Module 5: Digital Techniques and Electronic Instrument Systems

5.2 Numbering Systems

Number and Base Base: How many symbols (digits) are used in

the current numbering system? e.g. if the base of a number is 10, we use

(maximum) 10 symbols to display this number. if the base of a number is 8, we use 8 symbols to

display this number. (0, 1, 2, 3, 4, 5, 6, 7). Examples: 1578, 1808210, 1001012.

MSD and LSD In every numbering system, the value of a digit is

determined not only by its symbol, but also by its position. The first digit of a number is called Most Significant digit

(MSD). The last digit of a number is called Less Significant digit (LSD). Example: 184470287

MSD LSD

Binary System Rules 0 + 0 = 0 1 + 0 = 1 0 + 1 = 1 1 + 1 = 0 and 1

carry.

0 – 0 = 0 1 – 0 = 1 0 – 1 = 1 with 1

carry 1 – 1 = 0

Complementary subtraction:

Get the complementary of the subtrahend.

Add 1 Make addition Discard any carries

that exceed the size of the minuend.

Numbering System Conversions Decimal to digital:

I divide the decimal number by 2 I repeatedly divide the result by 2, until the dividend

becomes zero. The residue of each division in reverse order is the binary

number. Digital to decimal:

In the first digit d0 of the binary number I assign d0 *20. In the nth digit of the binary number I assign dn *2n. The sum: d0*20+ d1*21 + … + dn*2n is the decimal number.

Octal, Hex number conversion: The same methodology, but “2” is replaced with “8” and

“16” respectively.