intuitions. No prior knowledge needed. If the quiz feels hard, reread the lesson notes and try the previous topic first.
What should I learn before this?
Start with “Full Adder”, the previous topic in Computer Science. Each topic in a subject builds on the ones before it.
How do I track my progress?
Press “Mark as learned” on this page. Progress is stored only in this browser and is never sent anywhere. The Browse page can show topics you have learned or still need to learn.
Quick quiz
More in Computer Science
See all →Bitwise Operations
AND, OR, XOR and shifts act on every bit at once — the toolbox of low-level programming.
Two’s Complement
How computers store negative integers so that addition just works.
Floating-Point Numbers
A sign, an exponent and a mantissa let 32 bits represent huge and tiny values approximately.