Programming Logic Controllers

Number Systems and Codes - Chapter 3

Decimal

Each “place” in the decimal system represents a power of ten• Hundred’s• Ten’s• One’s• Tenths• Hundredth’s

Binary

Each “place” in the binary system represents a power of two• Four’s• Two’s• One’s• Half’s• Quarter’s

Hexadecimal

Each “place” in the hexadecimal system represents a power of sixteen• 0 – 9 = 0 – 9• A = 10• B = 11• C = 12• D = 13• E = 14• F = 15

Hexadecimal and Binary

• Hex numbers can be preceded with 0x• Hex is a shorthand way of writing

binary.• Each Hex digit is 4 binary digits• So hex 0x12 is binary 00010010• So hex 0xAF is binary 10101111

Octal

Each “place” in the octal system represents a power of eight• Each octal digit is 3 binary digits• Uses digits 0-7• Some systems indicate

octal numbers using a leading 0• Eg 023 = 010011• Eg 075 = 111101

Programming Notes

Many programming environments allow you to specify literal constants in other than decimal formats. The following is one somewhat common syntax.

0x prefix indicates hexadecimal0b prefixed indicate binary0 prefix indicates octal

Binary Coded Decimal - BCD

• 4 bits represent a single decimal digit• Only 0 – 9 are valid• BCD 0111001100010010 = decimal

7312

Gray Code

Special encoding system where two successive numbers are one “bit flip” apart• 00 = 0• 01 = 1• 11 = 2• 10 = 3Special uses like optical encoders

Binary Arithmetic

AdditionSubtractionMultiplicationDivision

Binary Fractions

• Hole around zero• Decimal .1 has no binary equivalent• 1/3 has no decimal equivalent (or

binary)• 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + …