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[[Image:020926-O-9999G-015.jpg|thumb|Pilots of the 332nd Fighter Group, "Tuskegee Airmen," the elite, all-African American 332nd Fighter Group at Ramitelli, Italy., from left to right, Lt. Dempsey W. Morgan, Lt. Carroll S. Woods, Lt. Robert H. Nelron, Jr., Capt. Andrew D. Turner and Lt. Clarence P. Lester.]]
'''Elementary [[arithmetic]]''' is the most basic kind of [[mathematics]]: it concerns the operations of [[addition]], [[subtraction]], [[multiplication]], and [[division]]. Most people learn elementary arithmetic in [[elementary school]].
The '''Tuskegee Airmen''' ([[International Phonetic Alphabet|IPA pronunciation]]: {{IPA|[təˈski.gi]}}<ref>See [http://inogolo.com/pronunciation/Tuskegee Pronunciation of Tuskegee].</ref>) was the popular name of a group of [[African American]] pilots who flew with distinction during [[World War II]] as the [[332nd Fighter Group]] of the [[United States Army Air Forces|US Army Air Corps]].
Elementary arithmetic starts with the [[natural numbers]] and the [[Arabic numerals]] used to represent them. It requires the memorization of addition tables and [[multiplication table]]s for adding and multiplying pairs of digits. Knowing these tables, a person can perform certain well-known procedures for adding and multiplying natural numbers. Other [[algorithm]]s are used for subtraction and division.
==Origins==
Though starting out with natural numbers, elementary arithmetic then moves on to [[vulgar fraction|fractions]], then [[decimals]] and [[negative numbers]], which can be represented on a number line.
[[Image:P-51C bomber escort.jpg|thumb|right|Aircraft of the 332d Fighter Group; the "redtails" of the Tuskegee Airmen. The nearest aircraft depicted is that of Lt. Lee Archer, the only ace among the Tuskegee Airmen.]]
Prior to the Tuskegee Airmen, no US military [[aviator|pilots]] had been African American. However, a series of legislative moves by the [[United States Congress]] in 1941 forced the Army Air Corps to form an all-black combat unit, much to the War Department's chagrin. In an effort to eliminate the unit before it could begin, the War Department set up a system to accept only those with a level of flight experience or higher education that they expected would be hard to fill. This policy backfired when the Air Corps received numerous applications from men who qualified even under these restrictions.
The US Army Air Corps had established the [[Psychological Research Unit 1]] at [[Maxwell Army Air Field]], [[Alabama]], and other units around the country for aviation cadet training, which included the identification, selection, education, and training of pilots, [[flight officer|navigator]]s and [[bombardier (rank)|bombardier]]s. Psychologists employed in these research studies and training programs used some of the first [[standardized tests]] to quantify [[IQ]], [[dexterity]], and [[leadership]] qualities in order to select and train the right personnel for the right role (bombardier, pilot, navigator). The Air Corps determined that the same existing programs would be used for all units, including all-black units. At Tuskegee, this effort would continue with the selection and training of the Tuskegee Airmen.
Nowadays people routinely use electronic [[calculator]]s, [[cash register]]s, and [[computer]]s to perform their elementary arithmetic for them. Before that, people used [[slide rule]]s, tables of [[logarithm]]s, [[nomogram|nomographs]], mechanical calculators, or they employed [[mental calculator|calculating prodigies]]. Otherwise, they just performed calculations by hand on paper, following the well-known rules of arithmetic.
==Training==
In ancient times, the instrument to perform arithmetical calculations was the [[abacus]]. In the 14th century Arabic numerals were introduced to Europe by [[Leonardo Pisano]]. These numerals were more efficient for performing calculations that [[Roman numeral]]s, because of the positional system, which also made multiplication by hand more efficient than the use of the abacus.
On [[19 March]] [[1941]], the 99th Pursuit Squadron (Pursuit being the pre-World War II descriptive for "Fighter") was activated at [[Chanute Field]] in [[Rantoul, Illinois]].<ref> Francis 1988, p. 15. Note: It was a lawsuit or the threat of a law suit from a rejected candidate that caused the USAAC to accept black applicants.</ref> Over 250 enlisted men were trained at Chanute in aircraft ground support trades. This small number of enlisted men was to become the core of other black squadrons forming at Tuskegee and Maxwell fields in Alabama– the famed Tuskegee Airmen.
[[Image:040315-F-9999G-024.jpg|thumb|left|Major James A. Ellison returns the salute of Mac Ross of Dayton, Ohio, as he passes down the line during review of the first class of Tuskegee cadets; flight line at US Army Air Corps basic and advanced flying school, Tuskegee, Alabama, 1941 with Vultee BT-13 trainers in the background.]]
In June 1941, the Tuskegee program officially began with formation of the [[99th Fighter Squadron]] at the [[Tuskegee Institute]], a highly regarded university founded by [[Booker T. Washington]] in [[Tuskegee, Alabama]].<ref> Thole 2002, p. 48. Note: The Coffey School of Aeronautics in Chicago was also considered.</ref> The unit consisted of an entire service arm, including ground crew, and not just pilots. After basic training at [[Moton Field]], they were moved to the nearby [[Tuskegee Army Air Field]] about 16 km (ten miles) to the west for conversion training onto operational types. The Airmen were placed under the command of Capt. [[Benjamin O. Davis Jr.]], one of the few African American [[United States Military Academy|West Point]] graduates. His father [[Benjamin O. Davis, Sr.]] was the first black general in the US Army.
During its training, the 99th Fighter Squadron was commanded by white and Puerto Rican officers, beginning with Capt. George "Spanky" Roberts. By 1942, however, it was Col. Frederick Kimble who oversaw operations at the Tuskegee airfield. Kimble proved to be highly unpopular with his subordinates, whom he treated with disdain and disrespect. Later that year, the Air Corps replaced Kimble with Maj. Noel Parrish. Parrish, counter to the prevalent racism of the day, was fair and open-minded, and petitioned Washington to allow the Tuskegee Airmen to serve in combat.{{Fact|date=April 2007}}
==The digits==
<big> 0 </big>, [[zero]], represents absence of objects to be counted. <br>
<big> 1 </big>, [[one]]. This is one stick: <big>I</big> <br>
<big> 2 </big>, [[two]]. This is two sticks: <big>I I</big> <br>
<big> 3 </big>, [[three]]. This is three sticks: <big>I I I</big> <br>
<big> 4 </big>, [[four]]. This is four sticks: <big>I I I I </big> <br>
<big> 5 </big>, [[five]]. This is five sticks: <big>I I I I I</big> <br>
<big> 6 </big>, [[six]]. This is six sticks: <big>I I I I I I</big> <br>
<big> 7 </big>, [[seven]]. This is seven sticks: <big>I I I I I I I</big> <br>
<big> 8 </big>, [[eight]]. This is eight sticks: <big>I I I I I I I I </big> <br>
<big> 9 </big>, [[nine]]. This is nine sticks: <big>I I I I I I I I I</big> <br>
There are as many digits as fingers on the hands: the word "digit" can also mean finger. But if counting the digits on both hands, the first digit would be one and the last digit would not be counted as "zero" but as "[[ten]]": <big> 10 </big>, made up of the digits one and zero. The number 10 is the first two-digit number.
In response, a hearing was convened before the [[House Armed Services Committee]] to determine whether the Tuskegee Airmen "experiment" should be allowed to continue. The committee accused the Airmen of being incompetent — based on the fact that they had not seen any combat in the entire time the "experiment" had been underway. To bolster the recommendation to scrap the project, a member of the committee commissioned and then submitted into evidence a "scientific" report by the [[University of Texas]] which purported to prove that Negroes were of low intelligence and incapable of handling complex situations (such as air combat). The majority of the Committee, however, decided in the Airmen's favor, and the 99th Pursuit Squadron soon joined two new squadrons out of Tuskegee to form the all-black [[332nd Fighter Group]].
If a number has more than one digit, then the rightmost digit, said to be the last digit, is called the "ones-digit". The digit immediately to its left is the "tens-digit". The digit immediately to the left of the tens-digit is the "hundreds-digit". The digit immediately to the left of the hundreds-digit is the "thousands-digit".
==Addition tableCombat==
[[Image:99th Fighter Squadron patch.jpg|thumb|right|Patch of the 99th Fighter Squadron]]
{| cellpadding="3" border="1"
The 99th was ready for combat duty during some of the Allies' earliest actions in the [[North African campaign]], and was transported to [[Casablanca]], [[Morocco]], on the ''[[USS Mariposa]]''. From there, they travelled by train to [[Oujda]] near [[Fes]], and made their way to [[Tunis]] to operate against the [[Luftwaffe]]. The flyers and ground crew were largely isolated by racial segregation practices, and left with little guidance from battle-experienced pilots. Operating directly under the [[Twelfth Air Force]] and the XII Air Support Command, the 99th FS and the Tuskegee Airmen were bounced around between three groups, the 33rd FG, 324th FG, and 79th FG. The 99th's first combat mission was to attack the small but strategic volcanic island of [[Pantelleria]] in the [[Mediterranean Sea]] between [[Sicily]] and [[Tunisia]], in preparation for the [[Allied invasion of Sicily]] in [[July]] [[1943]]. The 99th moved to Sicily while attached to the [[33rd Fighter Group]],<ref name="duc"/> whose commander, Col. [[William Momyer|William W. Momyer]], fully involved the squadron, and the 99th received a [[Distinguished Unit Citation]] for its performance in Sicily.
! +
[[Image:020903-o-9999b-098.jpg|thumb|left|Tuskegee Airmen in front of a <br />[[Curtiss P-40|P-40]].]]
! 0
The Tuskegee Airmen were initially equipped with [[Curtiss P-40|P-40 Warhawk]]s, later with [[P-47 Thunderbolt]]s, and finally with the airplane that they would become most identified with, the [[P-51 Mustang]].
! 1
! 2
! 3
! 4
! 5
! 6
! 7
! 8
! 9
|-
! 0
| 0 || 1 || 2 || 3 || 4 || 5 || 6 || 7 || 8 || 9
|-
! 1
| 1 || 2 || 3 || 4 || 5 || 6 || 7 || 8 || 9 || 10
|-
! 2
| 2 || 3 || 4 || 5 || 6 || 7 || 8 || 9 || 10 || 11
|-
! 3
| 3 || 4 || 5 || 6 || 7 || 8 || 9 || 10 || 11 || 12
|-
! 4
| 4 || 5 || 6 || 7 || 8 || 9 || 10 || 11 || 12 || 13
|-
! 5
| 5 || 6 || 7 || 8 || 9 || 10 || 11 || 12 || 13 || 14
|-
! 6
| 6 || 7 || 8 || 9 || 10 || 11 || 12 || 13 || 14 || 15
|-
! 7
| 7 || 8 || 9 || 10 || 11 || 12 || 13 || 14 || 15 || 16
|-
! 8
| 8 || 9 || 10 || 11 || 12 || 13 || 14 || 15 || 16 || 17
|-
! 9
| 9 || 10 || 11 || 12 || 13 || 14 || 15 || 16 || 17 || 18
|}
On [[27 January]] and [[28 January]] [[1944]], German [[Fw 190]] fighter-bombers raided [[Anzio Campaign|Anzio]], where the Allies had conducted amphibious landings on [[January 22]]. Attached to the 79th Fighter Group, eleven of the 99th Fighter Squadron's pilots shot down enemy fighters, including Capt. Charles B. Hall, who shot down two, bringing his aerial victory total to three. The eight fighter squadrons defending Anzio together shot down a total of 32 German aircraft, and the 99th had the highest score among them with 13.<ref name="kills">Haulman, Dr. Daniel L. ''Aerial Victory Credits of the Tuskegee Airmen''. AFHRA Maxwell AFB. [http://www.af.mil/shared/media/document/AFD-070207-059.pdf] Access date: [[16 February]] [[2007]].</ref>
What does it mean to add two natural numbers? Suppose you have two bags, one bag holding five apples and a second bag holding three apples. Grabbing a third, empty bag, move all the apples from the first and second bags into the third bag. The third bag now holds eight apples. This illustrates the combination of three apples and five apples is eight apples; or more generally: "three plus five is eight" or "three plus five equals eight" or "eight is the sum of three and five". Numbers are abstract, and the addition of three sugar cubes to a group of five sugar cubes also yields a bigger group of eight sugar cubes. Addition is a regrouping: two sets of objects which were counted separately are put into a single group and counted
together: the count of the new group is the "sum" of the separate counts of the two original groups.
The squadron won its second [[Distinguished Unit Citation]] on [[12 May]]-[[14 May]] [[1944]], while attached to the 324th Fighter Group, attacking German positions on Monastery Hill ([[Battle of Monte Cassino|Monte Cassino]]), attacking infantry massing on the hill for a counterattack, and bombing a nearby strong point to force the surrender of the German garrison to [[Moroccan]] [[Goumier]]s.
Symbolically, addition is represented by the "plus sign": <big> + </big>. So the statement "three plus five equals eight" can be written symbolically as
[[Image:100th Fighter Squadron patch.jpg|thumb|right|Patch of the 100th Fighter Squadron]]
:<math> 3 + 5 = 8.\ </math>
By this point, more graduates were ready for combat, and the all-black [[332nd Fighter Group]] had been sent overseas with three fighter squadrons: the [[100th Flying Training Squadron|100th]], [[301st Fighter Squadron|301st]] and [[302nd Fighter Squadron|302nd]]. Under the command of Col. [[Benjamin O. Davis, Jr.|Benjamin O. Davis]], the squadrons were moved to mainland [[Italy]], where the 99th FS, assigned to the group on [[1 May]], joining them on [[6 June]]. The Airmen of the 332nd Fighter Group escorted bombing raids into [[Austria]], [[Hungary]], [[Poland]] and [[Germany]].
The order in which two numbers are added does not matter, so that, for example, three plus four equals four plus three. This is the [[commutative]] property of addition.
Flying escort for heavy bombers, the 332nd racked up an impressive combat record, often entering combat against greater numbers of superior German aircraft and coming out victorious. Reportedly, the Luftwaffe awarded the Airmen the nickname, "Schwarze Vogelmenschen," or "Black Birdmen." The Allies called the Airmen "Redtails" or "Redtail Angels," because of the distinctive crimson paint on the vertical stabilizers of the unit's aircraft. Although bomber groups would request Redtail escort when possible, few bomber crew members knew at the time that the Redtails were black.{{Fact|date=April 2007}}
To add a pair of digits using the table, find the intersection of the row of the first digit with the column of the second digit: the row and the column intersect at a square containing the sum of the two digits. Some pairs of digits add up to two-digit numbers, with the tens-digit always being a 1. In the addition algorithm the tens-digit of the sum of a pair of digits is called the "carry digit".
[[Image:tuskegee airmen.jpg|thumb|left|Tuskegee Airmen gathered at a US base after a mission in the Mediterranean theater.]]
While it had long been said that the Redtails were the only fighter group who never lost a bomber to enemy fighters,<ref>[http://www.pingry.k12.nj.us/about/articles/2002-nov-11-tuskegee.html Lt. Col. Thomas E. Highsmith, Jr.; speech at The Pingry School, 8 November 2002]</ref> suggestions to the contrary, combined with Air Force records and eyewitness accounts indicating that at least 25 bombers were lost to enemy fire, resulted in the Air Force conducting a reassessment of the history of this famed unit in the fall of 2006.
The claim that the no bomber escorted by the Tuskegee Airmen had ever been lost to enemy fire first appeared on [[24 March]] [[1945]]. The claim came from an article, published in the [[Chicago Defender]], under the headline "332nd Flies Its 200th Mission Without Loss." Ironically, this article was published on the very day that, according to the [[28 March]] [[2007]] Air Force report, some bombers under 332nd Fighter Group escort protection were shot down.<ref> ''Report: Tuskegee Airmen lost 25 bombers''. The Associated Press, [[1 April]] [[2007]]. [http://www.usatoday.com/news/nation/2007-04-01-tuskegee-airmen_N.htm] Access date: [[1 April]] [[2007]].</ref><ref>[http://www.comcast.net/news/national/index.jsp?cat=DOMESTIC&fn=/2006/12/11/539246.html Comcast.net news; Access date: [[11 December]] [[2006]] (Article ID:539246)]</ref><ref>''Ex-Pilot Confirms Bomber Loss, Flier Shot down in 1944 was Escorted by Tuskegee Airmen''. Washington Post, [[17 December]] [[2006]], p. A18.</ref><ref>AP Story [[29 March]] [[2007]]</ref> The subsequent report, based on after-mission reports filed by both the bomber units and Tuskegee fighter groups as well as missing air crew records and witness testimony, was released in March 2007 and documented 25 bombers shot down by enemy [[fighter aircraft]] while being escorted by the Tuskegee Airmen.<ref>Report: ''Tuskegee Airmen lost 25 bombers''. The Associated Press, [[2 April]] [[2007]]
===Addition algorithm===
[http://aimpoints.hq.af.mil/display.cfm?id=17731] Access date: [[10 April]] [[2007]].</ref>
For simplicity, consider only numbers with three digits or less. Two add a pair of numbers (written in Arabic numerals), write the second number under the first one, so that digits line up in columns: the rightmost column will contain the ones-digit of the second number under the ones-digit of the first number. This rightmost column is the ones-column. The column immediately to its left is the tens-column. The tens-column will have the tens-digit of the second number (if it has one) under the tens-digit of the first number (if it has one). The column immediately to the left of the tens-column is the hundreds-column. The hundreds-column will line up the hundreds-digit of the second number (if there is one) under the hundreds-digit of the first number (if there is one).
A [[B-25]] bomb group, the [[477th Bombardment Group (Medium)]], was forming in the US but completed its training too late to see action. The 99th Fighter Squadron after its return to the United States became part of the 477th, redesignated the 477th Composite Group.
After the second number has been written down under the first one so that digits line up in their correct columns, draw a line under the second (bottom) number. Start with the ones-column: the ones-column should contain a pair of digits: the ones-digit of the first number and, under it, the ones-digit of the second number. Find the sum of these two digits: write this sum under the line and in the ones-column. If the sum has two digits, then write down only the ones-digit of the sum. Write the "carry digit" above the top digit of the next column: in this case the next column is the tens-column, so write a 1 above the tens-digit of the first number.
By the end of the war, the Tuskegee Airmen were credited with 109 Luftwaffe aircraft shot down,<ref name="kills"/> a patrol boat run aground by machine-gun fire, and destruction of numerous fuel dumps, trucks and trains. The squadrons of the 332nd FG flew more than 15,000 sorties on 1,500 missions. The unit received recognition through official channels and was awarded a [[Presidential Unit Citation (US)|Distinguished Unit Citation]] for a mission flown [[24 March]] [[1945]], escorting B-17s to bomb the [[Daimler-Benz]] tank factory at [[Berlin, Germany]], an action in which its pilots destroyed three [[Me-262]] jets in aerial combat. The 99th Fighter Squadron in addition received two DUCs, the second after its assignment to the 332nd FG.<ref name="duc"> ''Air Force Historical Study 82''. AFHRA Maxwell AFB. [http://afhra.maxwell.af.mil/numbered_studies/916794.pdf] Access date: [[16 February]] [[2007]].</ref> The Tuskegee Airmen were awarded several [[Silver Star Medal|Silver Stars]], 150 [[Distinguished Flying Cross (USA)|Distinguished Flying Cross]]es, 14 [[Bronze Star Medal|Bronze Stars]] and 744 [[Air Medal]]s.
If both first and second number each have only one digit then their sum is given in the addition table, and the addition algorithm is unnecessary.
In all, 992 pilots were trained in Tuskegee from 1940 to 1946; about 445 deployed overseas, and 150 Airmen lost their lives in training or combat.<ref>http://www.nationalmuseum.af.mil/factsheets/factsheet.asp?id=1356</ref>
==Postwar==
Then comes the tens-column. The tens-column might contain two digits: the tens-digit of the first number and the tens-digit of the second number. If one of the numbers has a missing tens-digit then the tens-digit for this number can be considered to be a zero. Add the tens-digits of the two numbers. Then, if there is a carry digit, add it to this sum. If the sum was 18 then adding the carry digit to it will yield 19. If the sum of the tens-digits (plus carry digit, if there is one) is less than ten then write it in the tens-column under the line. If the sum has two digits then write its last digit in the tens-column under the line, and carry its first digit (which should be a one) over to the next column: in this case the hundreds column.
[[Image:tuskegee airman poster.jpg|thumb|Color poster of a Tuskegee Airman]]
Far from failing as originally expected, a combination of pre-war experience and the personal drive of those accepted for training had resulted in some of the best pilots in the US Army Air Corps. Nevertheless, the Tuskegee Airmen continued to have to fight [[racism]]. Their combat record did much to quiet those directly involved with the group (notably bomber crews who often requested them for escort), but other units were less than interested and continued to harass the Airmen.
All of these events appear to have simply stiffened the Airmen's resolve to fight for their own rights in the US. After the war, the Tuskegee Airmen once again found themselves isolated. In [[1949]] the 332nd entered the yearly gunnery competition and won. After segregation in the military was ended in [[1948]] by President [[Harry S. Truman]] with [[Executive Order 9981]], the Tuskegee Airmen now found themselves in high demand throughout the newly formed [[United States Air Force]].
If none of the two numbers has a hundreds-digit then if there is no carry digit then the addition algorithm has finished. If there is a carry digit (carried over from the tens-column) then write it in the hundreds-column under the line, and the algorithm is finished. When the algorithm finishes, the number under the line is the sum of the two numbers.
Many of the surviving members of the Tuskegee Airmen annually participate in the Tuskegee Airmen Convention, which is hosted by [[Tuskegee Airmen, Inc]].
If at least one of the numbers has a hundreds-digit then if one of the numbers has a missing hundreds-digit then write a zero digit in its place. Add the two hundreds-digits, and to their sum add the carry digit if there is one. Then write the sum of the hundreds-column under the line, also in the hundreds column. If the sum has two digits then write down the last digit of the sum in the hundreds-column and write the carry digit to its left: on the thousands-column.
In 2005, four Tuskegee Airmen (Lt. Col. Lee Archer, Lt. Col. Robert Ashby, MSgt. James Sheppard, and TechSgt. George Watson) flew to Balad, Iraq, to speak to active duty airmen serving in the current incarnation of the 332nd, reactivated as first the 332d Air Expeditionary Group in 1998 and made part of the [[332d Air Expeditionary Wing]]. "This group represents the linkage between the 'greatest generation' of airmen and the 'latest generation' of airmen," said Lt. Gen. Walter E. Buchanan III, commander of the [[Ninth Air Force]] and US Central Command Air Forces, in an e-mail to the Associated Press.
===Example===
Say one wants to find the sum of the numbers 653 and 274. Write the second number under the first one, with digits aligned in columns, like so:
{|
| 6 || 5 || 3
|-
| 2 || 7 || 4
|}
==Legacy and honors==
Then draw a line under the second number and start with the ones-column. The ones-digit of the first number is 3 and of the second number is 4. The sum of three and four is seven, so write a seven in the ones-column under the line:
[[Image:Tuskegee Airmen + US Congressional Gold Medals, 2007March29.jpg|thumb|President George W. Bush presented the Congressional Gold Medal to about 300 Tuskegee Airmen on 29 March 2007 at the US Capitol.]]
On [[29 March]] [[2007]], about 350 Tuskegee Airmen and their widows were collectively awarded the [[Congressional Gold Medal]]<ref name=THOMAS>Library of Congress. [http://thomas.loc.gov/cgi-bin/query/D?c110:2:./temp/~c110J3sEbQ:: Resolved by the Senate (the House of Representatives concurring), That the Rotunda of the Capitol is authorized to be used on [[29 March]] [[2007]], for a ceremony to award a Congressional... (Engrossed as Agreed to or Passed by Senate)], [[7 March]] [[2007]]. </ref> at a ceremony in the [[United States Capitol rotunda|US Capitol rotunda]].<ref>Price, Deb. ''Nation to honor Tuskegee Airmen''. The Detroit News, [[29 March]] [[2007]]. [http://www.detnews.com/apps/pbcs.dll/article?AID=/20070329/NATION/703290308] Access date: [[29 March]] [[2007]].</ref><ref> ''Tuskegee Airmen Gold Medal Bill Signed Into Law''. Office of Congressman Charles B. Rangel. [http://www.house.gov/list/press/ny15_rangel/CBRStatementTuskegeeBillSigned04112006.html] Access date: [[26 October]] [[2006]].</ref><ref>
Evans, Ben. ''Tuskegee Airmen awarded Congressional Gold Medal''. Associated Press, [[30 March]] [[2007]].
[http://thetandd.com/articles/2007/03/30/news/doc460c7d58cd40f058827045.txt]
Access date: [[30 April]] [[2007]].</ref> The medal will go on display at the [[Smithsonian Institution]]; individual honorees will receive bronze replicas.<ref>AP Story [[29 March]] [[2007]]</ref>
The airfield where the airmen trained is now the [[Tuskegee Airmen National Historic Site]].<ref>Official NPS website: [http://www.nps.gov/tuai/ Tuskegee Airmen National Historic Site]</ref>
{|
| 6 || 5 || 3
|-
| <u>2 || <u>7 || <u>4
|-
| || || 7
|}
In 2006, California Congressman [[Adam Schiff]], and Missouri Congressman [[William Lacy Clay, Jr.]], have led the initiative to create a commemorative postage stamp to honor the Tuskegee Airmen.<ref>[http://schiff.house.gov/HoR/CA29/Newsroom/Press+Releases/2006/Schiff+Votes+to+Honor+Tuskegee+Airmen.htmSchiff Votes to Honor Tuskegee Airmen]</ref>
Next, the tens-column. The tens-digit of the first number is 5, and the tens-digit of the second number is 7, and five plus seven is twelve: 12, which has two digits, so write its last digit, 2, in the tens-column under the line, and write the carry digit on the hundreds-column above the first number:
== Film, media and other facts==
{|
* In 1945, the First Motion Picture Unit of the Army Air Corps produced ''[[Wings for This Man]]'', a "propaganda" short about the unit narrated by [[Ronald Reagan]].
| 1 || ||
* In 1996, [[HBO]] produced and aired ''[[The Tuskegee Airmen]]'', starring [[Laurence Fishburne]].
|-
* The Tuskegee Airmen are represented in the 1997 [[G.I. Joe]] action figure series.<ref>[http://www.mastercollector.com/neat/gijoe/hasbro/1997joes.html 1997 G.I. Joe Classic Collection]</ref>
| 6 || 5 || 3
* Television host [[Fred Rogers]]' foster brother was an instructor for the Tuskegee Airmen and taught Rogers how to fly.<ref>Garfield, Eugene. ''Mister Rogers on the Roots of Nurturing and the Untapped Role of Men in Professional Childcare''. Current Comments, [[25 September]] [[1989]]. [http://www.garfield.library.upenn.edu/essays/v12p270y1989.pdf#search=%22%22mr.%20rogers%22%20tuskegee%22]
|-
Access date: [[24 September]] [[2006]].</ref>
| <u>2 || <u>7 || <u>4
* In the book ''Wild Blue'', by [[Stephen Ambrose]], the Tuskegee Airmen were mentioned, and honoured.
|-
* The 2004 documentary film ''Silver Wings and Civil Rights: The Fight to Fly'', was the first film to feature the "Freeman Field Mutiny," the struggle of 101 African-American officers arrested for entering a white officer's club. [http://www.fight2fly.com/]
| || 2 || 7
* May 17, 2005, [[George Lucas]] is planning a film about the Tuskegee Airmen called ''Red Tails''. Lucas says, "They were the only escort fighters during the war that never lost a bomber so they were, like, the best."<ref>[http://www.filmfocus.co.uk/newsdetail.asp?NewsID=335 Exclusive: Lucas looks to the future]</ref>
|}
[[Image:Col Benjamin Oliver Davis, Jr.jpg|thumb|right|Col. [[Benjamin O. Davis, Jr.]], commander of the Tuskegee Airmen 332nd Fighter Group, in front of his [[P-47 Thunderbolt]] in Sicily.]]
Next, the hundreds-column. The hundreds-digit of the first number is 6, while the hundreds-digit of the second number is 2. The sum of six and two is eight, but there is a carry digit, which added to eight is equal to nine. Write the nine under the line in the hundreds-column:
==References==
{|
{{reflist}}
| 1 || ||
|-
| 6 || 5 || 3
|-
| <u>2 || <u>7 || <u>4
|-
| 9 || 2 || 7
|}
==Sources==
No digits (and no columns) have been left unadded, so the algorithm finishes, and
* Bucholtz, Chris and Laurier, Jim. ''332nd Fighter Group - Tuskegee Airmen''. London: Osprey Publishing, 2007. ISBN 1-84603-044-7.
: 653 + 274 = 927.
* Cotter, Jarrod. "Red Tail Project." ''Flypast, No, 248, March 2002''.
* Francis, Charles F. ''The Tuskegee Airmen: The Men who Changed a Nation''. Boston: Branden Publishing Company, 1988. ISBN 0-8283-1908-1.
* Hill, Ezra M. Sr. ''The Black Red Tail Angels: A Story of the Tuskegee Airmen''. Columbus, Ohio: SMF Haven of Hope. 2006.
* Leuthner, Stuart and Jensen, Olivier. ''High Honor: Recollections by Men and Women of World War II Aviation''. Washington, DC: [[Smithsonian Institution Press]], 1989. ISBN 0-87474-650-7.
* McKissack, Patricia C. and Fredrick L. ''Red Tail Angels: The Story of the Tuskegee Airmen of World War II''. New York: Walker Books for Young Readers, 1996. ISBN 0-80278-292-2.
* Ross, Robert A. ''Lonely Eagles: The Story of America's Black Air Force in World War II''. Los Angeles: Tuskegee Airmen Inc., Los Angeles Chapter, 1980. ISBN 0-917612-00-0.
* Sandler, Stanley. ''Segregated Skies: All-Black Combat Squadrons of WWII.'' Washington, DC: Smithsonian Institution Press, 1992. ISBN 1-56098-154-7.
* Thole, Lou. "Segregated Skies." ''Flypast, No, 248, March 2002''.
==External links==
==Successorship and Size==
{{Commons|Tuskegee Airmen}}
The result of the addition of one to a number is the ''successor'' of that number. Examples: <br>
* [http://www.amazon.com/dp/0802782922 "Red-Tail Angels": The Story of the Tuskegee Airmen of World War II]
the successor of zero is one, <br>
* [http://www.cbc.ca/asithappens/international/tuskegee_010814.html Tuskegee reunion: A whopping tale of coincidence]
the successor of one is two, <br>
* [http://www.imdb.com/title/tt0114745/ The Tuskegee Airmen (1995)]
the successor of two is three, <br>
* [http://www.shoppbs.org/sm-pbs-the-tuskegee-airmen--pi-1402874.html The Tuskegee Airmen] [[Public Broadcasting Service|PBS]] [[Documentary film]]hello
the successor of ten is eleven. <br>
Every natural number has a successor.
* [http://www.pbs.org/wnet/aaworld/reference/articles/tuskegee_airmen.html Reference Room: African American World, Articles, Tuskegee Airmen PBS [[Encyclopædia Britannica]]]
The predecessor of the successor of a number is the number itself. For example, five is the successor of four therefore four is the predecessor of five. Every natural number except zero has a predecessor.
* [http://www.aeromuseum.org/Exhibits/travel.html 99th Pursuit Squadron at Chanute Field]
* [http://www.blackaviation.com/blackhistory.html Articles about the Tuskegee Airmen] from the [[Chicago Defender]] newspaper, 1944, at Black Aviation Enterprises
* [http://tuskegeeairmen.org/ Tuskegee Airmen, Inc. - Official Web Site]
* [http://www.redtail.org/ The Red Tail Project]
* [http://www.army.mil/africanamericans/ African Americans in the U.S. Army]
* [http://dailynews.att.net/cgi-bin/news?e=pri&dt=070329&cat=news&st=newsd8o61bb00&src=ap AP Story March 28, 2007]
* [http://www.aaregistry.com/ National Museum of the United States Air Force: Eugene Jacques Bullard]
* [http://www.bahai.us/node/195 Honored Tuskegee Airmen include two Baha’is] Airmen Dempsey W. Morgan, far left in the header picture, and Myron Wilson are members of the [[Bahá'í faith]].
==See also==
If a number is the successor of another number, then the first number is said to be ''larger than'' the other number. If a number is larger than another number, and if the other number is larger than a third number, then the first number is also larger than the third number. Example: five is larger than four, and four is larger than three, therefore five is larger than three. But six is larger than five, therefore six is also larger than three. But seven is larger than six, therefore seven is also larger than three... therefore eight is larger than three... therefore nine is larger than three, etc.
* [[United States Colored Troops]]
* [[Buffalo Soldiers]]
* [[U.S. 2d Cavalry Division]]
* [[Freeman Field Mutiny]]
* [[U.S. 366th Infantry Regiment]]
* [[U.S. 761st Tank Battalion|761<sup>st</sup> Tank Battalion (aka Black panthers)]]
* [[Golden Thirteen]]
* [[The Port Chicago 50]]
* [[Bessie Coleman]]
* [[List of Congressional Gold Medal recipients]]
* [[Alfonza W. Davis]]
[[Category:African-American history]]
If two non-zero natural numbers are added together, then their sum is larger than either one of them. Example: three plus five equals eight, therefore eight is larger than three (8>3) and eight is larger than five (8>5). The symbol for "larger than" is >.
[[Category:Black history in the United States military]]
[[Category:History of Alabama]]
If a number is larger than another one, then the other is ''smaller than'' the first one. Examples: three is smaller than eight (3<8) and five is smaller than eight (5<8). The symbol for smaller than is <. A number cannot be at the same time larger and smaller than another number. Neither can a number be at the same time larger than and equal to another number. Given a pair of natural numbers, one and only one of the following cases must be true: <br>
[[Category:Groups of World War II]]
* the first number is larger than the second one,<br>
[[Category:Congressional Gold Medal recipients]]
* the first number is equal to the second one,<br>
[[Category:United States Army officers]]
* the first number is smaller than the second one.
[[Category:Tuskegee University]]
[[Category:Military units and formations of the United States in World War II]]
==Counting==
[[Category:People from Tuskegee, Alabama]]
To count a group of objects means to assign a natural number to each one of the objects, as if it were a label for that object, such that a natural number is never assigned to an object unless its predecessor was already assigned to another object, with the exception that zero is not assigned to any object: the smallest natural number to be assigned is one, and the largest natural number assigned depends on the size of the group. It is called ''the count'' and it is equal to the number of objects in that group.
The process of [[counting]] a group is the following:<br>
''Step 1:'' Let "the count" be equal to zero. "The count" is a variable quantity, which though beginning with a value of zero, will soon have its value changed several times.<br>
''Step 2:'' Find at least one object in the group which has not been labeled with a natural number. If no such object can be found (if they have all been labeled) then the counting is finished. Otherwise choose one of the unlabeled objects.<br>
''Step 3:'' Increase the count by one. That is, replace the value of the count by its successor.<br>
''Step 4:'' Assign the new value of the count, as a label, to the unlabeled object chosen in Step 2.<br>
''Step 5:'' Go back to Step 2.
When the counting is finished, the last value of the count will be the final count. This count is equal to the number of objects in the group.
Often, when counting objects, one does not keep track of what numerical label corresponds to which object: one only keeps track of the subgroup of objects which have already been labeled, so as to be able to identify unlabeled objects necessary for Step 2. However, if one is counting persons, then one can ask the persons who are being counted to each keep track of the number which the person's self has been assigned. After the count has finished it is possible to ask the group of persons to file up in a line, in order of increasing numerical label. What the persons would do during the process of lining up would be something like this: each pair of persons who are unsure of their positions in the line ask each other what their numbers are: the person whose number is smaller should stand on the left side and the one with the larger number on the right side of the other person. Thus, pairs of persons compare their numbers and their positions, and commute their positions as necessary, and through repetition of such conditional commutations they become ordered.
==Multiplication table==
{| cellpadding="3" border="1"
! x
! 0
! 1
! 2
! 3
! 4
! 5
! 6
! 7
! 8
! 9
|-
! 0
| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
|-
! 1
| 0 || 1 || 2 || 3 || 4 || 5 || 6 || 7 || 8 || 9
|-
! 2
| 0 || 2 || 4 || 6 || 8 || 10 || 12 || 14 || 16 || 18
|-
! 3
| 0 || 3 || 6 || 9 || 12 || 15 || 18 || 21 || 24 || 27
|-
! 4
| 0 || 4 || 8 || 12 || 16 || 20 || 24 || 28 || 32 || 36
|-
! 5
| 0 || 5 || 10 || 15 || 20 || 25 || 30 || 35 || 40 || 45
|-
! 6
| 0 || 6 || 12 || 18 || 24 || 30 || 36 || 42 || 48 || 54
|-
! 7
| 0 || 7 || 14 || 21 || 28 || 35 || 42 || 49 || 56 || 63
|-
! 8
| 0 || 8 || 16 || 24 || 32 || 40 || 48 || 56 || 64 || 72
|-
! 9
| 0 || 9 || 18 || 27 || 36 || 45 || 54 || 63 || 72 || 81
|}
When two numbers are multiplied together, the result is called a ''product''. The two numbers being multiplied together are called ''factors''.
What does it mean to multiply two natural numbers? Suppose there are five red bags, each one containing three apples. Now grabbing an empty green bag, move all the apples from all five red bags into the green bag. Now the green bag will have fifteen apples. Thus the product of five and three is fifteen. This can also be stated as "five times three is fifteen" or "five times three equals fifteen" or "fifteen is the product of five and three". Multiplication can be seen to be a form of repeated addition: the first factor indicates how many times the second factor should be added onto itself; the final sum being the product.
Symbolically, multiplication is represented by the ''multiplication sign'': <math> \times </math>. So the statement "five times three equals fifteen" can be written symbolically as
:<math> 5 \times 3 = 15.\ </math>
In some countries, and in more advanced arithmetic, other multiplication signs are used, e.g. <math>5\cdot3</math>. In some situations, especially in [[algebra]], where numbers can be symbolized with letters, the multiplication symbol may be omitted; e.g <math>xy</math> means <math>x \times y</math>. The order in which two numbers are multiplied does not matter, so that, for example, three times four equals four times three. This is the commutative property of multiplication.
To multiply a pair of digits using the table, find the intersection of the row of the first digit with the column of the second digit: the row and the column intersect at a square containing the product of the two digits. Most pairs of digits produce two-digit numbers. In the multiplication algorithm the tens-digit of the product of a pair of digits is called the "carry digit".
===Multiplication algorithm for a single-digit factor===
Consider a multiplication where one of the factors has only one digit, whereas the other factor has an arbitrary quantity of digits. Write down the multi-digit factor, then write the single-digit factor under the last digit of the multi-digit factor. Draw a horizontal line under the single-digit factor. Henceforth, the single-digit factor will be called the "multiplier" and the multi-digit factor will be called the "multiplicand".
Suppose for simplicity that the multiplicand has three digits. The first digit is the hundreds-digit, the middle digit is the tens-digit, and the last, rightmost, digit is the ones-digit. The multiplier only has a ones-digit. The ones-digits of the multiplicand and multiplier form a column: the ones-column.
Start with the ones-column: the ones-column should contain a pair of digits: the ones-digit of the multiplicand and, under it, the ones-digit of the multiplier. Find the product of these two digits: write this product under the line and in the ones-column. If the product has two digits, then write down only the ones-digit of the product. Write the "carry digit" as a superscript of the yet-unwritten digit in the next column and under the line: in this case the next column is the tens-column, so write the carry digit as the superscript of the yet-unwritten tens-digit of the product (under the line).
If both first and second number each have only one digit then their product is given in the multiplication table, and the multiplication algorithm is unnecessary.
Then comes the tens-column. The tens-column so far contains only one digit: the tens-digit of the multiplicand (though it might contain a carry digit under the line). Find the product of the multiplier and the tens-digits of the multiplicand. Then, if there is a carry digit (superscripted, under the line and in the tens-column), add it to this product. If the resulting sum is less than ten then write it in the tens-column under the line. If the sum has two digits then write its last digit in the tens-column under the line, and carry its first digit over to the next column: in this case the hundreds column.
If the multiplicand does not have a hundreds-digit then if there is no carry digit then the multiplication algorithm has finished. If there is a carry digit (carried over from the tens-column) then write it in the hundreds-column under the line, and the algorithm is finished. When the algorithm finishes, the number under the line is the product of the two numbers.
If the multiplicand has a hundreds-digit... find the product of the multiplier and the hundreds-digit of the multiplicand, and to this product add the carry digit if there is one. Then write the resulting sum of the hundreds-column under the line, also in the hundreds column. If the sum has two digits then write down the last digit of the sum in the hundreds-column and write the carry digit to its left: on the thousands-column.
===Example===
Say one wants to find the product of the numbers 3 and 729. Write the single-digit multiplier under the multi-digit multiplicand, with the multiplier under the ones-digit of the multiplicand, like so:<br>
{|
| 7 || 2 || 9
|-
| || || 3
|}
Then draw a line under the multiplier and start with the ones-column. The ones-digit of the multiplicand is 9 and the multiplier is 3. The product of three and nine is 27, so write a seven in the ones-column under the line, and write the carry-digit 2 as a superscript of the yet-unwritten tens-digit of the product under the line:
{|
| 7 || 2 || 9
|-
| _ || _ || <u>3</u>
|-
| || <sup><small>2</small></sup> || 7
|}
Next, the tens-column. The tens-digit of the multiplicand is 2, the multiplier is 3, and three times two is six. Add the carry-digit, 2, to the product 6 to obtain 8. Eight has only one digit: no carry-digit, so write in the tens-column under the line:
{|
| 7 || 2 || 9
|-
| _ || _ || <u>3</u>
|-
| || 8<sup><small>2</small></sup> || 7
|}
Next, the hundreds-column. The hundreds-digit of the multiplicand is 7, while the multiplier is 3. The product of three and seven is 21, and there is no previous carry-digit (carried over from the tens-column). The product 21 has two digits: write its last digit in the hundreds-column under the line, then carry its first digit over to the thousands-column. Since the multiplicand has no thousands-digit, then write this carry-digit in the thousands-column under the line (not superscripted):
{|
| || 7 || 2 || 9
|-
| _ || _ || _ || <u>3</u>
|-
| 2 || 1 || 8<sup><small>2</small></sup> || 7
|}
No digits of the multiplicand have been left unmultiplied, so the algorithm finishes, and
<math> 3 \times 729 = 2187</math>.
===Multiplication algorithm for multi-digit factors===
Given a pair of factors, each one having two or more digits, write both factors down, one under the other one, so that digits line up in columns.
For simplicity consider a pair of three-digits numbers. Write the last digit of the second number under the last digit of the first number, forming the ones-column. Immediately to the left of the ones-column will be the hundreds-column: the top of this column will have the second digit of the first number, and below it will be the second digit of the second number. Immediately to the left of the hundreds-column will be the hundreds-column: the top of this column will have the first digit of the first number and below it will be the first digit of the second number. After having written down both factors, draw a line under the second factor.
The multiplication will consist of two parts. The first part will consist of several multiplications involving one-digit multipliers. The operation of each one of such multiplications was already described in the previous multiplication algorithm, so this algorithm will not describe each one individually, but will only describe how the several multiplications with one-digit multipliers shall be coördinated. The second part will add up all the subproducts of the first part, and the resulting sum will be the product.
''First part.'' Let the first factor be called the multiplicand. Let each digit of the second factor be called a multiplier. Let the ones-digit of the second factor be called the "ones-multiplier". Let the tens-digit of the second factor be called the "tens-multiplier". Let the hundreds-digit of the second factor be called the "hundreds-multiplier".
Start with the ones-column. Find the product of the ones-multiplier and the multiplicand and write it down in a row under the line, aligning the digits of the product in the previously-defined columns. If the product has four digits, then the first digit will be the beginning of the thousands-column. Let this product be called the "ones-row".
Then the tens-column. Find the product of the tens-multiplier and the multiplicand and write it down in a row — call it the "tens-row" — under the ones-row, ''but shifted one column to the left''. That is, the ones-digit of the tens-row will be in the tens-column of the ones-row; the tens-digit of the tens-row will be under the hundreds-digit of the ones-row; the hundreds-digit of the tens-row will be under the thousands-digit of the ones-row. If the tens-row has four digits, then the first digit will be the beginning of the ten-thousands-column.
Next, the hundreds-column. Find the product of the hundreds-multiplier and the multiplicand and write it down in a row — call it the "hundreds-row" — under the tens-row, but shifted one more column to the left. That is, the ones-digit of the hundreds-row will be in the hundreds-column; the tens-digit of the hundreds-row will be in the thousands-column; the hundreds-digit of the hundreds-row will be in the ten-thousands-column. If the hundreds-row has four digits, then the first digit will be the beginning of the hundred-thousands-column.
After having down the ones-row, tens-row, and hundreds-row, draw a horizontal line under the hundreds-row. The multiplications are over.
''Second part.'' Now the multiplication has a pair of lines. The first one under the pair of factors, and the second one under the three rows of subproducts. Under the second line there will be six columns, which from right to left are the following: ones-column, tens-column, hundreds-column, thousands-column, ten-thousands-column, and hundred-thousands-column.
Between the first and second lines, the ones-column will contain only one digit, located in the ones-row: it is the ones-digit of the ones-row. Copy this digit by rewriting it in the ones-column under the second line.
Between the first and second lines, the tens-column will contain a pair of digits located in the ones-row and the tens-row: the tens-digit of the ones-row and the ones-digit of the tens-row. Add these digits up and if the sum has just one digit then write this digit in the tens-column under the second line. If the sum has two digits then the first digit is a carry-digit: write the last digit down in the tens-column under the second line and carry the first digit over to the hundreds-column, writing it as a superscript to the yet-unwritten hundreds-digit under the second line.
Between the first and second lines, the hundreds-column will contain three digits: the hundreds-digit of the ones-row, the tens-digit of the tens-row, and the ones-digit of the hundreds-row. Find the sum of these three digits, then if there is a carry-digit from the tens-column (written in superscript under the second line in the hundreds-column) then add this carry-digit as well. If the resulting sum has one digit then write it down under the second line in the hundreds-column; if it has two digits then write the last digit down under the line in the hundreds-column, and carry over the first digit to the thousands-column, writing it as a superscript to the yet-unwritten thousands-digit under the line.
Between the first and second lines, the thousands-column will contain either two or three digits: the hundreds-digit of the tens-row, the tens-digit of the hundreds-row, and (possibly) the thousands-digit of the ones-row. Find the sum of these digits, then if there is a carry-digit from the hundreds-column (written in superscript under the second line in the thousands-column) then add this carry-digit as well. If the resulting sum has one digit then write it down under the second line in the thousands-column; if it has two digits then write the last digit down under the line in the thousands-column, and carry the first digit over to the ten-thousands-column, writing it as a superscript to the yet-unwritten ten-thousands-digit under the line.
Between the first and second lines, the ten-thousands-column will contain either one or two digits: the hundreds-digit of the hundreds-column and (possibly) the thousands-digit of the tens-column. Find the sum of these digits (if the one in the tens-row is missing think of it as a zero), and if there is a carry-digit from the thousands-column (written in superscript under the second line in the ten-thousands-column) then add this carry-digit as well. If the resulting sum has one digit then write it down under the second line in the ten-thousands-column; if it has two digits then write the last digit down under the line in the ten-thousands-column, and carry the first digit over to the hundred-thousands-column, writing it as a superscript to the yet-unwritten ten-thousands digit under the line. However, if the hundreds-row has no thousands-digit then do not write this carry-digit as a superscript, but in normal size, in the position of the hundred-thousands-digit under the second line, and the multiplication algorithm is over.
If the hundreds-row does have a thousands-digit, then add to it the carry-digit from the previous row (if there is no carry-digit then think of it as a zero) and write the single-digit sum in the hundred-thousands-column under the second line.
The number under the second line is the sought-after product of the pair of factors above the first line.
===Example===
Let our objective be to find the product of 789 and 345. Write the 345 under the 789 in three columns, and draw a horizontal line under them:
{|
| 7 || 8 || 9
|-
| <u>3 || <u>4 || <u>5
|}
''First part.'' Start with the ones-column. The multiplicand is 789 and the ones-multiplier is 5. Perform the multiplication in a row under the line:
{|
| || 7 || 8 || 9
|-
| <u> || <u>3 || <u>4 || <u>5
|-
| 3 || 9<sup><small>4 || 4<sup><small>4 || 5
|}
Then the tens-column. The multiplicand is 789 and the tens-multiplier is 4. Perform the multiplication in the tens-row, under the previous subproduct in the ones-row, but shifted one column to the left:
{|
| || || 7 || 8 || 9
|-
| <u> || <u> || <u>3 || <u>4 || <u>5
|-
| || 3 || 9<sup><small>4 || 4<sup><small>4 || 5
|-
| 3 || 1<sup><small>3 || 5<sup><small>3 || 6
|}
Next, the hundreds-column. The multiplicand is once again 789, and the hundreds-multiplier is 3. Perform the multiplication in the hundreds-row, under the previous subproduct in the tens-row, but shifted one (more) column to the left. Then draw a horizontal line under the hundreds-row:
{|
| || || || 7 || 8 || 9
|-
| <u> || <u> || <u> || <u>3 || <u>4 || <u>5
|-
| || || 3 || 9<sup><small>4 || 4<sup><small>4 || 5
|-
| || 3 || 1<sup><small>3 || 5<sup><small>3 || 6
|-
| <u>2 || <u>3<sup><small>2 || <u>6<sup><small>2 || <u>7 || <u> || <u>
|}
''Second part.'' Now add the subproducts between the first and second lines, but ignoring any superscripted carry-digits located between the first and second lines.
{|
| || || || 7 || 8 || 9
|-
| <u> || <u> || <u> || <u>3 || <u>4 || <u>5
|-
| || || 3 || 9<sup><small>4 || 4<sup><small>4 || 5
|-
| || 3 || 1<sup><small>3 || 5<sup><small>3 || 6
|-
| <u>2 || <u>3<sup><small>2 || <u>6<sup><small>2 || <u>7 || <u> || <u>
|-
| 2 || 7<sup><small>1 || 2<sup><small>2 || 2<sup><small>1 || 0 || 5
|}
The answer is
:<math> 789 \times 345 = 272205. </math>
'''See also:''' [[subtraction]], [[long division]], [[unary numeral system]], [[binary arithmetic]], [[number line]], [[equals sign]], [[plus and minus signs]], [[0 (number)|0]].
[[Category:Arithmetic]]
| 