Numeration Systems

Digital computers communicate with external devices through ports: sets of terminals usually arranged in groups of 4, 8, 16, or more (4 bits = 1 nybble, 8 bits = 1 byte, 16 bits = 2 bytes). These terminals may be set to high or low logic states by writing a program for the computer that sends a numerical value to the port. For example, here is an illustration of a micro controller being instructed to send the hexadecimal number F3 to port A and 2C to port B:

Suppose we wished to use the upper four bits of port A (pins 7, 6, 5, and 4) to drive the coils of a stepper motor in this eight-step sequence:

Step 1:

0001

Step 2:

0011

Step 3:

0010

Step 4:

0110

Step 5:

0100

Step 6:

1100

Step 7:

1000

Step 8:

1001

As each pin goes high, it drives a power MOSFET on, which sends current through that respective coil of the stepper motor. By following a “shift” sequence as shown, the motor will rotate a small amount for each cycle.

Write the necessary sequence of numbers to be sent to port A to generate this specific order of bit shifts, in hexadecimal. Leave the lower four bit of port A all in the low logic state.

The numeration system we use in our daily lives is called base ten, also called decimal or denary. What, exactly, does “base ten” mean?

Given the following base-ten number, identify which digits occupy the “one’s place,” “ten’s place,” “hundred’s place,” and “thousand’s place,” respectively:

Observe the following sequence of numbers:

00

01

02

03

04

05

06

07

08

09

10

11

12

13

14

15

16

17

18

19

20

21

.

.

.

What pattern(s) do you notice in the digits, as we count upward from 0 to 21 (and beyond)? This may seem like a very simplistic question (and it is!), but it is important to recognize any inherent patterns so that you may understand counting sequences in numeration systems with bases other than ten.