NUMBER LOGIC / JD · 2026.10.19
Base Ten, Base Eight and Base Sixteen Compared
Understand that the same quantity can be written with different digit symbols.
Place value sets the base
In base ten, 205 means 2×100 + 0×10 + 5. In base eight, 205₈ means 2×64 + 0×8 + 5 = 133 in decimal. The symbol string is not enough: its base matters.
Hexadecimal for compact binary
Base sixteen uses the digits 0–9 and A–F, where A=10 through F=15. Decimal 255 is FF₁₆, because 15×16 + 15 = 255. One hexadecimal digit corresponds to four binary bits.
Avoid invalid digits
The number 8 cannot appear as a digit in a valid base-eight numeral, while F is perfectly valid in hexadecimal. Conversions should preserve numerical value, not digit appearance.
Practice conversions
Convert decimal 31 into binary, octal and hexadecimal. Then reverse each calculation using powers of the given base. The number dictionary shows binary facts for 1–3,000.