**MATH**

____________________________________

**MATHEMATICS OF COMPUTERS(can you count to "one" ?)
UNDERSTANDING DECIMAL,
HEXADECIMAL,
AND BINARY NUMBERS**

Since computers use BINARY ( BI means two,

as in BI- cycle, for a TWO wheeled bike,

or BI-Plane, for a plane with two wings - one

on top of the other, typically the older wooden

planes from the parent, Wright Brothers first

design ).

BI-nary numbers only have ( 2 ) TWO numbers.

a ZERO ( 0 ) and a ONE ( 1 ).

Before we start doing the mathematics of BINARY,

it is very important to understand ordinary,

everyday DECIMAL numbers. You might think that

you already know DECIMAL numbers, since you

have been using them for years, but there are a

few CONCEPTS that most people take for granted

since they "use" the numbers every day, without thinking

about some of the things that happen in simple counting!

First, DEC-imal is based on ( 10 ) TEN numbers,

( which you are thinking, I already know that ),

and if I ask you to count all the fingers on both your

hands, you will count, 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10.

Similarly, if I ask you "What are the TEN numbers in the

decimal system?" you probably make the mistake

of saying that the numbers are:

1, 2, 3, 4, 5, 6, 7, 8, 9, and 10

Wrong.

It is very common to start counting at ONE (1 ).

ALSO, " 10 " ( TEN ) is not a number, it is two numbers,

a ONE ( 1 ) and a ZERO ( 0 ).

For thousands of years, entire Civilizations, such as

the Roman Empire, were missing the most important

number -- ZERO (0 ). This crippled their mathematics,

and made Accounting, and Science and Engineering

very difficult, and sometimes, impossible.

Although we take for granted that everyone in the

world uses DECIMAL ( 10 ) based mathematics, this is

NOT the general rule if you look for hundreds and

thousands of years into the past.

In a survey of 1920, Eskimos never counted more than

the number 5 in everyday use. In their lifetime, it was almost silly

to think of having 25 igloos, or 25 kayaks, or 25 harpoons.

If you asked an Eskimo, " how many igloos do you have",

they would probably reply " ONE ". To see

counting numbers in 5,000 languages, CLICK HERE.

http://www.zompist.com/numbers.shtml

In Australia, a number of Aboriginal languages had

two counting numbers, for example, the Watchandies

tribe, used cooteon for one and utaura for two.

Three would be cooteon/utaura ( one/two ) while

four would be ataura/utaura. (two/two)

Any number more, was booltha, or "MANY"

Again, the idea idea of having 25 spears, or

25 boomarangs, was impractical and not needed.

Other Civilizations used FIVE (5) numbers in counting,

or (3 ) THREE numbers, or ( 6 ) numbers, so, Historically,

10 ( TEN ) DECIMAL numbers is only a recently used

System. Some languages, such as Yumbri, had NO numbers

at all, just the words, neremoy (little) and nakobe (much).

The area of the Yumbri has ancient ruins such as these.

The discovery of ( 0 ) ZERO in " OUR " computer

world is recent as well, although ancient, and what were

termed " Primitive " Civilizations, have been using it for

thousands of years, even if they used a system of 5, 20, or 60.

MAYAN symbols, used either pictures of complex

characters , or just the heads of characters, or

a simplified character set of 3 symbols: a DOT = 1

a LINE = 5, and a CLAMSHELL = 0. Groups of

numbers are each 20 times bigger, 1's, 20's and 400's.

Only 4 MAYAN " CODICES" or books survived, and the

Dresden book showed the orbits of the planets. In the

picture below, from page 9, the Goddess of CORN is

shown on the left. By using the extremely accurate calenders,

the Mayans knew when to plant seeds. To see the actual books

CLICK HERE. http://kidbots.com/MATH/codex.html

A good LINK on the web to view different Mayan

NUMBERS is at CLICK HERE.

http://www.pauahtun.org/Calendar/numbers.html

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COMPUTER MATHEMATICS

_________________________________

OK, back to DECIMAL, ( 10 ) based number systems...

The TEN ( 10 ) numbers in DECIMAL, are:

0, 1, 2, 3, 4, 5, 6, 7, 8, and 9

NOTE ! ! ! There is NO number 10 ( TEN ) in the DECIMAL,

or " 10 " ( TEN ) based counting system !!!

The biggest counting number you can use in a TEN ( 10 )

based counting system is TEN minus ONE equals " 9 ".

Lets look at ordinary DECIMAL counting in detail and point

out some obvious, but interesting things that happen.

If you start counting, you get:

0

1

2

3

4

5

6

7

8

9 -At which point, you run out of numbers.

Now, you ADD 9 PLUS 1, in the ONES column,

NOTE WHAT HAPPENS ! !

9

+1

______

= 0

YUP... 9 plus 1 = ZERO !!!

You probably add NINE plus ONE and get

the answer ZERO every day, without thinking

about it, but it is a VERY important point !

There is one more step to do ...

CARRY A ONE (1) into the next column,

and start counting all over again at ZERO ( 0 ).

1 0

1 1

1 2

1 3

1 4

1 5

1 6

1 7

1 8

1 9

AGAIN, at this point, you add a ONE (1) to the NINE (9)

1 9

+ 1

_____

= 0 AGAIN, 9 + 1 = 0 ( ZERO ), in the ones column,

Now you CARRY a ONE into the next column , and

start counting from ZERO again...

2 0

2 1

2 2

2 3 etc.

When you count up to 99, again you run out of numbers,

and you take the last (9 ) nine, CARRY a one over to

the next column, and reset the NINE ( 9) to Zero

97

98

99

9 9

1+ ( add (1) ONE to the last 9 )

___

( RESET the last 9 to a ZERO ( 0 ) )

EQUALS

9 0 ( Again, note that 99 + 1 = 90 ! ! )

1 + ( and once more, we CARRY a 1 )

___

Then the FIRST ( 9), adds to one, RESETS to ZERO (0 ),

and CARRIES a One to the next COLUMN.

EQUALS

1 0 0

OK, lets try this again in the QUADernary system....

In a QUAD ( 4 ) system, there are 4 numbers,

0, 1, 2, and 3

if you start counting, you get:

0

1

2

3 ( and you run out of numbers )

So you CARRY a (1) ONE into the next column, and reset

the (3 ) three to a Zero, and start counting again.

3

+1

___

1 0 Equals ( NOTE 3 + 1 = 0 )

( 10 q, in Decimal, equals our 4 )

Then you start counting again:

10 one 4 + zero 1's = 4 decimal

11 one 4 + one 1's = 5 d

12 one 4 + two 1's = 6 d

13 ( And you run out of numbers)

So you carry a ONE (1) over to the next column, and reset

the ( 3) three to a ZERO ( 0 ), and start counting again.

20 (Which in Decimal, is our 8 )

21 two 4's + one 1's = 8+1= 9 d

22 two 4's + two 1's = 8+2= 10 d

23 ( And you run out of numbers)

So you carry a ONE (1) over to the next column, and reset

the ( 3) three to a ZERO ( 0 ), and start counting again.

Can you guess what the next 9 numbers will be in the

sequence? Try writing them down on paper.**

Here is another analysis of counting numbers and COLUMNS.

In the DECIMAL counting system, when you count to (9), NINE,

you run out of numbers, and you carry a ( 1 ), ONE, into the

next column. This new column in decimal is the " TENS" (10's)

column, and each number is multiplied by ( 10 ) TEN.

When you count to 99, you run out of counting numbers in the

last column, and you carry over a ( 1 ), ONE, into the next

( 10's) TENS column, from the ( first column, the " ONES" ),

and then you run out of numbers in the TENS column, so

you carry over another ( 1 ) ONE into the HUNDREDS column,

and RESET the ( 9 ) NINEs to ZERO ( 0 ).

In ordinary, everyday Decimal, if you read a number like

384

You are really saying 3 ( Times ONE HUNDREDS )

PLUS

8 ( times TENs )

PLUS

4 ( Times ONEs )

3 x 100 + 8 x 10 + 4 x 1

Note that in the ( 10 ) TEN based, DECIMAL system, the

biggest digit is (9) , BUT, the columns that you

carry the ONE ( 1 ) over to, ARE EACH (x 10 ) TIMES TEN.

so that you have:

HUNDREDS, TENS, ONES columns. To get another column,

if you run out of numbers at 999, you multiply the last

column, the 100's column by 10 ( 100 x 10 = 1000 ),

and you get the thousands column.

You then get the columns:

1000 100 10 1

Thousands, Hundreds, Tens, Ones,

(10x10x10x1), (10x 10x1 ), (10x 1 ), (x1)

as each column is TEN times Bigger than the last one.

If I write down that I have " 260 Dollars " in DECIMAL ( 10 )

I have 2 times the HUNDREDS,

plus 6 times TENs,

plus ZERO times ONES.

2 6 0

Hundreds, Tens, Ones,

(10x 10x1 ), (10x 1 ), (x1)

2 x 100 + 6 x 10 + 0 x 1

200 + 60 + 0

This is a total of ( 260 ) Dollars.

In the example of the QUAD ( 4 ) based system, if I say

I have ( 1 2 3 ) dollars, this would be, noting that columns

each are ( 4 ) times bigger than the previous, or

SIXTEENS, FOURS, ONES columns,

( 4x4x1 ) (4x1 ) ( x1 )

123 Dollars =

1 2 3 Dollars

(4x4x1 ) (4x1 ) ( x1 )

16 4 1

I have ONE ( times 16 )

plus 2 ( times 4 )

plus 3 ( times 1 )

EQUALS 16 plus 8 plus 3

EQUALS (27 ) Twenty Seven Dollars in our DECIMAL system.

Lets try another counting system, this time ( 9 ).

The numbers would count from ZERO (0 ) thru (8 )

(Remember, the highest number in a counting system

is ONE LESS than the SYSTEM number - for example,

a 17 NUMBER system would have the highest number

as 16, but each CARRY OVER column, would be 17

times bigger than the previous.)

In a ( 9) NINE based system, you would count:

0, 1, 2, 3, 4, 5, 6, 7, 8 and then run out of counting

numbers. So you carry over from the " ONEs " column,

into the next column, and since this is a ( 9) based system,

the next column is ( TIMES 9 ) or " 9's " column. You then

RESET the 8, to a zero, carry over a ( 1 ) into the next

column ( the 9 times 1 = "9's ) column, and start over

counting:

10, 11, 12, 13, 14, 15, 16, 17, 18, and again, run out of

numbers, since there is no number higher than 8.

You reset the 8 to a ( 0 ) ZERO, carry over a ( 1 ) ONE

into the ( 9's ) NINEs column, and keep on counting.

20, 21, 22, 23, 24, 25, 26, 27, 28, and you run out of

numbers again. So you reset the 8 to ( 0 ) ZERO, carry

over a ONE ( 1 ) into the next column, ( the 9's column)

and keep counting, 30, 31, 32, 33, 34, 35, etc.

In this system of NINEs, if I say I have 121 Dollars,

The COLUMNS are each ( 9) NINE times bigger, or

81's column, 9's column, 1's column

(9x9x1) (9x1) (x1 )

1 2 1 dollars

121 (base 9) would be

ONE x 81's, plus 2 x 9's plus 1 x 1's or

(81 ), plus 18, plus 1 =

81 + 18 + 1 = 100 Dollars. ( In our Decimal )

I f you have paid attention and done the mathematics

above, you will quickly and easily understand BINARY.

Binary is a ( 2 ) number system, so the biggest number you

can count to, is ( 2 ) - 1 = 1.

You Count:

0,

1,

AGAIN, just like we added 1 + 9 above in the DECIMAL

review above, we now add

1 ( Binary )

+

1 ( Binary )

____

WATCH WHAT HAPPENS ! !

1

+

1

____

= 0

YES, 1 + 1 = 0 ( ZERO ) !

Just as we did in adding 9 +1 ( = 0 ) above

we now have to CARRY a (1) ONE into the next

column.

Since this is a TWO (2 ) based system, each column in the

system of CARRY OVER, is ( 2 ) times bigger, so that the

next column is ( 2 ) x 1 = 2, so that the columns are

Two's Ones

( 2x1 ) ( x 1 )

and after counting : 0, 1, you RESET the ONE ( 1 ) to ZERO,

and carry over to the ( TWOs ) 2's column.

you now count:

0

1

1 0 ( 10 b ( binary ) equals our Decimal 2 )

you then keep counting:

0

1

10 , you add a one in the ONEs column,

11, and you run out of counting numbers, so you

RESET the ONEs column 1 to a ZERO, and CARRY over a one

to the TWO's column.

11

+1

_____=

10

+1

Here, we CARRY a new 1 into the next cloumn

But at this point you have1 plus 1 in the TWOs column, and

since there is no bigger number than ( 1 ) ONE to count to,

you RESET the (1) in the TWO's column to ZERO, and Carry

over a ONE into the next column. Since this is a TWO (2 )

Binary based system, each column is ( 2 ) times as big

as the last one, so the new column would be:

( 2 x 2 x1 )

Giving three columns,

4 2 1

( 2 x 2 x1), (2 x1 ), (x 1 )

and with the third new column of CARRY OVERS, you can

keep on counting:

0 = 0 Decimal

1 = 1 (decimal)

10 = 2 (d)

11 = 3 (d) since 1x2 + 1x1 = 3

100, Now you can add a ONE in the ONEs column.

101, and when you add another ONE in the ONEs

column, you RESET to ZERO, and CARRY

110, Now you can add a ONE in the ONEs column,

111, And Now you run out of numbers in the ONEs

so you ZERO the ONEs, Carry, ZERO the

TWOs, carry, ZERO the FOURs, and carry

GIVING you a new column, ( 2x2x2x1 ) or the EIGHTs column.

1000, Now you can add ONE in the ONEs column,

1001, Zero the One in the ONEs, Carry over to TWOs,

1010, Now you can add ONE in the ONEs column,

1011, Now you have to Zero the ONEs, carry to the

TWO's, Zero the TWO's column, and carry

into the FOUR's column,

1100, and keep counting.

When you reach 1111, you Zero all the ONEs, and Carry into the

fifth new column, the ( 2x 2 x 2 x 2 x1 ) ( or 16's DECIMAL )

column, giving you columns:

16 8 4 2 1

(2x2x2x2x1 ), ( 2x2x2x1), ( 2x2x1), ( 2x1 ), ( x 1)

.

Now if I say in BINARY, that I have 01010 Dollars, I have:

0 1 0 1 0 dollars

16 8 4 2 1 columns

(2x2x2x2x1 ), ( 2x2x2x1), ( 2x2x1), ( 2x1 ), ( x 1)

0 (times 16s ) plus

1 ( times 8's ) plus

0 ( times 4's ) plus

1 ( times 2's ) plus

0 ( times 1's )

EQUALS 0 +8 + 0 + 2 + 0 = 10 Dollars ( In our Decimal ).

So that in Binary, examples would be:

0001 = ONE (1 ) in Decimal

1000 = EIGHT ( 8) in Decimal

1001 = NINE ( 9 ) in Decimal ( Note 8+1=9)

0010 = TWO ( 2 ) in Decimal

1010 = TEN ( 10 ) in Decimal (Note 8+2=10)

0100 = FOUR ( 4 ) in Decimal

0111 = SEVEN ( 7 ) in Decimal (4+2+1=7)

and

1111 =

1 1 1 1

( 1x8 + 1x4 +1x2 + 1x1 )

= 8 + 4 + 2 + 1

= FIFTEEN ( 15 ) Decimal.

(( TRY to read this sentence properly :

There are only 10 kinds of people

- those who know Binary,

and those who do not.))***

Since " modern" typical computers used 8 bits as

a standard grouping, common numbers inside a

Mailbox, or MEMORY LOCATION, would typically look like:

00001010

11111110

10111011

11101100

00101100

11101010

00001010

11111110

11101011

11101100

01011100

11101010

which, when you have go through thousands of memory

locations BY HAND, writing them all down, is extremely

confusing. It is very easy to make mistakes in copying

row after row of BINARY numbers, particulary when you

know that after the 80-386 and 80-486 typical " IBM"

types of computers, the typical memory group doubled

into 16 BITs of Binary ( 110010110101011 )

and 32 Bits of Binary ( 100010110101011101010100101011)

and now, with the newest chips, it is 64 bit groups, so that

a list of memory locations now looks like:

1000101001010111000101001010111000101011010111000101001010111010

1000101101010111000101001010111100010100101011100010100101011110

1001010111001101100101011100010110010101110101011001010111000101

1110010110010101110001011001010111000101100101011100010110010101

1001010111000001100101111100000110010101110110011001011111010001

1000101001010111000101001010111000101001010111000101001010111010

1000101001010111000111001010111100010100101011100010100101011110

1001010111010101100101011100010110010101110001011001110111100011

1101010111110101110001011011010111000101100101011100010110010101

1001010111000001100101011100000110010101110000011001010111011001

ETC.

Do you think you could copy 10,000 lines of these 64 BIT Binary

numbers, and not make a single mistake?

Since everyone DID make mistakes in copying these Binary numbers

over and over again, a system of SHORTHAND, was developed

to turn the confusing mess of ZEROs and ONEs ( 0s and 1s )

into something easier to understand and work with.

Since most computers, particulary the 8086 ( the first INTEL chip

of the series ), the 8088 ( a cheaper version of the 8086 released

at a later date, and used by IBM in their first PC computer ),

the 80186 ( a more powerful version of the 8086, NOT used by

IBM but used by many CLONE computers), the 80286

( Known commonly as the " 286 " or " AT " ( Advanced

Technology )), and the 80386, all used the 8 BIT BINARY memory

locations, the common HEXADECIMAL Shorthand developed

in changing the 8 bit BINARY locations into TWO (2 ), CHARACTERS.

HEX is 6 and DECIMAL is 10 ( 6 + 10 = 16 ), so HEXADECIMAL is a

mathematical system with 16 NUMBERS, and you count :

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, - hey, I just ran out of characters to count with-

there ARE NO NUMBERS to count with higher than ( 9 ) nine

in our system... How do I write down numbers higher than 9

in a 16 BASED system, when I only have ZERO ( 0 ) to NINE (9)?

What the engineers did was to use LETTERS OF THE ALPHABET,

A, B, C, D, E, and F, to replace the missing numbers,

10, 11, 12, 13, 14, and 15, that do not exist in our DECIMAL system.

So in a 16 ( HEXaDECimal ) based system, the highest number is

( 16 - 1 = 15 ), and you count from ZERO ( 0 ) to ( 15 ) like:

0 =0 ( decimal )

1 =1

2 =2

3 =3

4 =4

5 =5

6 =6

7 =7

8 =8

9 =9

A =10

B =11

C =12

D =13

E =14

F =15 (decimal )

As it turns out, in BINARY, if you remember from above,

15 (decimal ) in Binary is 1111 ( b inary )

and since an eight digit binary number like 10001010

is TWO GROUPS OF FOUR (4 ), 1000 and 1010,

you can write each of the groups of ( 4 ) FOUR, in

only two HEX (adecimal ) numbers...

FOR EXAMPLE, if an 8 bit binary number is 11110001

this would be two groups -- 1111 and 0001

which in BINARY, would be 1111 (b )= 15 Decimal,

and 0001 (b ) = 1, so you write down the HEX adecimal

Shorthand of ( 15 , 1 ) ( d ecimal )

as ( F , 1 ) in ( HEX or h ).

In order to use this shorthand, you can first change groups

of (4 ) four BINARY DIGITS into DECIMAL, THEN,

you write the HEX (adecimal, or h ) number in place of the

Decimal Numbers...

EXAMPLES:

8 BIT MEMORY, at LOCATION 378 ( HEX ), has in it:

10100111 ,

First, break the 8 bits into TWO groups of 4:

1010 and 0111

These BINARY GROUPS, in DECIMAL ( from the tables above)

would be

( 10 ) and ( 7 ).

Looking at the table above of HEX, we see that

( 10 ) d (decimal ) is A in HEX, and ( 7 ) d, is ( 7 ) in HEX.

SO that now 10100111 can be written as

A 7

Thus, long lists of 8 bit BINARY code like:

11110000

00011000

11001110

00000001

11111111

can be written in HEX Shorthand as:

F0

18

CE

01

FF

which is a LOT easier to write down and to copy,

and to check over long lists to see if there are any mistakes.

The earliest computers that you would use in a home

used a keypad like on a telephone, with the HEXADECIMAL

numbers on them, and when you programmed them, you

typed in lists of HEX, which a tiny program on the computer

changed into 8 Binary bits that the computer would use in its

memory locations.

NOTE: Sometimes you do not have ZEROs at the front of

a binary number. This is the same situation as in every day

decimal. For Example, you would NOT say

" I have ZERO ZERO ZERO twenty six Dollars"

You would just say " I have twenty six dollars".

In Binary, a number might leave OFF the ZEROs at the

front, so that 00000111 may be written as just 111.

To convert to HEXadecimal, you just add the missing

ZEROs at the front, so that you have all 8 characters...

So that 111 becomes 0000 0111, which, in HEX

becomes 0 ( for 0000)

and 7 (for 0111)

THUS, 111 in binary = 07 in HEX, using TWO hex characters.

In the same way, Zero (0) in two digit Hex would be:

0000 0000 as the computer sees the 8 bits, or in HEX,

0000 = 0 and 0000 = 0, so HEX is 00 in 2 digits. This is

the smallest number in two digit HEX, while 1111 1111,

where 1111= F, 1111= F, so the largest two digit Hex

is FF.

In the programming of a typical robot computer, I

can use BINARY or DECIMAL numbers to put in the "Mailbox"

"Memory" " RAM ( Random access memory) locations.

The BASIC programming language uses HEXadecimal

counting, very often to locate a MEMORY location, so that

instead of saying put 0000 0001 in MEMORY location

0000001101111000, it says put the value ONE (1) in

memory location 378 ( Hex ).

It is a lot easier to say "Put 1 in 378" than to say

"load00000001in0000001101111000". And it is a lot

less confusing. This is why HEX is used a great deal

in typical computing. Note that HEX location 378

is actually a FOUR digit Hex number, 0378, with the

leading zero dropped. To continue counting after you

get to 1111 1111 in binary, or two digit HEX FF, just

keep multiplying the last column by 2 in binary, so that

1111 is 8s, 4s, 2s, and 1s, the next colum would be

16s, then 32's, then 64s, then 128s, then 256s, etc.

256 128 64 32 16 8 4 2 1

1 1 1 1 1 1 1 1 1

So that, 100000000 binary is 1x256=256 decimal

110000000 binary is 1x256 + 1x128 = 384 d

100000010 binary is 1x256 + 1x2 = 258 d

HEXadecimal "shorthand" 0378 is

0000 for the first 0

0011 for the 3

0111 for the 7 , and

1000 for the 8.

If you put these together into a SINGLE BINARY, you get:

0000001101111000. IF you remove the leading zeros, you get

1101111000, giving columns:

1 1 0 1 1 1 1 0 0 0 (b)

512, 256, 128, 64, 32, 16, 8, 4, 2, 1 (d)

= 512+256+64+32+16+8 (d)

= 888 in Decimal

= 378 in Hexadecimal

= 1101111000 in Binary

= 1570 in Octal (see below)

You can use this method to find any Binary or Hex number!

There is a built in CALCULATOR in most Windows Systems

that will CONVERT from one counting system to another, but

the Times, Divide, Decimal , and MINUS buttons are ridiculous,

so that I use a FREE calculator with intelligent buttons! To download

the FREE calculator, CLICK HERE.

Numbers used in computing can change from DECIMAL,

to BINARY, to HEXADECIMAL, constantly, so that each

programming language, like BASIC, QBASIC, BASICA,

GWBASIC, etc. uses a way to let you know what number

is being used, and how to ENTER what kind of number in

to the programs. In a textbook, you might find things like

0135 ( h ) for Hexadecimal, or 3120 ( d ) for decimal.

In many BASIC programs, Hexadecimal might be written

as &378H, where the "&H" (Ampersand) sign and H, is used to tell

the computer that the number is in HEX. A great many

of the first computers did NOT use groups of 4 binary

digits, but used 5 digits, 6 digits, 9 digits, 10 digits, 15 digits,

etc., so that the HEX shorthand is not the ONLY one

commonly used today.

OCTAL, or 8 is often used, even

today. Since OCTAL uses base 8, the largest number is

7, or BINARY 111, so that all computers using 3, 6, 9, 15

etc. bits, would group the binary digits in sets of 3, and

use OCTAL Shorthand instead of HEX - You often see the

OCT key on a calculator today.

EXAMPLE:

A computer uses 9 bits instead of 8, so the data looks like:

001111011

Which breaks into 3 binary groups, 001 111 and 011

And these binary groups in DECIMAL are 1, 7, and 3,

(( note that in Octal, which has the biggest number as a 7

which is less than Decimal 9, the Octal and Decimal numbers

all appear the SAME - you do not have to use A, B, C, etc. ))

THUS, 001111011, can be written simply in OCTAL shorthand

as 173 (octal )

** ANSWERS to above questions: To continue counting in QUAD,

We so far had, 0, 1, 2, 3, 10,11,12,13, 20,21,22, and 23. At this point

there is no number higher than 3 in a QUAD system, so we ZERO the 3,

and carry, giving, 30,31, 32, 33, and again, zero the 3, and carry, giving

33 +1 =

30

+1 however, there is no number more than 3, so 3+1 =0, and we

carry the one into a NEW column, the (4x4x1) column, giving us

100 as the next number (= 16d) , and keep counting, 101, 102, 103,

(zero and carry again ) 110, 111, 112, 113, etc.

*** You SAY in English, " There are only TWO kinds of people, those

who know Binary, and those who do not." This is because 10 in BINARY

is TWO, not " TEN".

****

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IMAGES CLICK HERE

Robot Cartoons CLICK HERE

Robots CLICK HERE

About kidbots CLICK HERE

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Kidbots.com is a Free website, with Free

information, on how to USE Free computers

to make Free ROBOT controllers and

Free ROBOTS for kids.