,SCIENTIFIC NOTATION
,SCIENTIFIC NOTATION IS THE WAY OF
WRITING NUMBERS AS A PRODUCT OF A
COEFFICIENT AND A POWER OF TEN  A  #AJ 
AND IT EXISTS BECAUSE THE UNIVERSE
REFUSES TO STAY AT A HUMAN SCALE4 ,A
HYDROGEN ATOM IS #J4JJJJJJJJJAJF METERS
ACROSS4 ,A MOLE OF ANYTHING IS
#FJB1BJJ1JJJ1JJJ1JJJ1JJJ1JJJ1JJJ THINGS4
,BOTH OF THOSE SENTENCES ARE UNREADABLE4
,REWRITE THEM AS #A4JF  #AJ_<;P ;M AND
#F4JBB  #AJ1 AND SUDDENLY THEY ARE NOT
ONLY LEGIBLE  THEY ARE COMPARABLE4 ,THE
EXPONENT IS THE SIZE2 THE COEFFICIENT IS
THE DETAIL2 THE NOTATION IS THE LENS
THAT LETS A HUMAN LOOK DIRECTLY AT
SCALES WE CANNOT TOUCH4
,ALMOST EVERY DISCIPLINE THAT MEASURES
THE PHYSICAL WORLD ADOPTED IT BECAUSE IT
DOES THREE JOBS AT ONCE3 IT COMPRESSES
ABSURD DIGIT STRINGS1 IT MAKES MAGNITUDE
VISIBLE AT A GLANCE1 AND  QUIETLY1
IMPORTANTLY  IT ENCODES HOW PRECISE A
NUMBER ACTUALLY IS4 ,THIS IS THE STORY
OF HOW THE NOTATION CAME TO BE1 HOW IT
WORKS1 AND HOW IT ESCAPED PAPER ENTIRELY
TO BECOME THE WAY COMPUTERS STORE THE
UNIVERSE4
,THE PROBLEM OF INCONVENIENT NUMBERS
,THE TROUBLE STARTS AS SOON AS YOU LEAVE
THE HUMAN SCALE IN EITHER DIRECTION4
,THE MASS OF AN ELECTRON IS ABOUT
#J4JJJJJJJJJJJJJJJJJJJJJJJJJJJJJJIAA
KILOGRAMS  THIRTY ZEROS PAST THE DECIMAL
POINT BEFORE THE ACTUAL INFORMATION
BEGINS4 ,THE MASS OF THE ,SUN IS ROUGHLY
#A1IHI1JJJ1JJJ1JJJ1JJJ1JJJ1JJJ1JJJ1JJJ1JJJ
KILOGRAMS  THE SAME PROBLEM IN THE OTHER
DIRECTION4 ,THESE NUMBERS ARE NOT EXOTIC
EDGE CASES2 THEY ARE ROUTINE WORKING
VALUES IN PHYSICS AND CHEMISTRY4
,WRITTEN LONGHAND1 SUCH NUMBERS ARE
WORSE THAN INCONVENIENT  THEY ARE
DANGEROUS4 ,MISCOUNT A ZERO IN A DRUG
DOSAGE OR AN ORBITAL PARAMETER AND THE
ERROR IS NOT SMALL2 IT IS A FACTOR OF
TEN1 WHICH IN THE PHYSICAL WORLD IS
OFTEN THE DIFFERENCE BETWEEN WORKING AND
CATASTROPHE4 ,ENGINEERS OF THE
SLIDE-RULE ERA HAD A NAME FOR THIS
OCCUPATIONAL HAZARD3 THE 8DECIMAL POINT
ERROR40 ,THE FIX WAS NOT MORE CARE  IT
WAS BETTER NOTATION4
,ARCHIMEDES COUNTS THE SAND
,THE FIRST PERSON KNOWN TO GRAPPLE WITH
THIS PROBLEM SYSTEMATICALLY WAS
,ARCHIMEDES OF ,SYRACUSE1 AROUND #BAF
,,BC1 IN A SHORT WORK ADDRESSED TO ,KING
,GELON CALLED ,THE ,SAND ,RECKONER
"<,PSAMMITES">4.<CITE3 ,ARCHIMEDES1 ,THE
,SAND ,RECKONER "<,PSAMMITES">1 ;C4 #BAF
,,BC  .<,THOMAS ,HEATH'S TRANSLATION1
,THE ,WORKS OF ,ARCHIMEDES "<#AHIG">.>.>
,THE PREMISE WAS A PROVOCATION3 A COMMON
BELIEF HELD THAT THE GRAINS OF SAND ON
ALL THE WORLD'S BEACHES WERE EFFECTIVELY
INFINITE  BEYOND NUMBERING4 ,ARCHIMEDES
SET OUT TO PROVE OTHERWISE1 AND TO DO IT
HE FIRST HAD TO SOLVE THE HARDER PROBLEM
OF HIS AGE3 ,GREEK NUMERALS TOPPED OUT
AT THE MYRIAD "<#AJ1JJJ">1 AND THERE WAS
NO NOTATION FOR ANYTHING LARGER4
,HIS SOLUTION WAS THE GERM OF EVERY
SCALED NOTATION THAT FOLLOWED4 ,HE
GROUPED NUMBERS INTO ORDERS3 THE FIRST
ORDER RAN UP TO A MYRIAD OF MYRIADS
"<#AJ;X">2 THE SECOND ORDER RAN UP TO
#AJ;X SQUARED "<#AJ;V">2 AND SO ON
THROUGH PERIODS OF ORDERS  EFFECTIVELY
INDEXING NUMBERS BY THE POWER OF A LARGE
BASE RATHER THAN WRITING EACH ONE OUT4
,WITH THIS MACHINERY HE BUILT A
GEOMETRIC MODEL OF THE COSMOS "<ON THE
HELIOCENTRIC-SCALE ESTIMATES OF
,ARISTARCHUS1 NO LESS"> AND COUNTED THE
SAND NEEDED TO FILL IT4 ,HIS ANSWER3 NO
MORE THAN ABOUT #AJ;V GRAINS  AN
ABSURDLY LARGE NUMBER1 BUT A FINITE1
WRITABLE ONE4 ,THE POINT WAS NEVER THE
COUNT4 ,IT WAS THAT THE COUNT COULD BE
WRITTEN AT ALL4
,DECIMALS1 EXPONENTS1 LOGARITHMS
,THREE MORE INVENTIONS HAD TO ARRIVE
BEFORE THE MODERN FORM COULD EXIST4 ,THE
FIRST WAS THE DECIMAL FRACTION3 ,SIMON
,STEVIN'S #AEHE BOOKLET ,DE ,THIENDE
"<8,THE ,TENTH0"> TAUGHT ,EUROPE TO
WRITE FRACTIONAL QUANTITIES IN DECIMAL
FORM AND  CRUCIALLY FOR OUR STORY 
MARKED EACH DECIMAL POSITION WITH A
SMALL CIRCLED INDEX1 TREATING THE PLACE
VALUE ITSELF AS A NUMBERED THING4.<CITE3
,SIMON ,STEVIN1 ,DE ,THIENDE "<,THE ,ART
OF ,TENTHS">1 ,LEIDEN1 #AEHE  FIRST
SYSTEMATIC ,EUROPEAN TREATMENT OF
DECIMAL FRACTIONS2 .<#AFJH ,ENGLISH
TRANSLATION.>.>
,THE SECOND WAS EXPONENT NOTATION4 ,REN
,DESCARTES' ,LA ;,GOMTRIE "<#AFCG">
ESTABLISHED THE NOW-UNIVERSAL CONVENTION
OF RAISED SUPERSCRIPTS FOR POWERS  ;X1
;X1 ;X  REPLACING THE CLUMSY VERBAL AND
LETTER-HYBRID NOTATIONS OF EARLIER
ALGEBRAISTS4.<CITE3 ,FLORIAN ,CAJORI1 ,A
,HISTORY OF ,MATHEMATICAL ,NOTATIONS1
VOL4 ,I1 ,OPEN ,COURT1 #AIBH  .<SCAN AT
,INTERNET ,ARCHIVE.>.> ,ONCE POWERS HAD
A COMPACT WRITTEN FORM1 8TEN TO THE
POWER OF0 WAS ONE SMALL STEP FROM
BECOMING THE NOTATION'S ENGINE4
,THE THIRD WAS THE LOGARITHM4 ,JOHN
,NAPIER'S ,MIRIFICI ,LOGARITHMORUM
,CANONIS ,DESCRIPTIO "<#AFAD"> GAVE
SCIENCE A WAY TO TURN MULTIPLICATION
INTO ADDITION  AND AS A SIDE EFFECT MADE
THE EXPONENT1 RATHER THAN THE DIGITS1
THE OPERATIVE PART OF A NUMBER4 ,THE
SLIDE RULE "<,WILLIAM ,OUGHTRED'S
REFINEMENT1 #AFBJ;S"> PUT THAT IDEA IN
EVERY ENGINEER'S POCKET FOR THREE
CENTURIES3 A PHYSICAL DEVICE THAT
COMPUTED BY ADDING EXPONENTS4
,SLIDE-RULE DISCIPLINE  CARRY THE
MANTISSA1 TRACK THE ORDER OF MAGNITUDE
SEPARATELY  IS PRECISELY THE MENTAL
MODEL SCIENTIFIC NOTATION
ENCODES4.<CITE3 ,JOHN ,NAPIER1 ,MIRIFICI
,LOGARITHMORUM ,CANONIS ,DESCRIPTIO1
,EDINBURGH1 #AFAD  .<SCAN AT ,INTERNET
,ARCHIVE.>.>
,THE NOTATION SETTLES
,THERE WAS NO SINGLE INVENTION DATE FOR
A  #AJ4 ,THROUGH THE NINETEENTH CENTURY1
ASTRONOMERS AND PHYSICISTS INCREASINGLY
WROTE UNWIELDY VALUES AS A DECIMAL PART
TIMES AN EXPLICIT POWER OF TEN1 AND
TEXTBOOKS GRADUALLY CONVERGED ON THE
CONVENTION3 A COEFFICIENT WITH EXACTLY
ONE NONZERO DIGIT BEFORE THE DECIMAL
POINT1 AND AN INTEGER EXPONENT4 ,BY THE
EARLY TWENTIETH CENTURY THE FORM WAS
STANDARD IN SCIENTIFIC LITERATURE1 AND
BY MID-CENTURY IT WAS ENTRENCHED IN
SCHOOL CURRICULA AS STANDARD FORM "<THE
,BRITISH NAME"> OR SCIENTIFIC NOTATION
"<THE ,AMERICAN ONE">4 ,FLORIAN
,CAJORI'S ,A ,HISTORY OF ,MATHEMATICAL
,NOTATIONS "<#AIBH"> TRACES THE LONG
COALESCENCE OF THE SUPERSCRIPT-POWER
IDEA THAT MADE IT POSSIBLE4.<CITE3
,FLORIAN ,CAJORI1 ,A ,HISTORY OF
,MATHEMATICAL ,NOTATIONS1 VOL4 ,I1 ,OPEN
,COURT1 #AIBH  .<SCAN AT ,INTERNET
,ARCHIVE.>.>
,THERE IS NO SINGLE INVENTOR OF
SCIENTIFIC NOTATION4 ,IT IS A
SEDIMENTARY INVENTION3 ,ARCHIMEDES'
SCALING ORDERS1 ,STEVIN'S DECIMAL
PLACES1 ,DESCARTES' SUPERSCRIPTS1 AND
,NAPIER'S LOGARITHMS EACH DEPOSITED A
LAYER BEFORE THE MODERN FORM SURFACED IN
NINETEENTH-CENTURY PRACTICE4
,ANATOMY OF THE NOTATION
.<FIGURE3 ,THE PARTS OF A
SCIENTIFIC-NOTATION NUMBER3 A
COEFFICIENT BETWEEN #A AND #AJ1
MULTIPLIED BY A POWER OF TEN4.>
,THE CONVENTION IS EXACT3
- ,COEFFICIENT A3 A DECIMAL NUMBER WITH
#A ;D _\A_\ @< #AJ  EXACTLY ONE NONZERO
DIGIT LEFT OF THE POINT4 - ,BASE3 ALWAYS
#AJ  THE LADDER IS OUR DECIMAL SYSTEM
ITSELF4 - ,EXPONENT ;N3 AN INTEGER4
,POSITIVE COUNTS LEFTWARD SHIFTS
"<#AJ">2 NEGATIVE COUNTS RIGHTWARD
"<#AJ_<;P">4
,SO #F4JBB  #AJ SAYS 8TAKE #F4JBB AND
MOVE THE POINT TWENTY-THREE PLACES
RIGHT10 AND #A4JF  #AJ_<;P SAYS 8MOVE
TEN PLACES LEFT40 ,THE FORM WHERE THE
COEFFICIENT OBEYS THE RANGE RULE IS
CALLED NORMALIZED SCIENTIFIC NOTATION2
ALLOWING THE COEFFICIENT TO DRIFT
PRODUCES USEFUL VARIANTS "<ENGINEERING
NOTATION1 BELOW">4
,THE PAYOFF IS INSTANT COMPARISON4 ,TWO
NUMBERS IN NORMALIZED FORM CAN BE
ORDERED WITHOUT READING DIGITS AT ALL3
WHICHEVER HAS THE BIGGER EXPONENT IS
BIGGER1 AND ONLY TIES NEED THE
COEFFICIENT CONSULTED4 #I4AA  #AJ_< KG
AND #A4FG  #AJ_<;W KG ARE SEPARATED BY
FOUR ORDERS OF MAGNITUDE  ROUGHLY TEN
THOUSAND TO ONE  BEFORE YOU HAVE READ A
SINGLE SIGNIFICANT DIGIT4
,A LADDER OF EVERYTHING
,,METERS                                
,,BYTES #AJ_<;U  A PROTON               
#AJ;P   A BYTE  ONE CHARACTER #AJ_<;P  A
HYDROGEN ATOM                        
#AJ   A KILOBYTE  A PARAGRAPH #AJ_<;W  
A VIRUS                                
#AJ;V   A MEGABYTE  A MINUTE OF MUSIC
#AJ_<;T   A POLLEN GRAIN                
#AJ;Y   A GIGABYTE  A FEATURE FILM #AJ;P
A HUMAN                                
#AJ  A TERABYTE  A LAPTOP DRIVE #AJ;W   
,EARTH  #AB1GDB KM                      
#AJ;U  A PETABYTE  A DATA-CENTER RACK
#AJ   THE ,SOLAR ,SYSTEM                
#AJ;X  AN EXABYTE  ALL WORDS EVER SPOKEN
#AJ;V   A LIGHT-YEAR                    
#AJ  A ZETTABYTE  THE EARLY INTERNET #AJ
THE ,MILKY ,WAY                         
@9#AJ  THE HUMAN DATASPHERE  ,H#BJJ ,,ZB
#AJ;V   THE OBSERVABLE UNIVERSE
,THE OBSERVABLE UNIVERSE IS ,H#H4H 
#AJ;V ;M ACROSS2 HUMANITY'S ACCUMULATED
DATA IS ,H#B  #AJ BYTES4 ,TWO ENTIRELY
DIFFERENT QUANTITIES  YET ON THE SAME
LADDER THEIR EXPONENTS NEARLY COINCIDE4
,THE FULL SPAN OF THE NOTATION'S
TERRITORY FITS ON ONE LADDER4 ,EACH RUNG
IS TEN TIMES THE ONE BELOW3 FROM THE
PROTON AT #AJ_<;U ;M THROUGH ATOMS
"<#AJ_<;P">1 CELLS "<#AJ_<;U">1 PEOPLE
"<#AJ;P">1 PLANETS "<#AJ;W">1 STAR
SYSTEMS "<#AJ">1 GALAXIES "<#AJ">1 UP TO
THE OBSERVABLE UNIVERSE AT ROUGHLY #H4H 
#AJ;V ;M4 ,THE SAME LADDER READS IN
BYTES ON THE COMPUTING SIDE  KILO-1
MEGA-1 GIGA-1 TERA-1 PETA-1 EXA-1
ZETTA-1 YOTTA-  AND THE TWO COLUMNS MEET
AT THE TOP1 WHERE THE OBSERVABLE
UNIVERSE AND HUMANITY'S ACCUMULATED DATA
LIVE AT EERILY SIMILAR EXPONENTS4
,CHARLES AND ,RAY ,EAMES IMMORTALIZED
THIS VIEW IN THEIR #AIGG FILM ,POWERS OF
,TEN1 WHICH ZOOMS BY A FACTOR OF TEN
EVERY TEN SECONDS FROM A ,CHICAGO PICNIC
BLANKET TO THE EDGE OF THE OBSERVABLE
COSMOS AND BACK  ARGUABLY THE MOST
EFFECTIVE PIECE OF MATHEMATICS
COMMUNICATION EVER FILMED4.<CITE3
,CHARLES AND ,RAY ,EAMES1 ,POWERS OF
,TEN1 FILM1 ,EAMES ,OFFICE1 #AIGG 
.<EAMESOFFICE4COM.>.>
,ARITHMETIC IN THE NOTATION
,WORKING IN SCIENTIFIC NOTATION  RATHER
THAN CONVERTING IN AND OUT  IS WHERE IT
EARNS ITS KEEP3
,OPERATION  ,RULE  ,EXAMPLE   ,MULTIPLY 
MULTIPLY COEFFICIENTS1 ADD EXPONENTS 
"<#C#AJ;T">"<#B#AJ"> "7 #F#AJ;W  
,DIVIDE  DIVIDE COEFFICIENTS1 SUBTRACT
EXPONENTS  "<#H#AJ;V">"<#B#AJ"> "7
#D#AJ;T   ,ADD _/ SUBTRACT  EQUALIZE
EXPONENTS FIRST  #E#AJ;T "6 #C#AJ "7
#E#AJ;T "6 #J4C#AJ;T "7 #E4C#AJ;T  
,RAISE TO A POWER  POWER THE
COEFFICIENT1 MULTIPLY EXPONENT 
"<#B#AJ"> "7 #D#AJ;V   ,ROOT  ROOT THE
COEFFICIENT1 DIVIDE EXPONENT 
"<#I#AJ;V"> "7 #C#AJ   ,RENORMALIZE 
NUDGE COEFFICIENT BACK TO #A1AJ">1
ADJUST EXPONENT  #FB#AJ;W "7 #F4B#AJ;X
,THE RENORMALIZATION STEP IS THE ONLY
SUBTLETY3 ARITHMETIC CAN PUSH THE
COEFFICIENT OUT OF RANGE "<#FB  #AJ;W">1
AND THE FIX IS TO SLIDE THE DECIMAL
POINT AND COMPENSATE IN THE EXPONENT4
,EVERY FLOATING-POINT UNIT IN EVERY
COMPUTER PERFORMS EXACTLY THIS DANCE1
BILLIONS OF TIMES PER SECOND4
,HONEST PRECISION3 SIGNIFICANT FIGURES
,SCIENTIFIC NOTATION CARRIES A SECOND1
QUIETER PAYLOAD3 IT ENCODES PRECISION4
,IN LONGHAND1 8#J4JJCJ0 AND 8#J4JJC0 ARE
EASY TO CONFUSE AND THE TRAILING ZERO'S
MEANING IS AMBIGUOUS4 ,IN SCIENTIFIC
FORM THE AMBIGUITY EVAPORATES3 #C4J 
#AJ_< DECLARES TWO SIGNIFICANT FIGURES1
#C  #AJ_< DECLARES ONE1 AND #C4JJ  #AJ_<
DECLARES THREE4 ,THE COEFFICIENT'S
DIGITS ARE1 BY CONVENTION1 EXACTLY THE
DIGITS THE MEASUREMENT EARNED4.<.<CITE3
,,BIPM_/,,JCGM1 ,GUIDE TO THE
,EXPRESSION OF ,UNCERTAINTY IN
,MEASUREMENT "<,,JCGM #AJJ3#BJJH"> 
.<OFFICIAL ,,PDF.>.>
,THIS IS WHY THE NOTATION IS NOT MERELY
COSMETIC IN SCIENCE4 ,AVOGADRO'S NUMBER
WRITTEN #F4JBB  #AJ ASSERTS FOUR
SIGNIFICANT DIGITS2 WRITING IT
#F4JBBADJGF  #AJ "<THE EXACT ,,SI
DEFINITION SINCE #BJAI"> ASSERTS TEN4
,THE EXPONENT CARRIES THE SCALE2 THE
COEFFICIENT CARRIES THE CONFIDENCE4
,MEASUREMENT SCIENCE  UNCERTAINTY1 ERROR
BARS1 LABORATORY REPORTING  IS
IMPOSSIBLE TO WRITE CLEANLY WITHOUT THIS
SEPARATION4
,ENGINEERING NOTATION AND THE ,,SI
PREFIXES
,A CLOSE COUSIN RELAXES THE COEFFICIENT
RULE3 ENGINEERING NOTATION PINS THE
EXPONENT TO MULTIPLES OF THREE  SO EVERY
VALUE LANDS ON AN ,,SI PREFIX BOUNDARY4
#DG  #AJ IS NOT NORMALIZED SCIENTIFIC
FORM1 BUT IT IS INSTANTLY 8#DG KILO10
WHICH IS HOW ENGINEERS ACTUALLY TALK
ABOUT IT4 ,MODERN CHIP DATASHEETS1 FOR
INSTANCE1 QUOTE SUPPLY CURRENTS IN M,A
AND GATE DELAYS IN NS  ENGINEERING
NOTATION WEARING ITS PREFIX
DIRECTLY4.<CITE3 ,,BIPM1 ,THE
,INTERNATIONAL ,SYSTEM OF ,UNITS "<,,SI
,BROCHURE">1 #ITH EDITION1 #BJAI 
.<BIPM4ORG.>.>
,EXPONENT  ,PREFIX  ,SYMBOL  ,A FAMILIAR
EXAMPLE   #AJ_<  PICO  ;P  PICOSECOND
LASER PULSES   #AJ_<;Y  NANO  ;N 
NANOMETER-SCALE TRANSISTORS   #AJ_<;V 
MICRO    MICROAMP SENSOR CURRENTS  
#AJ_<  MILLI  ;M  MILLIMETER WAVELENGTHS
#AJ;P      THE HUMAN SCALE   #AJ  KILO 
;K  KILOGRAMS1 KILOMETERS   #AJ;V  MEGA 
;,M  MEGAPIXELS   #AJ;Y  GIGA  ;,G 
GIGABYTES1 GIGAHERTZ   #AJ  TERA  ;,T 
TERABYTE DRIVES1 TERAFLOPS   #AJ;U  PETA
;,P  PETASCALE SUPERCOMPUTING   #AJ;X 
EXA  ;,E  EXASCALE3 #AJ;X ,,FLOPS   #AJ 
ZETTA  ;,Z  THE GLOBAL DATASPHERE  
#AJ;T  YOTTA  ;,Y  YOTTABYTE-SCALE
TOTALS ,IN ,NOVEMBER #BJBB THE ,,SI
ADDED FOUR MORE  RONNA "<#AJ;W"> AND
QUETTA "<#AJ;P"> ABOVE1 RONTO
"<#AJ_<;W"> AND QUECTO "<#AJ_<;P"> BELOW
PARTLY BECAUSE DATA VOLUMES AND
ATOMIC-SCALE MEASUREMENT HAD FINALLY
OUTGROWN YOTTA4.<CITE3 #BGTH ,,CGPM1
,NOVEMBER #BJBB  ,RESOLUTION #C ADOPTING
RONNA1 QUETTA1 RONTO1 AND QUECTO
PREFIXES  .<BIPM4ORG.>.> ,THE LADDER1 IT
TURNS OUT1 STILL NEEDED MORE RUNGS4
;,E-NOTATION3 WHEN KEYBOARDS CAN'T
SUPERSCRIPT
,COMPUTERS INHERITED THE NOTATION BUT
NOT THE TYPOGRAPHY3 THERE IS NO #AJ KEY1
AND NO SUPERSCRIPT SHIFT4 ,THE FIX WAS
THE LETTER4 ,,FORTRAN "<#AIEG">
ESTABLISHED THE CONVENTIONS STILL
PRINTED TODAY3 #A4JF;;,E-#AJ FOR
SCIENTIFIC NOTATION IN SINGLE PRECISION1
AND ;,D  #A4E;;,D-#JH  FOR DOUBLE
PRECISION1 A LIVING FOSSIL IN ,FORTRAN
CODEBASES TO THIS DAY4 ,,ALGOL TRIED
SOMETHING TYPOGRAPHICALLY NOBLER3 A
DEDICATED SUBSCRIPT-TEN SYMBOL 1 SO
#F4JBDHF#BC MEANT #AJ  BEAUTIFUL IN
PRINT1 BUT A CHARACTER THAT SCARCELY ANY
REAL INPUT DEVICE COULD TYPE1 AND ONE OF
THE ERA'S STANDING CHARACTER-ENCODING
HEADACHES4.<CITE3 ;,P4 ,NAUR "<ED4">1
,REVISED ,REPORT ON THE ,ALGORITHMIC
,LANGUAGE ,,ALGOL #FJ1 #AIFC  .<FULL
TEXT.>.>
,C'S PRINTF CEMENTED .0E AS THE FORMAT
SPECIFIER "<AND .0G FOR 8PICK THE TIDIER
FORM0">1 AND VIRTUALLY EVERY LANGUAGE
SINCE ACCEPTS #A;;E-#AJ1 #F4JB;E#BC1 OR
#D4G;,E#C AS LITERALS  ,,ALB AMONG THEM4
,THE ;,E IS PURE CONVENTION3 IT STANDS
FOR EXPONENT1 AND THE NUMBER IT WRITES
IS IDENTICAL TO ITS SUPERSCRIPTED
ANCESTOR4
,FLOATING POINT3 SCIENTIFIC NOTATION IN
BINARY
,THE DEEPEST INHERITANCE RUNS ONE LEVEL
LOWER1 INTO THE HARDWARE ITSELF4 ,A
FLOATING-POINT NUMBER IS SCIENTIFIC
NOTATION  JUST WITH BASE #B INSTEAD OF
BASE #AJ4 ,SIGN BIT1 EXPONENT FIELD1
MANTISSA3 EXACTLY OUR ;S1 ;N1 AND A1
PACKED INTO A REGISTER4
,AND THE MACHINE THAT DID IT FIRST IS
ONE OF COMPUTING HISTORY'S GREAT BURIED
FACTS3 ,KONRAD ,ZUSE'S ,Z#A "<#AICH">
AND ,Z#C "<#AIDA">  MECHANICAL AND RELAY
MACHINES RESPECTIVELY  COMPUTED IN
#BB-BIT BINARY FLOATING POINT3 #A SIGN
BIT1 A #G-BIT TWO'S-COMPLEMENT EXPONENT1
AND A #AD-BIT MANTISSA WITH A HIDDEN
LEADING #A1 THE SAME NORMALIZATION TRICK
,,IEEE #GED STANDARDIZED FOUR DECADES
LATER4 ,ZUSE WAS SOLVING BALLISTIC AND
STRUCTURAL CALCULATIONS2 DECIMAL
FIXED-POINT SIMPLY COULD NOT SPAN THE
SCALES INVOLVED1 SO HIS MACHINES
NORMALIZED EVERY VALUE INTO
SIGN-PLUS-EXPONENT-PLUS-FRACTION FORM 
SCIENTIFIC NOTATION1 IMPLEMENTED IN
TELEPHONE RELAYS4.<CITE3 ;,R4 ,ROJAS1
8,KONRAD ,ZUSE'S ,LEGACY3 ,THE
,ARCHITECTURE OF THE ,Z#A AND ,Z#C10
,,IEEE ,ANNALS OF THE ,HISTORY OF
,COMPUTING #AI"<#B">1 #AIIG 
.<DOI3#AJ4AAJI_/#HE4EHFJFG2 AUTHOR'S
,,PDF.>.>
,,IEEE ,STANDARD #GED "<#AIHE1 REVISED
#BJAI"> GENERALIZED THE SCHEME3 A
BINARY#CB PACKS SIGN1 #H-BIT BIASED
EXPONENT1 AND #BC-BIT MANTISSA "<#BD
EFFECTIVE BITS ;,H #G DECIMAL DIGITS">
INTO A WORD2 BINARY#FD WIDENS THAT TO
#AE#AF DIGITS4.<CITE3 ,,IEEE ,STANDARD
#GED-#BJAI1 ,,IEEE ,STANDARD FOR
,FLOATING-,POINT ,ARITHMETIC 
.<STANDARDS4IEEE4ORG.>.> ,EVERY
NORMALIZATION RULE FROM THE TABLE ABOVE 
THE SLIDING POINT1 THE EXPONENT
BOOKKEEPING1 THE COEFFICIENT THAT MUST
LIVE IN RANGE  RUNS IN SILICON1 ROUGHLY
#AJ;X TIMES PER SECOND ACROSS THE
WORLD'S COMPUTERS AT ANY INSTANT4 ,EVEN
THE FAILURE MODES INHERIT FAITHFULLY3
OVERFLOW "<#A;E#CJI  ,INFINITY">1
UNDERFLOW TO ZERO1 AND THE NOTORIOUS
#J4A "6 #J4B ARE ALL BASE-#B SHADOWS OF
THE SAME NOTATION'S LIMITS4
,THE COMMONEST SCIENTIFIC-NOTATION BUG
IN HISTORY WASN'T HUMAN AT ALL3
FLOATING-POINT CANNOT REPRESENT MOST
DECIMAL FRACTIONS EXACTLY1 SO #J4A "6
#J4B EVALUATES TO #J4CJJJJJJJJJJJJJJJD 
A ROUNDING ARTIFACT THAT HAS CONFUSED
NEARLY EVERY PROGRAMMER WHO HAS EVER
TYPED IT4
,VARIANTS IN THE WILD
.<FIGURE3 ,ONE VALUE  A HYDROGEN ATOM'S
DIAMETER  WRITTEN FOUR WAYS4.>
,THE SAME NUMBER CAN LEGALLY WEAR
SEVERAL COSTUMES3 LONGHAND DECIMAL1
NORMALIZED SCIENTIFIC1 ENGINEERING
NOTATION1 OR ;,E-NOTATION  EACH A
DIFFERENT ANSWER TO A DIFFERENT
CONSTRAINT4 ,A FEW MORE CONVENTIONS
WORTH KNOWING BY NAME3
- ,ORDER OF MAGNITUDE3 THE EXPONENT
ALONE1 USED AS A ROUNDED CLAIM  8WITHIN
AN ORDER OF MAGNITUDE0 MEANS WITHIN A
FACTOR OF ABOUT TEN4 - ,SCIENTIFIC ;,E
NOTATION3 THE TEXTUAL FORM
"<#A4JF;;E-#AJ">1 NORMALIZED COEFFICIENT
ASSUMED4 - ,HEXADECIMAL FLOAT
"<#JX#A4HP"6#C IN ;,C">3 SCIENTIFIC
NOTATION WITH BASE #B MANTISSA AND
EXPLICIT BINARY EXPONENT  THE HARDWARE'S
DIALECT SURFACED IN SOURCE4 - ,MAGNITUDE
NOTATION IN PHYSICS3 ASTRONOMERS
ADDITIONALLY REPURPOSE 8MAGNITUDE0 FOR
LOGARITHMIC BRIGHTNESS SCALES  RELATED
SPIRIT1 DIFFERENT BEAST4
,SEE ALSO
- ,FLOATING POINT  ,ENGINEERING NOTATION
,ORDER OF MAGNITUDE  ,SIGNIFICANT
FIGURES - ,Z#A  ,Z#C  ,KONRAD ,ZUSE 
,PLANKALK;L  THE FIRST FLOATING-POINT
MACHINES - ,,IEEE #GED  ,,FORTRAN 
,,ALGOL  ;,C - ,HISTORY OF ,MATHEMATICS 
,POWERS OF ,TEN  ,,UNIVAC ,I
,QUIZ
- .< .> ,WRITE THE MASS OF AN ELECTRON
"<#J4JJJJJJJJJJJJJJJJJJJJJJJJJJJJJJIAA
KG"> IN SCIENTIFIC NOTATION4 - .< .> ,IN
A  #AJ NORMALIZED FORM1 WHAT IS THE
ALLOWED RANGE OF THE COEFFICIENT8 - .<
.> ,WHO ESTIMATED THE GRAINS OF SAND
NEEDED TO FILL THE UNIVERSE1 AND IN WHAT
WORK8 - .< .> ,WHICH #AEHE TEXT BY
,SIMON ,STEVIN ESTABLISHED DECIMAL
FRACTIONS IN ,EUROPE8 - .< .> ,HOW DO
YOU ADD #E  #AJ;T AND #C  #AJ  AND
WHAT'S THE RESULT8 - .< .> ,WHAT DO THE
TRAILING ZEROS IN #C4JJ  #AJ_< TELL YOU
THAT #C  #AJ_< DOES NOT8 - .< .> ,WHAT
IS ENGINEERING NOTATION'S RULE FOR THE
EXPONENT1 AND WHY8 - .< .> ,WHAT DID
,,FORTRAN USE THE LETTERS ;,E AND ;,D
FOR8 - .< .> ,WHICH #AICH MACHINE
COMPUTED IN BINARY FLOATING POINT  AND
HOW MANY BITS WAS ITS WORD8 - .< .>
,WHAT DO THE #BJBB ,,SI PREFIXES RONNA
AND QUETTA DENOTE8
#A  "9"9#I4AA  #AJ_< KG"9"9  #B  "9"9#A
;D _\A_\ @< #AJ"9"9  EXACTLY ONE NONZERO
DIGIT BEFORE THE POINT  #C 
"9"9,ARCHIMEDES"9"91 IN "9,THE ,SAND
,RECKONER"9 "<@9#BAF ,,BC">2 HIS BOUND
WAS ,H#AJ;V GRAINS  #D  "9"9"9,DE
,THIENDE"9"9"9 "<8,THE ,TENTH0">  #E 
"9"9EQUALIZE EXPONENTS FIRST"9"93
#E#AJ;T "6 #J4C#AJ;T "7 "9"9#E4C 
#AJT"9"9  #F  "9"9THREE SIGNIFICANT
FIGURES"9"9  A CLAIM ABOUT MEASUREMENT
PRECISION  #G  "9"9EXPONENTS MUST BE
MULTIPLES OF #C"9"91 SO VALUES MAP ONTO
,,SI PREFIXES "<KILO1 MEGA1 @&">  #H 
"9"9,E"9"9 MARKS THE EXPONENT IN SINGLE
PRECISION1 "9"9,D"9"9 IN DOUBLE
PRECISION "<.*#A4E;;,D-#JH.*">  #I 
"9"9,ZUSE'S ,Z#A"9"9  A "9"9#BB-BIT"9"9
FLOATING-POINT WORD "<SIGN1 #G-BIT
EXPONENT1 #AD-BIT MANTISSA "6 HIDDEN
#A">  #AJ  "9"9#AJW"9"9 AND "9"9#AJP"9"9
RESPECTIVELY  ADOPTED ,NOVEMBER #BJBB1
EXTENDING THE LADDER BEYOND YOTTA4