**Videos**

**About this topic**

All computers – from large mainframes to hand-held micros – ultimately can do one thing: detect whether an electrical signal is “on” or “off”. Computer programs in BASIC and Pascal are converted by various pieces of systems software into sequences of bits (*B*inary dig*IT*s) which correspond to sequences of on/off (equivalently TRUE/FALSE or 1/0) signals. Proficiency in the binary number system is essential to understanding how a computer works.

Since binary numbers representing moderate values quickly become rather lengthy, bases eight (octal) and sixteen (hexadecimal) are frequently used as short-hand. Octal numbers group binary numbers in bunches of 3 digits and convert the triplet to a single digit between 0 and 7, inclusive. For example, 1001010110_{2 }= 001 001 010 110_{2 }= 1126_{8}. Hexadecimal numbers group binary numbers by fours, and convert the quadruplet to a single digit in the range 0, 1, 2 …, 9, A, B, C, D, E, F. For example, 10110110100101_{2 }= 0010 1101 1010 0101_{2 }= 2DA5_{16}.

**References**

Many pre-Algebra textbooks cover bases other than 10. From the computer science point of view, most books covering Assembly Language also cover binary, octal and hex number systems. The texts cited for the Boolean Algebra category cover computer number systems.

**Sample Problems**

Solve for
One method of solution is to convert 3676 An easier solution, less prone to arithmetic mistakes, is to convert from octal (base 8) to hexadecimal (base 16) through the binary (base 2) representation of the number: 3676 = 0111 1011 1110 = 7BE |

Solve for
The rightmost digit becomes |

In the ACSL computer, each “word” of memory contains 20 bits representing 3 pieces of information. The most significant 6 bits represent Field A; the next 11 bits, Field B; and the last 3 bits represent Field C. For example, the 20 bits comprising the “word” 18149 E 1 B 7 D = 1110 0001 1011 0111 1101 = 1110 00 01 1011 0111 1 101 Field B = 01 1011 0111 1 = 011 0110 1111 = 3 6 |