Learning paths

Learning paths

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Learning path

Full curriculum

Full curriculum
  1. Amplitude quantization and quantization error
  2. Information and data
  3. Binary representation
  4. Bits, bytes and integer encodings
  5. Digital registers
  6. Bitwise operations and bit masks
  7. Digital symbols, bit rate and symbol rate
  8. Cryptographic hash functions
  9. Introduction to electric circuits
  10. Digital-to-analog conversion

Unit content

Binary representation

A positional number system represents a number using digits whose values depend on position. Ordinary decimal notation uses powers of ten; binary uses only the digits $0$ and $1$ and powers of two.

For example,

$$1011_2=1\cdot2^3+0\cdot2^2+1\cdot2^1+1\cdot2^0=11_{10}.$$

Place values

Moving one binary position to the left multiplies the place value by two:

... 16  8  4  2  1
     1  0  1  1

The pattern 1011 therefore selects $8+2+1$.

Converting to binary

A nonnegative integer can be written as a sum of powers of two. For example,

$$13=8+4+1,$$

so

$$13_{10}=1101_2.$$

Why binary is useful

Physical digital systems can reliably distinguish two states, so binary symbols map naturally to electronic representations.

Binary notation is still just a representation of a number. The same numerical value can be written in decimal, binary, hexadecimal or another base without changing the underlying quantity.