These notes include more information than is needed for MGF 1107.

*MGF 1107 students can ignore everything about binary
fractions, the binary version of scientific notation, and hexadecimal numbers when reading what follows. We only use
binary integers in MGF 1107.*

Web links on using binary numbers

This is done by applying the definition of the place values
for a binary number, evaluating each one in base ten. For
example, we convert **1001 _{two}**
from binary to decimal as follows:

1*8 + 0*4 + 0*2 + 1*1 = 8 + 0 + 0 + 1which is

Fractions are handled with negative exponents
for each fractional place value, just as with decimal numbers,
so we convert **0.11 _{two}**
from binary to decimal by recognizing that

1*(1/2) + 1*(1/4) = 0.5 + 0.25gives

The integer part is obtained (working from right to left) by repeated
division by two (which shifts the binary-point to the left), keeping
the remainder as the next binary digit and using the integer quotient
for the next step. We stop when the quotient is zero.
(*The easiest way to get the integer part is to use the Hex
conversion feature of a calculator and convert hexadecimal to binary, but we are not using calculators in this unit.*)

The fractional part is obtained (now working from left to right) by repeated multiplication by two (which shifts the binary-point to the right), keeping the integer part as the next binary digit and using the fractional part for the next step. There are no easy shortcuts for the fractional part, but watch for any repeating pattern.

We can write the result as a normalized floating-point binary number with the convention that the mantissa (the significand) is between 1 and 2 -- that is, the mantissa is of the form 1.ffffff.

**Examples: **

**1.** Convert **11 _{ten}** to binary:

The integer is converted by

11 / 2 = 5 r 1 (1) in "ones" place 5 / 2 = 2 r 1 (1) 2 / 2 = 1 r 0 (0) 1 / 2 = 0 r 1 (1)and is

(This is 1.011 x 2

**2.** Convert **5.75 _{ten}** to binary:

The integer part is found by

5 / 2 = 2 r 1 (1) in "ones" place 2 / 2 = 1 r 0 (0) 1 / 2 = 0 r 1 (1)and is

The fractional part is found by

0.75 x 2 = 1.5 (1) 0.5 x 2 = 1.0 (1) 0 x 2 = 0 (0) so it terminatesso the number is

(This is 1.0111 x 2

**3.** Convert **0.1 _{ten}** to binary:

The fractional part is found by

0.1 x 2 = 0.2 (0) 0.2 x 2 = 0.4 (0) 0.4 x 2 = 0.8 (0) 0.8 x 2 = 1.6 (1) 0.6 x 2 = 1.2 (1) 0.2 x 2 = 0.4 (0) which repeats 0.2 above ... 0.4 x 2 = 0.8 (0) etc ... 0.8 x 2 = 1.6 (1) etc ...so the number is

(This is 1.10011001.... x 2

**Decimal** - **Hexadecimal** - **Binary Table**

0 1 2 3 4 5 6 7 (decimal) 0 1 2 3 4 5 6 7 (hexadecimal) 0 1 10 11 100 101 110 111 (binary) 8 9 10 11 12 13 14 15 (decimal) 8 9 A B C D E F (hexadecimal) 1000 1001 1010 1011 1100 1101 1110 1111 (binary) 16 17 .... (decimal) 10 11 .... (hexadecimal) 10000 10001 .... (binary)

**Historical note:**

Way back in the 1950s, the ILLIAC was built at the University of
Illinois (Urbana-Champaign) using the "sexadecimal" number system.
However, they counted ... 8, 9, K, S, N, J, F, L instead of using
... 8, 9, A, B, C, D, E, F as we do today.
They chose those letters (which were remembered using the mnemonic
**K**ing **S**ized **N**umbers **J**ust **F**or **L**aughs)
because they proved the most convenient ones when they converted a
teletype to be used for punching the paper tape used for I/O.
*(This info from J. Sutherland Frame, professor emeritus of Mathematics
at Michigan State University, an early programmer of the 1957 clone
of the ILLIAC known as
MISTIC, both of which are in the
ORDVAC family of computers.)*

(An aside: Fans of "2001: A Space Odyssey" will recall that the HAL-9000
computer was built in Champaign-Urbana. This reflects the historic
role the Univ. of Illinois had in the early development of computers
with the construction of the ILLIAC.
In addition, although Everyone knows that HAL is rot25 of IBM, the serial
number reflects the sequence used by Control Data, whose 6000 series
machines were *the* supercomputer in the mid-60s when "2001" was made.
The Cray-1 would have been an 8000 series machine, but CDC chose the "Star"
design that led to the Cyber-205 and Cray formed his own company. Kubrick
guessed about right in placing a 9000 series machine that could talk in
the early 90s - HAL was born on 1/12/92 - but someone forgot the mouse.)

This material is © Copyright 2000, by James Carr.