Understanding Binary, Hexadecimal, Decimal (Base-10), and more
Based on Corey Schafer's video on YouTube. If you like this content, support the original creators by watching, liking and subscribing to their content.
Base-10 numbers are computed by multiplying each digit by a power of 10 based on its position and summing the results.
Briefing
Binary and hexadecimal become much easier once they’re treated as “base systems” built on the same positional idea as everyday base-10 numbers. In base 10, each digit’s place value is a power of 10: the rightmost digit is multiplied by 10^0 (ones), the next by 10^1 (tens), then 10^2 (hundreds), and so on. That’s why 1,234 works out as 1×10^3 + 2×10^2 + 3×10^1 + 4×10^0—each digit contributes its face value times the appropriate power of 10.
The key insight is that binary and hexadecimal use the same method, just with a different base. For binary, the base is 2, so place values are powers of 2. The transcript walks through 1010₂: 1×2^3 + 0×2^2 + 1×2^1 + 0×2^0 = 8 + 0 + 2 + 0 = 10 in decimal. A second example, 1111₂, becomes 1×2^3 + 1×2^2 + 1×2^1 + 1×2^0 = 8 + 4 + 2 + 1 = 15. Once the powers-of-the-base pattern is clear, any binary string can be converted by multiplying each digit (0 or 1) by its corresponding power of 2 and summing.
Hexadecimal uses base 16, which introduces a practical twist: a single digit must represent values from 0 up to 15. Since decimal digits only go 0–9, hexadecimal extends into letters: 10 becomes A, 11 becomes B, continuing through C (12), D (13), E (14), and F (15). The conversion method still relies on positional powers—now powers of 16. The transcript demonstrates BA55₁₆ by mapping B=11 and A=10, then computing 11×16^3 + 10×16^2 + 5×16^1 + 5×16^0 = 11×4096 + 10×256 + 5×16 + 5 = 47701 in decimal.
A real-world application ties the math to how computers store color. RGB values are often written in hexadecimal as three pairs of digits: the first pair is red, the second is green, and the third is blue. For white, the hex form is FF FF FF. Converting one pair shows the pattern: F=15, so 15×16^1 + 15×16^0 = 240 + 15 = 255. Repeating for all three channels yields 255 255 255.
Overall, the takeaway is procedural: treat each digit as a coefficient, multiply by the base raised to the digit’s position index, and sum. With that rule, binary (base 2) and hexadecimal (base 16) stop feeling like special cases and become straightforward variants of the same positional-number system.
Cornell Notes
The transcript explains how base-10 positional notation generalizes to binary and hexadecimal. In base 10, each digit is multiplied by a power of 10 based on its place (10^0 for ones, 10^1 for tens, etc.), and the results are added. Binary replaces the base with 2, so place values become powers of 2; for example, 1010₂ converts to 10 by summing 1×2^3 and 1×2^1. Hexadecimal replaces the base with 16 and uses letters for digits 10–15 (A=10 through F=15); for example, BA55₁₆ becomes 47701 by summing coefficients times powers of 16. The same conversion logic also explains RGB hex color values like FF FF FF turning into 255 255 255.
How does base-10 positional notation determine the value of a number like 1,234?
Why does converting 1010₂ to decimal work the same way as converting 1,234 to decimal?
What changes in hexadecimal compared with binary, and how are digits 10–15 represented?
How is BA55₁₆ converted to decimal?
How do hex RGB values like FF FF FF translate into decimal RGB values?
Review Questions
- If a binary number has digits dₙ…d₀, what is the general decimal conversion formula using powers of 2?
- Convert 1101₂ to decimal using the positional power method.
- Convert C3₁₆ to decimal by mapping C to 12 and applying powers of 16.
Key Points
- 1
Base-10 numbers are computed by multiplying each digit by a power of 10 based on its position and summing the results.
- 2
Binary uses the same positional method but with base 2, so place values are powers of 2.
- 3
Hexadecimal uses base 16 and extends digit symbols: A=10 through F=15, so each hex character becomes a coefficient.
- 4
Converting any base-N number to decimal follows the same pattern: sum (digit value)×N^(position index).
- 5
RGB hex colors split into three two-digit hex pairs (red, green, blue), each converting to a 0–255 decimal value.
- 6
Once the powers-of-the-base rule is understood, binary and hexadecimal conversions become mechanical rather than memorization-based.