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JCM inVentures Inc.

info@jcminventures.com

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Objective 1.1:  Describe the binary numbering systems

Resources:

For this section you should have the following resources:

bulletVulcan Logic Trainer
bullet22 ga wire
bulletVirtual Vulcan Logic Simulator
bulletMulti-media logic Simulator

The BIT ( Binary Digit )

A binary number, at its most basic level, is called a BIT ( BIT being short for Binary-digIT).  A BIT is a value which can hold two possible states.  These states can be many different things.  They can be...

bulletTRUE or FALSE
bulletON or OFF
bulletYES or NO
bulletUP or DOWN
bulletACTIVE or INACTIVE

Numerically they are expressed as ONE (1) or ZERO (0).  Every piece of information stored in a computer system ultimately breaks down to one of these two BITS.

In digital electronics, these bits are represented by switching on or off a small amount of electric current to a device which is capable of sensing that current. 

Experiment:  Wire up your own digital BIT:

On your Vulcan logic trainer or Virtual Vulcan, connect switch1 to lamp10.  Turn on the Vulcan and set the switch ON or OFF.  The lamp shows you the digital state of that switch!

Logic Probes

So there you have the basics of our entire information system revolution!  The lonely little BIT.  The instrument we use in Digital electronics to measure this binary digit is called the LOGIC PROBE.  You will find a logic probe connection on your Vulcan trainer on the "header" socket shown attached to the green wire in the following illustration.  A logic HIGH is shown as a RED lamp in the logic trainer, and a logic LOW is shown as a GREEN lamp.  An absent logic level ( neither HIGH or LOW) is indicated by NO lamp lighting.

The LOGIC PROBE connection

Binary Weighted Numbers 

Binary values can represent conditions of various elements such as:

bullet

robot forward / robot backward

bullet

light sensor on / light sensor off

bullet

touch sensor touching / touch sensor not touching

bullet

water touching sensor / water not touching sensor

Any phenomenon which has two possible conditions may be represented by a BIT.

You can also combine many bits together to create a number!  If you need to have a computer recognize a quantity, you can place the bits in a sequence of "binary weighted numbers".  This is called the Base-2 numbering system.  

(Base 2 numbering system)

Now, you may have noticed that the BIT doesn't carry much weight.  Sure, it's fine if you want to describe one of two quantities like UP or DOWN, YES or NO.  What do you do if you need more than that.  How do you turn the bits into  numbers?

Well, you take a large number of bits, and turn them into something larger.  You can take eight BITS, for instance, and create a number called a BYTE!

Each bit in the byte represents a weighted value, and the value of that bit determines if that value should be included in the number or not.  0 means, don't include that weighting, 1 means DO include that weighting.

Bit Most significant            

Least significant

Weighting 128 64 32 16 8 4 2 1

The weighting of the BIT increases by a factor of two from the BIT to it's right.  Showing numbers in binary format is called the BASE-2 numbering system.  

Click on image to download MMlogic simulation.

Try clicking on the illustration above to download the Multimedia logic simulation.  When you press any key on the keypad, the binary value which is associated with that number will show up on the lights.  Remember, if the light is ON, then add on its associated weighting.

A partial binary equivalence table is shown below...

Decimal

 

Binary
0

 

0000
1

 

0001
2

 

0010
3

 

0011
4

 

0100
5

 

0101
6

 

0110
7

 

0111
8

 

1000
9

 

1001

Try This:

Wire up the Vulcan trainers binary switches to the first four lamps as shown.  The binary value shown in the lamps is decoded in the red number display on the top of the Vulcan.

As you set in various binary combinations ( 0 to 9 ), you should see the actual number show up in the number display.

For more information on binary numbering systems, try these sites...

http://www.nwu.edu/musicschool/links/projects/midi/pages/undstdbh.html 

http://whatis.techtarget.com/WhatIs_Definition_Page/0,4152,213816,00.html 

http://www.usbyte.com/common/Binary%20System.htm 

 

Base-8 ( Octal ) Numbering system

More to come:

Until I can make it back to update this section, visit:

http://whatis.techtarget.com/WhatIs_Definition_Page/0,4152,212687,00.html 

 

 

Base-16 ( Hexadecimal ) Numbering system

More to come!

In the meantime, visit these great links:

http://webopedia.internet.com/TERM/h/hexadecimal.html 

http://ag.arizona.edu/classes/ABE547/lab4/HexadecimalCounting/ 

http://whatis.techtarget.com/WhatIs_Search_Results_Exact/1,282033,,00.html?query=hexadecimal 

 

 

Converting between numbering systems

I'm off to do some work on the LOGIC GATES page...  If you want to learn how to convert between numbering systems, try these links...

http://jimvb.home.mindspring.com/binary.htm 

http://www.jaworski.com/htmlbook/dec-hex.htm

http://www.cut-the-knot.com/binary.html

http://www.studdard.com/tutorial/base/binary-cvt.html

http://acc6.its.brooklyn.cuny.edu/~gurwitz/core5/nav2tool.html