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Hello. I’m Professor Von Schmohawk
and welcome to Why U.
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In the previous lectures, we explored some
examples of the earliest number systems
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which were used primarily for counting objects.
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These counting numbers are called
“natural numbers”.
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The natural numbers start at one
and can count to arbitrarily large quantities.
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As we have seen, Roman numerals are one of
many possible ways to represent natural numbers.
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The Roman system was eventually replaced with
the modern decimal number system
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which uses “positional notation”
and only ten numeric symbols.
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The decimal number system was found to be
superior to the ancient Roman system
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because of the simple rules it uses
to create numbers.
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In the decimal system there are ten numeric
symbols, 0 through 9, called “digits”.
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Depending on the column they occupy
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these digits represent the quantity of ones
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tens
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hundreds
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thousands
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and so on
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which make up the number.
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In positional notation,
the column occupied by a digit
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determines the “multiplier” for that digit.
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For example, in the decimal system
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the value of the right-most digit
is multiplied by 1.
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The digit in the next column to the left
is multiplied by 10.
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The next digit is multiplied by 100
and so on.
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The value of a number is the sum of
all its digits times their multipliers.
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For example, the value of
the decimal number 1879
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is 1 times 1000
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plus 8 times 100
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plus 7 times 10
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plus 9 times 1.
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In any positional notation, each column’s
multiplier differs from the adjacent column
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by a constant multiple called
the “base” of the number system.
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In the decimal system, each column multiplier
is ten times the previous column.
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Therefore the decimal system is called a
“base-10” number system.
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There are an infinite number of columns in
the decimal number system
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with each column multiplier being ten times
bigger than the column to the right.
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However, when writing a number,
the zeros in front are normally not written.
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We can count up to 9
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using only the ones column.
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Once we reach 9
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the ones column starts over at 0
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and the tens column increments.
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As we continue counting
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the tens column counts the number of times
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that the ones column
has passed from 9 to 0.
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In other words, the tens column registers
the number of tens which we have counted.
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This continues until we reach 99.
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At that point
the ones and tens columns start over at 0
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and the hundreds column increments.
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The positional notation system is simple.
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Every time a column passes from 9 to 0
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the next column to the left increments.
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How is it that we ended up with a number system
based on multiples of ten?
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There is not any good reason
for choosing ten over some other number
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other than the fact that people have ten fingers
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and probably originally communicated quantities
using their fingers.
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But what if we were cartoon characters
with four digits on each hand?
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Is it possible that in cartoon land
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everyone uses a number system
based on multiples of eight instead of ten?
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How would a base-8 or “octal”
number system work?
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In octal there are only eight numeric symbols
instead of ten as in decimal.
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Instead of 0 through 9
the symbols 0 through 7 are used.
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The symbols 8 and 9 are not needed.
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Counting in octal is very similar
to counting in decimal.
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Since there are no symbols for 8 or 9
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the highest quantity which can be represented
in the ones column is 7.
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Counting an eighth item
causes the ones column to start over at 0
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and the next column to increment.
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So the second column
counts the number of eights.
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Therefore in octal
the number following 7 is 10
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which looks just like the decimal number ten.
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After octal "10" comes octal "11", "12", and so on
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until we get to octal "17".
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At that point, we go to octal "20".
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The second column has now counted
two “eights” or sixteen.
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We continue like this until we get to the
highest number we can represent with two digits
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octal "77".
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At that point, the ones and eights columns
start over at 0 and the third column increments.
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The 1 in the third column represents
eight “eights” or sixty-four.
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Each column multiplier is
eight times the previous one.
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Every number which can be written in decimal
can also be written in octal
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although after counting to 7
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the way the quantities are represented
is completely different.
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It is easy to convert an octal number to decimal
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when you consider how positional notation works.
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Let’s take for example, the octal number "1750".
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As in decimal, the value of the octal number
is the sum of all its digits times their multipliers.
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So the number "1750" represents
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1 times 512
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plus 7 times 64
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plus 5 times 8
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plus 0 ones
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which adds up to the quantity
which in decimal is called one-thousand.
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You may sometimes see a small subscript 8 or 10
after an octal or decimal number
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in case there may be some confusion
about which base is being used.
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Digital computers use electronic circuits
called “flip-flops” to represent numbers.
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Each flip-flop can store a single bit
which can represent either a 0 or a 1.
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Multiple bits can be combined to represent
a base-2 or “binary” number.
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In the binary number system
0 and 1 are the only two numeric symbols.
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Since binary is base-2
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each column multiplier is two times
the multiplier of the previous digit.
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And just like decimal or octal numbers
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the value of a binary number is sum of
all its digits times their multipliers.
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Since the digits are either 1 or 0
the calculation is simple.
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We just add the multipliers
of all the columns which contain ones.
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For example, the binary number 11010
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represents 1 sixteen
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plus 1 eight
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plus 1 two
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which is equal to twenty-six.
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Even though digital computers
store numbers in binary
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it can be quite tedious to write down
or remember large binary numbers.
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For instance, the number one-million
in binary is
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one one one one
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zero one zero one
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one zero zero one
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one zero one one
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zero zero zero zero.
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Early in the history of digital computers
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engineers found that it was easier
to use octal notation
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than to deal with long strings
of ones and zeros.
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Three binary digits can be represented
by a single octal symbol.
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It is easy to memorize the eight possible
combinations of three binary bits.
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To convert a multiple-digit octal number
to binary
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each octal digit in the number
is converted to a 3-bit binary equivalent
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and the binary digits are all combined
into a single binary number.
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Any leading zeros can be removed.
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To convert a binary number to octal
we do the same thing in reverse.
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To convert this binary number back to octal
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we split it into 3-bit groups
starting from the right
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and each 3-bit group is then converted
to its equivalent octal symbol.
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So the octal equivalent to this binary number
is "3654660"
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a lot easier to remember
than all those ones and zeros.
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Today, computer storage is normally organized
into 8-bit groups called "bytes".
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Because of this, many computer engineers
prefer to use base-16
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otherwise known as “hexadecimal” or “hex”
instead of octal.
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With hexadecimal, every group of four bits
converts to a single hex symbol.
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Two hex symbols represent exactly one byte.
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Even fewer digits than octal are required
to represent a given number
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and it's just as easy to convert
back and forth to binary.
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Hexadecimal numbers
use sixteen numeric symbols.
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The symbols 0 through 9
are used just as in decimal
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but six more symbols are needed.
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Instead of making up new symbols,
the letters A through F are used
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to represent what we call
ten through fifteen in decimal.
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Counting in hexadecimal
works the same way as in decimal or octal
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except that hex uses sixteen symbols per digit.
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Because each column multiplier is
sixteen times larger than the previous column
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hexadecimal can represent large numbers
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with fewer digits than octal or decimal.
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When counting in hexadecimal
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after getting to F which is decimal 15
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we go to "10" which is decimal 16
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then "11", "12", and so on.
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Once we reach 1F
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we go to "20" which is decimal 32.
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When we get to the largest number which we
can represent with two hex digits, FF
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we go to "100" which is decimal 256
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and so on.
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As we mentioned
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using hex notation, four binary bits
can be represented by a single hex symbol.
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Each of the sixteen possible combinations
of four bits
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is equivalent to a single hex digit.
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Let's convert the same binary number as before
to hex.
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Starting from the right,
we group the digits into groups of four.
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Each group of binary digits is then converted
to its equivalent hex symbol.
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So we have seen how the same natural number
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can be represented in base-2
using two numeric symbols
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base-8 using eight symbols
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base-10 using ten symbols
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and base-16 using sixteen symbols.
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But no matter how we choose to write
this natural number
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it still represents the same quantity.
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As you have seen
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we use the same basic rules for counting in binary
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octal
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decimal
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and hexadecimal.
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The only difference is that each base has
a different number of numeric symbols.
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So using positional notation
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we can create a number system
using any natural base we like.
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Try creating one of your own.
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Who knows, it might catch on!
