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Lesson Directive // Binary NumbersREF_CORE

Base-2 System

Value\text{Value}==∑i=0n\sum_{i=0}^{n}bib_i⋅\cdot2i2^i

Hover over a variable in the formula above, or see glossary below:

bib_i
Binary Digit (Bit)
0 or 1

Unlike our normal Base-10 system which uses digits 0-9, binary only uses 0 and 1. Each position represents a power of 2 instead of a power of 10.

INSIGHT: Computers use binary because it perfectly maps to electrical switches being OFF (0) or ON (1).
Detailed Theory & ReferencesEXT_DOC

Binary Numeral System

The binary numeral system, or base-2 number system, represents numeric values using two different symbols: typically 0 (zero) and 1 (one).

Positional Notation

In a positional numeral system, the base is the number of unique digits, including zero, used to represent numbers. In base-10, we use 0-9 and each position represents a power of 10 (1s, 10s, 100s). In binary (base-2), we use 0 and 1, and each position represents a power of 2 (1s, 2s, 4s, 8s, 16s, etc.).

Conversion to Decimal

To convert a binary number to decimal, you sum the powers of 2 corresponding to each 1 bit. For example, 110121101_2:

  • 1×23=81 \times 2^3 = 8
  • 1×22=41 \times 2^2 = 4
  • 0×21=00 \times 2^1 = 0
  • 1×20=11 \times 2^0 = 1
  • 8+4+0+1=138 + 4 + 0 + 1 = 13

References

AI NOTICE

AI Assistance Disclaimer: This module uses AI-assisted educational models and interactive visual representations to help explain scientific and mathematical concepts. For formal research or academic evaluation, please verify formulas and data against standard primary reference materials.