Counting

Got it! This site "robinsnyder.com" uses cookies. You consent to this by clicking on "Got it!" or by continuing to use this website. Note: This appears on each machine/browser from which this site is accessed.

Why is counting important?

In English class, spelling words is fundamental (as is grammar which builds on words).
In music class, notes, scales, sharps, flats, etc., are fundamental.
In sporting events, skills for that sport are fundamental. Some volleyball skills: serve, bump, set, spike, block.
In computer class, counting is a fundamental skill.

Computer people often start counting at 0 and not at 1.

The first decimal number is 0.
The second decimal number is 1.
The third decimal number is 2.
... and so on ...

There is a difference between a value and a representation of a value.

The value is an abstract idea. A representation is a way to make the value (in reality) concrete.

Plato in his philosophy takes the position that the abstract is more "real" than the concrete.

An abstract value has many possible concrete representations.

There is one abstract notion of "five". There are many representations and meanings related to "five".

five (English)
fünf (German)
fif (Old English)
fyue (Middle English)
fiue (Middle English)
fife (radio communications term to avoid confusion with word "fire")
cinco (Spanish, as in "Cinco de Mayo", the 5th of May)
quinque (Latin as in "quintuplets")
пять (Russian as "pyat")
πεντε (Greek as "pende", as in "pentagon" with five sides)
पंज (Hindi as "panj", related to "punch" as in the drink with five ingredients)
5 (Arabic numeral)
5th (ordinal number)
V (Roman numeral, upper case)
v (Roman numeral, lower case)
ε (Greek epsilon, historical five in gematria)
ה (Hebrew heh, historical five in gematria)
0101b (binary)
05d (decimal)
05o (octal)
05h (hexadecimal)
✋ (Unicode U+270B, #9995, raised hand)
tally marks (not part of common Unicode range)
2+3 (mathematical expression, infinite number of such)

Base 10, or decimal, uses 10 digits, and is human friendly.
Base 2, or binary, uses 2 digits, and is machine friendly.
Base 16, or hexadecimal, uses 16 digits, and is partly human, partly machine friendly.
Base 8, or octal, uses 8 digits, but unlike hexadecimal, is not used very much anymore.

Base 10 uses 10 digits, from 0 to 9.

0 1 2 3 4 5 6 7 8 9

Question: What base 10 number comes after 9?

In base 10, the number 10 comes after 9.

Question: What rule is used to go from 9 to 10?

The 9 goes to 0, with a carry of 1 in the next decimal place.

1 9 + 1 ---- 1 0

In base ten, adding 1 to 9 is 0 with a carry of 1, which is brought down to get 10, or ten.

Note: 9 is the tenth and last digit in base 10.

Consider the number 156.

156 = 100 + 50 + 6 = 1*100 + 5*10 + 6*1 = 1*102 + 5*101 + 6*100

This positional number system, or Arabic numbering system, is much better than the previously used Roman numeral system.

Digit 0, for 10⁰, or 1, is the first digit.
Digit 1, for 10¹, or 10, is the second digit.
Digit 2, for 10², or 100, is the third digit.
Digit 3, for 10³, or 1000, is the fourth digit.
Digit 4, for 10⁴, or 10000, is the fifth digit.

How much difference does it make on a bank check when you add one zero: $1, $10, $100, $1,000, $10,000, etc.

Base 16 uses 16 digits, where 0 to 9 are the first 10 digits.

0 1 2 3 4 5 6 7 8 9

Question: After 9, what are the other 6 hexadecimal digits?

The 6 hexadecimal digits after 9 are A, B, C, D, E and F

So base 16 uses 16 digits, where A, B, C, D, E, and F have decimal values 10 through 15.

0 1 2 3 4 5 6 7 8 9 A B C D E F

A network adapter card is used to connect a computer to a network.

Manufacturer's of network adapter cards have arranged for every network card has an address that is unique.

Network adapter card addresses are expressed in hexadecimal, such as 08-00-5A-E4-2B-F7.

Question: What base 16 number comes after F?

In base 16, the number after F is 10.

Question: What rule is used to go from F to 10 in base 16?

The F goes to 0, with a carry of 1 in the next hexadecimal place.

1 F + 1 ---- 1 0

In base sixteen, adding 1 to F is 0 with a carry of 1, which is brought down to get 10, or one-zero hex, or sixteen decimal.

Note: F is the sixteenth and last digit in base 16.

Base 2 uses 2 digits.

0 1

Question: What base 2 number comes after 1? The base 2 number after 1 is 10.

Question: What rule is used?

The 1 goes to 0, with a carry of 1.

1 1 + 1 ---- 1 0

In base two, adding 1 to 1 is 0 with a carry of 1, which is brought down to get 10, or one-zero binary, or two decimal.

Note: 1 is the second and last digit in base 2.

1 1 1 9 F 1 + 1 + 1 + 1 --- --- --- 1 0 1 0 1 0

When more than one base is being used, what is 10?

Whenever multiple bases are being used, add a suffix to the number to indicate the base of the number.

1d is one in decimal (the default).
1h is one in hexadecimal, or hex.
1b is one in binary.

Note that the abstract notion of "one" does not change and is independent of the base in which the idea is expressed.

So the ambiguity can be resolved using a suffix.

1 1 1 9 d F h 1 b + 1 d + 1 h + 1 b ----- ----- ---- 1 0 d 1 0 h 1 0 b

10d is ten, 10h is sixteen, and 10b is two.

Note: Programming language notation may differ as to how to specify the base to be used.

Question: How many values can be represented with 1 bit?

2 values can be represented with 1 bit

1 bit can represent any 2 values, but not more than 2, and everyone using and communicating using those bits must agree on what those 2 values are.

1 bit can represent 2 values:

present for class (1b)
absent from class (0b)

What if you are late to class? That cannot be represented with one bit unless one of the above two values is not used.

2 bits can represent 4 values:

present for class (11b)
late for class (01b)
departed early (10b)
absent from class (00b)

Note: If you leave for an extended period of time in the middle of class, then this might be approximated by marking you either "late for class" or "departed early" as there are not enough bits used to handle more than 4 cases.

Inside the computer are bits and bytes. To really understand what is going on, you need to know something about bits and bytes.

A bit, for binary digit, can represent two values.

The two values of a bit are often written as 0 and 1.

You cannot actually see a 0 or 1 inside a computer.

How are the 2 values of a bit represented?

John Tukey (American mathematician and statistician) , working with John von Neumann, coined the term "bit" as "binary digit". The classical "bit" has one of two values, which can be represented as 0 (usually taken as false) and 1 (usually taken as true).

This term was first used in an article by Claude Shannon in 1948.

More: John Tukey

Here are some ways to represent 2 values of a bit.

low or high voltage level
low or high light level
low or high magnetic field
low or high reflection level

A original standard CD (Compact Disc) can hold 650 MB of data, or about 5 billion bits. Each bit is either a burned hole or not a burned hole at a certain position on the CD that either reflects light or does not reflect light. That is how the 5 billion bits are created and accessed.

A byte is made up of 8 bits.

Question: How many values can a byte represent?

A byte can represent 256 values.

Since each bit can represent 2 values, 8 bits can represent 256 values as follows.

2*2*2*2*2*2*2*2 = 28 = 256

A nibble is half a byte

Yes, the technical term for four bits, or half a byte, is a nibble.

A group of 4 bits make a nibble that can represent 16 values.
A hexadecimal digit represents one nibble, or 4 bits.

Computer engineers are often concerned with the upper nibble (upper 4 bits) and lower nibble (lower 4 bits) of a byte.

A group of 16 bits (i.e., 2 bytes) is called a word or WORD.
A group of 32 bits is called a dword, for double-word or DWORD.
A group of 64 bits is called a qword, for quad-word or QWORD.

Here are the base 2 numbers from 0d to 15d.

0000 0100 1000 1100 0001 0101 1001 1101 0010 0110 1010 1110 0011 0111 1011 1111

You need to know these 16 numbers in base 2.

The hard way to learn the first 16 numbers in base 2 is to memorize them.

The easier way, and what I do, is to write down the list starting at 0000 and ending at 1111.

Start with 0000.
Add 1 to get 0001.
Add 1, to get 0 with a carry of 1, to get 0010.
Add 1 to get 0011.
Add 1 to get 0100 (i.e., two carries of 1, as in going from 99 to 100 in base 10).
... and so on ...

To help you see the pattern, here is the sequence where 1 is added to each binary number starting at 0000.

0000 0100 1000 1100 + 1 + 1 + 1 + 1 ---- ---- ---- ---- 0001 0101 1001 1101 0001 0101 1001 1101 + 1 + 1 + 1 + 1 ---- ---- ---- ---- 0010 0110 1010 1110 0010 0110 1010 1110 + 1 + 1 + 1 + 1 ---- ---- ---- ---- 0011 0111 1011 1111 0011 0111 1011 1111 + 1 + 1 + 1 + 1 ---- ---- ---- ----- 0100 1000 1100 10000

Here is the list from 0000b to 1111b.

10 16 2 10 16 2 0d 0h 0000b 8d 8h 1000b 1d 1h 0001b 9d 9h 1001b 2d 2h 0010b 10d Ah 1010b 3d 3h 0011b 11d Bh 1011b 4d 4h 0100b 12d Ch 1100b 5d 5h 0101b 13d Dh 1101b 6d 6h 0110b 14d Eh 1110b 7d 7h 0111b 15d Fh 1111b

Write the numbers from 0 to 15 (decimal) in both hexadecimal and binary. Include suffixes for each.

The bits of a byte can be numbered from 0 to 7, just as post office mailboxes can have numbers. Each box can contain a 0 or a 1.

7 6 5 4 3 2 1 0 --------------------------------- | ? | ? | ? | ? | ? | ? | ? | ? | ---------------------------------

Bit 0, for 2⁰, or 1, is the first bit.
Bit 1, for 2¹, or 2, is the second bit.
Bit 2, for 2², or 4, is the third bit.
Bit 3, for 2³, or 8, is the fourth bit.
Bit 4, for 2⁴, or 16, is the fifth bit.
Bit 5, for 2⁵, or 32, is the sixth bit.
Bit 6, for 2⁶, or 64, is the seventh bit.
Bit 7, for 2⁷, or 128, is the eighth bit.

So, Ch is 1100b is 1*8 + 1*4 + 0*2 + 0*1 is 12d.

dec bin hex base 2 exponential notation 0d 0000b 0h 0*23 + 0*22 + 0*21 + 0*20 1d 0001b 1h 0*23 + 0*22 + 0*21 + 1*20 2d 0010b 2h 0*23 + 0*22 + 1*21 + 0*20 3d 0011b 3h 0*23 + 0*22 + 1*21 + 1*20 4d 0100b 4h 0*23 + 1*22 + 0*21 + 0*20 5d 0101b 5h 0*23 + 1*22 + 0*21 + 1*20 6d 0110b 6h 0*23 + 1*22 + 1*21 + 0*20 7d 0111b 7h 0*23 + 1*22 + 1*21 + 1*20 8d 1000b 8h 1*23 + 0*22 + 0*21 + 0*20 9d 1001b 9h 1*23 + 0*22 + 0*21 + 1*20 10d 1010b Ah 1*23 + 0*22 + 1*21 + 0*20 11d 1011b Bh 1*23 + 0*22 + 1*21 + 1*20 12d 1100b Ch 1*23 + 1*22 + 0*21 + 0*20 13d 1101b Dh 1*23 + 1*22 + 0*21 + 1*20 14d 1110b Eh 1*23 + 1*22 + 1*21 + 0*20 15d 1111b Fh 1*23 + 1*22 + 1*21 + 1*20

A magic number is a number that appears in many places as if it has some special type of meaning.

Many magic numbers in computers are actually base 2 numbers of the form 2ⁿ or 2ⁿ-1.

n 2n 2n 0d 1d 0000000000000001b 1d 2d 0000000000000010b 2d 4d 0000000000000100b 3d 8d 0000000000001000b 4d 16d 0000000000010000b 5d 32d 0000000000100000b 6d 64d 0000000001000000b 7d 128d 0000000010000000b 8d 256d 0000000100000000b 9d 512d 0000001000000000b 10d 1024d 0000010000000000b 11d 2048d 0000100000000000b 12d 4096d 0001000000000000b 13d 8192d 0010000000000000b 14d 16384d 0100000000000000b 15d 32768d 1000000000000000b 16d 65536d 10000000000000000b

Notice that the 1 moves to the left one place when the number is multiplied by 2 (in base 2). This is similar to the number 12 multiplied by 10 being 120 in base 10.

n 2n-1 2n-1 1d 1d 0000000000000001b 2d 3d 0000000000000011b 3d 7d 0000000000000111b 4d 15d 0000000000001111b 5d 31d 0000000000011111b 6d 63d 0000000000111111b 7d 127d 0000000001111111b 8d 255d 0000000011111111b 9d 511d 0000000111111111b 10d 1023d 0000001111111111b 11d 2047d 0000011111111111b 12d 4095d 0000111111111111b 13d 8191d 0001111111111111b 14d 16383d 0011111111111111b 15d 32767d 0111111111111111b 16d 65535d 1111111111111111b

n 10n 10n 0 0d 00001d 1 10d 00010d 2 100d 00100d 3 1000d 01000d 4 10000d 10000d

n 10n-1 10n-1 1 9d 00009d 2 99d 00099d 3 999d 00999d 4 9999d 09999d

An earlier Excel version had the following limitations, much later increased.

An Excel spreadsheet can have from 1 to 65526 rows (i.e., 16 bits are used).
An Excel spreadsheet can have from 1 to 256 columns (i.e., 8 bits are used).

The column after Z is AA, then AB, etc. This is a base 26 representation where A is column 1, I is column 9, V is column 22, Z is column 26, AA is column 27, etc. Thus, column IV is

9*26 + 22 = 234 + 22 = 256.

Often, software limits picked by software developers correspond to the above magic numbers, either 1 to 2ⁿ or 0 to 2ⁿ-1.

Here is the idea behind the NCAA basketball tournament.

Initially, there are 64 teams, or 2⁶ teams.
After the first round, there are 32 teams, or 2⁵ teams.
After the second round, there are 16 teams, or 2⁴ teams (sweet sixteen).
After the third round, there are 8 teams, or 2³ teams (elite eight).
After the fourth round, there are 4 teams, or 2² teams (final four).
After the fifth round, there are 2 teams, or 2¹ teams.
After the sixth round, there is 1 team, or 2⁰ teams, and this team is declared the national champion for that year.

So, for a n-round tournament, you start with 2ⁿ teams so that after n rounds, you have 2^n-n = 2⁰ = 1 team left.

Note: To have more than 64 teams in the tournament, one must add another level for some of the teams.

A bit, or binary digit, can represent 2 values.

8 bits make a byte that can represent 256 values.

Suffixes are used to specify the number of bytes, as a power of 2.

1024 bytes make a KB (Kilobyte).

1,000 = 103 1,024 = 210

Since 1000 and 1024 are close in size, one can approximate a KB with 1000.

1024 KB make a MB (Megabyte) (slightly more than a million).

1024 MB make a GB (Gigabyte) (about a billion, or thousand million).

A standard CD holds more than half a GB (i.e., 650 MB).

1024 GB make a TB (Terabyte) (about a trillion, or million million).

Since you waited this long to take the course (i.e., rather than, say, many years ago) you have additional suffixes to learn.

A standard DVD holds several GB.

During February, 2000, the US began using the space shuttle to map over 80% of the entire globe.

The data would take about 15,000 CD's, or about 7 to 10 TB (terabyte) of data.

1024 TB make a PB (Petabyte) (about a quadrillion).

Many years ago, companies such as Wal-Mart passed the PB (petabyte) size for databases.

Information about every purchase made at Wal-Mart ends up in a large database that is used in a variety of ways to improve the competitive position of Wal-Mart.

Other companies do much the same thing.