Why 62(1+2)=1?

This question was initially asked in the Yahoo website, and spread rapidly through Facebook recently, especially in Taiwan, it even debated by parliamentary members. As reported by Sin Chew Daily, a popular Malaysian Chinese Newspaper, there are more than 3 million people responded to it. About 2 million voted for the answer 9, and about 1.5 million voted for the answer 1. Actually, similar questions have been asked long time ago, but human tend to be forgetful or ignorant of history.

Some mathematics experts said the answer is 9, because they opined that calculation should follow the order of operations called BODMAS, which stands for Bracket first, O for orders (powers and square roots, etc.), Divide and Multiply (from left to right), Add and Subtract (from left to right), or PEMDAS stands for Parentheses, Exponents, Multiplication and Division, Addition and Subtraction. Since “x” and “¸” are of the same level of operation order, they must be carried out from left to right as or as if they appear in the equation, so, the answer obtained is 9. But, this is wrong!  

There are various suggestions that the question should be presented in more clearly defined ways such as:

        By adding an “x” just before the bracket:  6¸2x(1+2) = 6¸2x3 = 3x3 = 9

        By adding a bracket just before the other:  (6¸2)(1+2) = 3x3 = 9

        By adding a square bracket after “¸”:  6¸[2(1+2)] = 6¸[2x3] = 6¸6 = 1

By doing so, the answer will be definite. However, there is nothing wrong with the original question as it is written 6¸2(1+2). The above additions although could remove doubts and useful for primary students, but are sometimes redundant. There should be no issue about the correctness of the question if there is only one answer and basic principles are followed.  

If the operation is based on BODMAS rules and the first “2” in front of the bracket is merely treated as a number:

         

(1)   However, if you type on a calculator with the equation’s numbers and symbols exactly as written, it should give the correct answer as 1. The calculator must be correct although it does not provide explanation. If the calculator created and programmed by human is wrong, what else can we trust?    

        Unfortunately, it is really found that: If use Casio fx-570MS and fx-350MS calculators the answer is 1. If use Casio FC-100V and fx570ES calculators the answer is 9, this is wrong. All teachers and people must be informed or warned that something wrong with calculators. The MS series and ES series do not agree each other probably due to certain mechanism, programming error or conceptual mistake. The mathematician, the software engineer, programmer, school teachers, education ministry, global system, or who to be blamed? But not the designer of “¸” and ( ).

         

(2)   In fact, the first “2” is not just a number, but a factor for “(1+2)”, so 2(1+2) must be wholly performed first as follows:

The first “2” can be a factor because it is true that when 6¸(2+4) is factorized will becomes 6¸2(1+2) or 6¸[2(1+2)], but usually the [  ] is omitted and treated as redundant.

(3)   We know that 6¸2(1+2) can be written as follows using an “over” sign, or fraction bar, i.e. the horizontal line representing division and separate the numerator and denominator. This is the important concept learned in primary and secondary schools.

It is not correct to write and calculate as follows.

               

(4)   If represent (1+2) with y, i.e. y = (1+2), and the first “2” is a factor of y, therefore 2y is bound together as one term, then,

         

Substitute y = (1+2) back to it:

(5)   Similarly, equalize a bracket to a multiplication is not always true, as can be seen in the following examples.

       

(6)   Let consider a scientific example, the ideal gas law, PV = nRT can be rearranged to n = PV¸RT or n = VP¸TR, etc.

       

So, those who have never properly studied the calculation of ideal gas law will probably give the wrong answer 9!

(7)   One obscure rule which may override BODMAS is the factorization and defactorization or expansion, which can always be a priority to be performed first when necessary or sometimes done later or just ignored. Under the situation of the question, defactorization will be of priority.

        If we defactorize/expand the equation,

The other way round, if we factorize the following equation, it should give the same answer of 1.

This means the first “2” is a factor. It can be bracketed, although always is omitted, and the above equation becomes,

         

If we substitute a = 1, b = 2, c = 3 in the question and carry out factorization, and substitute back the values of a, b and c:

       

If we try to factorize the above equation the following way,

       

        The trick here is, we cannot factorize two terms separated by a “¸” sign. Otherwise any number can be the answer.

e.g.

(8)   The main issue now is the meaning of “¸” and its substitute/alternative namely the fraction bar “¾” as well. “¸” is the division symbol used to separate between two immediate terms side-by-side. The term before ¸ is the numerator, in this case is 6. The term after ¸ is the denominator, in this case is 2(1+2). In the presence of “¸”, we cannot split the second term 2(1+2) into two separate terms/parts as 2 and (1+2) and use the former part for division but leave the second part for multiplication. Therefore, 2(1+2) is not exactly same meaning as 2x(1+2), but more than that.

(9)   One example of equivalence for further understanding the meaning of “¸” and ( ).

       

The following answer is incorrect:

This proves that we cannot simply drop the bracket and change it to multiplication, and “force” it to follow the BODMAS.

(10) One money-counting matter about money mongers. A billionaire plans to donate 6 million ringgits to two families, each family has a single parent and two children, the money is to be distributed evenly per person.

If you calculate as follows, each person will get 1 million ringgits, total 6 million ringgits.

If you calculate as follows, each person will get 9 million ringgits, total 54 million ringgits!

Where is the money come from? It is not a magic, it is just a mental trick! That’s only sort of one of many ways money mongers become rich.

Some conclusions

The back and forth of mathematical operations must always give consistent results. Otherwise, the Laws of Conservation do not apply, or, creation and destroy of energy and mass, and even natural counting will be a magic or just miracle.

It is always a common perception that a scientific theory or equation needs mathematics to backup. Now, we can see that even a simple arithmetic mathematics need scientific concepts and measurements to support the validity.  

A mathematic operation is not just following an incomplete set of rules like BODMAS and ignoring certain hidden logics. A question or scientific equation must have its meaning and purpose, so the answer must also be meaningful and purposeful, and fulfill what is being asked and sought for, otherwise rules are not just rules but also for abuse, facts will be misinterpreted, ethics will twist around, etc. This is what happened to the mass-energy equivalence equation: E = mc².      

Hew Nam Fong, Senior Lecturer, HELP CAT, 12-May-2011.    http://hewnf.blogspot.com/

Post 3. Twenty Clues to Debunk the Video #3 Footage of UFO on the Dome of the Rock in Jerusalem

1^st published on 3 March 2011. Last edited on 21 March 2011.

Based on various analyses of the video and its still-image, the following clues are provided to determine whether the Video #3 footage of UFO on the Dome of the Rock is a fake or genuine event.  

1.   Soon after the reveal of the UFO footages on Youtube, expert was called by an Israel main evening paper (Yedioth Achronoth) to declare the UFO event as a hoax based on the Video #3. But supporters argued that the action was a government cover-up of UFO event, yet still many people believe this video is a strong proof and it supports the other three videos. The fact is that, the Video #1 was posted first on 29 January 2011, whereas the Video #2 and Video #3 were posted one day later on 30 January 2011, and Video #4 was posted on 2 February 2011. By simple logics, if Video #1 came first is a fake, very surely Video #3 will be a fake, in other words, if Video #3 is real then Video #1 is unlikely a fake came before it, otherwise there should have been a leak prior to the internet posting, or, extreme coincidence applies; however, if the Video #3 is a fake, it does not prove or disprove the Video #1, #2 and #4, instead a separate set of thorough analyses is required for judgment.      

   

2.   Presumably the Video #3 was recorded or posted by Wendy Johnson, her friend or the name is just a disguise. However, about the second week after posting, the video and/or its duplicates has been temporarily withdrawn from some Youtube websites, for a few days, and a notice was given stating the copyright of the clip was being claimed. It is suspicious whether the posting has been regretted, or the consequence and benefit issues were being considered.   

3.   The still images of various scenes captured in the Video #3 are similar to those found in an internet panoramic photo (see Post 3 image 1 and the website: http://upload.wikimedia.org/wikipedia/commons/d/da/Jerusalem_night_7088.JPG.), which is presumably taken from a position quite far away, probably from between the north-west, and facing towards the Mount of Olives at far background. If the video was recording an authentic event, then it must have been done from the same hot spot, or at least from the same direction as for the photo. Interestingly, the entire video shooting was never framing beyond the range as indicated by the rectangular box drawn on the internet photo (see Post 3 image 2), if there is even one still image happen to be outside the photo frame, then the photo definitely cannot be considered as a possible source for the scenes of the Video #3.   

        Post 3 image 1                                                                                                                                         Pot 3 image 2

  

4.   An example of fully zoom-out video still image is that at approximately second 38.5 (see Post 3 image 3), with its dimension about half the size of the internet photo as represented by the smaller rectangular box (see post 3 image 4). If it is a real-scene video shooting, the video viewer (the person who recorded the video) could be at a position of about 50 m closer than the photo viewer (the person who took the photo) if they were using the zoom lens of same power. If both were from exactly the same spot, the video camera must be using a more powerful zoom lens, probably between 70 mm and 210 mm, but quite uncommon for a video camera.  Certainly, they were not very close to the Dome of the Rock because there are quite a number of buildings seen in the foreground; and, to obtain this broad range of view, the camera or video camera should be using a wide-angle zoom mode. Based on the above clues, the viewers are estimated to be about 300-500 m away or slightly farther from the Dome of the Rock, and that should be outside the walls of the Temple Mount (which has a trapezium shape with dimensions of west 488 m, east 470 m, north 315 m and south 280 m) [the Dome of the Rock is located approximately at the centre but closer to the north wall of the Temple Mount]. Certainly a video camera is able to zoom-out and zoom-in in order to produce broad views and close-up views of the UFO and the Dome of the Rock when considering it as a genuine scene; if it is an imitation instead, the video camera would just be a few inches away from a computer monitor, by moving it back and forth in front of a screen will also give the similar zooming effect. Moreover, the zoom-out still images looked rather like a close-up resulted from a “zoom-in” effect applied onto a photo.   

        Post 3 image 3                                                                                                                                          Post 3 image 4

 

5.   The video movie started with a “medium lens” at about second 7 of the video time (see Post 3 image 5), and represented by the lower medium box on the internet photo (see Post 3 image 6), it shows some distortion with obvious slanting of the left side such as the window, this could be due to the effect resulted by tilting of camera, especially if using a lens of low quality. The video was hunting for targets during the first 4 seconds (seconds 7-11 of the video time), then it focused to and around the Dome of the Rock for 10 seconds (seconds 11-21, represented by the upper medium box, see Post 3 image 6), example of still image shown is at about second 16.5 (see Post 3 image 7). At second 21 the descending of UFO was then noticed and recorded, and just before second 27 it zoomed in within less than 1 second (approx. 0.6-0.8 second) to have the maximum close-up view of the orb, still image at second 27.5 (see Post 3 image 8) is represented by the small box on the internet photo. At seconds about 34.5-36.5, it was fully zoomed out and remained in this mode until the end of the movie, still image at about second 43.5 (see Post 3 image 9) is represented by the big box on the internet photo; with this maximum width of view considered as the relative size of 1x, the size during medium lens is about 1.5x, and when it is in close-up, the size of the objects are about 3x. It is quite unusual for a video camera to zoom-in so fast in less than 1 second and without temporary decrease in resolution. The resolutions of the still images are quite consistent, and these zoom-in and zoom-out images are looked like directly “extracted” right the way from the big picture. 

                                                                                            Post 3 image 6

          Post 3 image 5                                                                                                                        Post 3 image 7

  














        Post 3 image 8                                                                                                                        Post 3 image 9

6.   It is quite curious that for 14 seconds after started the video shooting (seconds 7 to 21 of the video time), the UFO in the sky above the Dome of the Rock was not noticed even though there were a few persons around. If the viewer was using a hand-phone video camera or handy video camera which does not need to see into the kind of view finder of an SLR camera, she should be able to see the UFO which took about 10 seconds in descending as recorded by Video #1 and #2. Instead, the video focused to the Dome of the Rock, like getting ready to expect something dropping from the sky.

7.   When the video camera was fixed at a specific zoom state, such as at medium lens during the seconds 7-27, still images at seconds 7, 9.5, 11.5, 13.5, 16, 24.5 are represented by boxes on the internet photo (see Post 3 image 10), in the order of moving from the lowest to the top. There are some variations in the range and dimension of the frames, for example, being broader or taller, sometimes slightly trapezoid, or with some distortion. These effects should not be obvious when recording distant objects without adjusting the zoom, but if the video camera is just a few inches from the screen of the photo, together with hand-shaking and tilting, it will result in unparallel alignment between the lens and the photo or the screen, the above effect would then be enhanced.

                                                                                            Post 3 image 10

8.  The internet photo is likely taken by using a wide-angle zoom lens of 28/35 mm, less likely more than 50 mm lens. If the video camera was exactly at the same spot, then it must be having a powerful telephoto zoom lens in order to get a frame of about 4x larger (see Post 3 image 6). Alternatively, the video camera should be at a closer location (about 50 m), if it is so, the viewer then should be at a lower building, may be 10-20 m lower, otherwise this building will be seen in the internet photo. Yet, this building must be taller (or it is on the slope shoulder of a mountain) than those of the foreground buildings, so that when looking down from such a higher level would be able to see the roofs, the foreground buildings together with the whole Dome of the Rock at far background. Assuming at that position and far distance, by just looking a little up into the sky, the tourists should be able to see the glowing orb even before it was descending. The size of the UFO is about 5 m in diameter, and at that distance should be looked slightly smaller than the moon. But the Video #3 was started by searching for targets around the foreground, more likely a pretended intention rather than incidental action, furthermore, it only recorded the last second (second 21) of the UFO descending.  

9.   Usually, a video camera or mobile phone camera is not equipped with a six-star filter, yet, the 6-star shapes of lights in the video or its still images (Post 3 image 11) are exactly the same as those found in the internet photo (Post 3 image 1). Besides, the starred lights also did not show any motion or perspective change with respect to the space dimension of the buildings, but they looked rather like fixed on two-dimension scenes. Moreover, for the largest six-star light at the location slightly to the left of the centre of the photo, one of the arms (the right arm) did not show continuation, instead, it is ending with a spot. Interestingly, when this photo is video-recorded by using a hand-phone camera, it becomes continuous (Post 3 image 12) and is the same as that found in the Video #3 still image.    

        Post 3 image 11                                                                                                                        Post 3 image 12

10. If the UFO seen in the Video #3 was glowing, it should increase the brightness of the surface of the Dome of the Rock, but it did not, rather just looked like a white orb, which is not radiantly shining, and does not seem to be “hardly look into it” (as said by one of the woman of Video #3). Besides, there is also no sign of showing reflection from other buildings and atmosphere air surrounding it. Among the other three video footages, only the Video #4 is radiantly shining and spinning, and likely to be “hardly look into it”; however, it was only posted two days later than the Video #3. 

11. After the UFO descended close to the Dome of the Rock, it is obviously staying at a fixed distance to it. There are two types of shaking effects, first type is the video camera vibration effect, it can be due to either really hand shaking or merely created by computer software. For second type, the orb and the photo scene are probably bound together, and they wobble in tandem but in irregular manners. Although the video shows shaking effect and wobbling movement of the orb, in fact it did not drift away or hover around as observed in the Video #1, #2 and #4, this is one of the major differences between them.   

12. Practically, when a camera is in the zoom-in mode or using a telephoto lens, unsteady hand shaking normally will cause more severe shaking effect to the video as compared to when it is in a zoom-out mode. But the camera-shaking effects in the Video #3 are quite consistent whether it is in the zoom-out mode and zoom-in mode. However, the resolutions during the zoom-in and zoom-out filming are quite similar, it is rather unusual to have this consistency no matter how steady is the holding state. But not surprise that this result can be mimicked by using a camera moving back and forth and “surfing” around on the photo.   

13. Among the women speaking English, one woman said: “hardly look into it”, another woman: “we ha-n’ (haven’t), we seen in Mississippi like this, but never like that, never like that…” the voice is ending with reducing volume and like fading away, either the woman turned her head to the back or she was turning while walking away from the video camera or a tape recorder. Guessing one of the possibility, the women were watching one or more videos (Video #1, #2, #3 and #4) displayed individually or together at the same time and same place, what she saw in Mississippi was like one of them (among the four videos) and not like another one; this conversation was then dubbed to the Video #3.  

  

14. Before the UFO blasted off, there was also a conversation in Chinese between a man and a woman. The man was saying: “…why don’t take photo? Let’s take a photo! 怎不拍照?咱照个相吧!”, the woman: “my hand is taking my camera 我手拿着我的照相机”, and the man: “I think to take photo with Triumphal Arch 我想和凯旋门照”, certainly these are irrelevant to the UFO sighting. Supporters of the Video #3 may argue that the Chinese tourists might be facing a different direction and did not notice the amazing UFO event. However, a possible reason is that, the conversation was genuinely recorded somewhere else (maybe in Europe) where the American women were watching a video display of the UFO and then the composite sound effects were dubbed to the video clip to produce the final Video #3. Dubbing soundtracks is a common traditional technique and can be easily done. 

15. Unlike the Videos #1, #2 and #4, the Video # 3 does not show two flashes therefore did not surprise them to scream out before its blasting off, but they “aoouu…woouu…” after, and the story then ended soon.  

16. In Video #3, the time length for the hovering of UFO above the Dome of the Rock is 22.5 seconds, but in Video #1 is 23 seconds, these seem to be not tally between them. The time length of an event must be exactly the same, it can’t miss even a fraction of second in reality while another person gain in the space. In physical nature, there is no such time paradox in reality and neither there is such theory, unless it is merely technological glitches due to the optical and digital devices.

   

17. As the UFO was shooting up into the sky at second 44, the video was still staying on recording the Dome of the Rock. If there were “actual witnesses” at a far perspective of 300-500 m away, they should be able to trace where it went to. However, the frame of the video did not move away from the scene of the Dome of the Rock, it did not follow the UFO up higher above the range indicated by the box in the internet photo; moreover, the recording discontinued in about 3 seconds after the UFO blasted off, and the story just stopped there!   

 

        Post 3 image 13                                                                                                                       Post 3 image 4

Post 2. The Four Video Clips of UFO Above The Dome of The Rock in Jerusalem

1^st published on 1/3/2011. Last edited:on 2/3/2011.

So far four video clips of UFO sighted above the Dome of the Rock have been posted on the Youtube websites. They are supposedly recorded at the same time about 01:00 am on 29 January 2011, but from different locations and angles, thus provide different perspectives and descriptions of the event. They are numbered as Video #1, #2, #3 and #4 according to some websites.

Video #1                                       

Posted by Eligael, 1 minute 45 seconds (1:45), uploaded on 29 Jan 2011.

Website: http://www.youtube.com/watch?v=zQ-bNOy_CKQ&feature=related.

It was recorded by Eligael Gidlovitch from Tel Aviv, who was the night tourist together with another guy at the Promenade of Armon Hanatziv in Jerusalem, which is at a distance of about 2.5 km more or less southwest of the Dome of the Rock. This video focused on the white-glowing UFO orb from a far distance, it also framed-in the guy who is seen video shooting the same event using a mobile phone camera.

Video #2

Posted by truthPacifist, 1 minute 30 seconds (1:30), uploaded on 30 Jan 2011.

Website: http://www.youtube.com/watch?v=-K3Ih-o4qiQ&feature=related.

It has more or less the same vantage point but with a closer view of the UFO event as compared to the Video #1. Video #2 and Video #1 have been incorporated into a synchronized clip in a few of Youtube sites. There is a comment saying that Video #2 is produced by the man seen in the Video #1, but a detailed examination may disagree with this point of view.  

Video #3

Posted by 50nFit, 48 seconds (0:48), uploaded on 30 Jan 2011.

Website: http://www.youtube.com/watch?v=rY2FFEufsuY&feature=related.

In some websites it is labeled as Video #2. It provides the most close-up view of the UFO orb. There was a group of female tourists at the scene of video shooting, with a few of Chinese-speaking tourists nearby but presumably did not notice the UFO event. Since there are quite some buildings framed in the foreground, it is likely that they were outside the platform of the Temple Mount where the Dome of the Rock situated. Some debunkers thought that the view of the Dome of the Rock in this clip is similar to that found in the panoramic photo taken from an Internet image of the following sources:

http://i158.photobucket.com/albums/t120/PunJedi/jersnight.jpg.    

http://upload.wikimedia.org/wikipedia/commons/d/da/Jerusalem_night_7088.JPG.

Video #4

Posted by supergia007, 2 min and 35 seconds (2:35), uploaded on 2 Feb 2011.

Website: http://www.youtube.com/watch?v=jczjyV4v3i0&feature=related.

It was recorded by a group of people consists of at least one woman and two men, they were likely from the direction between southwest and west of the Dome of the Rock, and at a closer location to the scene as compared to that of witnesses in Video #1 and #2. It shows a relatively close-up look of the UFO orb, and has been edited to produce zoomed-in and slow motion plays.

Post 1. The Most Amazing UFO Event or Hoax Happened on the Temple Mount in Jerusalem on 29 January 2011

1^st published on 28/2/2011. Last edited on 1/3/2011.

Video clips showing a glowing UFO on the Temple Mount in Jerusalem have been posted on Youtube websites since January 2011. This UFO orb has been said hovering above and around the Dome of the Rock and recorded from 4 different perspectives at about 01:00 am on 29 January 2011 (based on the time and date stated by Video #1), or late night at approx. 0100 am on 28 January supposed to be morning at approx. 0100 am on 29; however, in most websites this UFO visit was reported to be on 28 January, because these European and American sources are a few hours behind the eastern times, e.g. London was at 11:00 pm, New York was at 06:00 pm on 28 January. Anyhow, the time should be reported as the local time where the event occurred.  

        Post 1 Video #1                                                                                                                        Post 1 Video #2

        Post 1 Video #3                                                                                                                        Post 1 Video #4

If it is a genuine UFO, it is considered as the most incredible UFO event ever recorded, because it happened on the Dome of the Rock, and on the Temple Mount, both of which are the most religious shrines of Islam, Judaism and Christian, and have been controversial for thousands of years in the past and present. Flying over the Dome of the Rock or docking something on it is considered forbidden, and making such movie by human may trigger new conflicts among religions in the Middle East region and eventually will spread world-widely.

The governments and many non-government agents including the dominant mass media have been skeptical about the event and remained silent at the beginning; in spite of that, there are reports and UFO believers suspecting this as another cover-up of UFO. However, this UFO event has inspired a huge quantum of comments and arguments in some mainstream media world-wide, ranging from TV, newspapers and Internet. The news spread very fast especially on the e-world, for example, about 1 week after the postings in Youtube only less than 100,000 visitors surfed to the Youtube sites, but after 2 weeks there were more than 2 million of people have viewed the video clips, and the number is still increasing. Virtual internet war has sparked off between the supporters and debunkers of this UFO event, and apparently most people believe it is a real UFO footage.      

If it is a hoax, a computer graphic artifact may also commit a non-respecting act of the religious shrine, yet, the virally spreading of the videos has not been prohibited, therefore, it adds challenge to the doubt of it being a fake.

The former British Ministry of UFO investigator, Nick Pope, commented that if it is not a real UFO, it would be either a well-planned and coordinated hoax, or one of drones created by man. Nevertheless, and to the best of knowledge, none of the developed countries in the current world has achieved such extreme technology in manipulating vehicle against the strong gravity force on the Earth.       

If it is true, it is thought to be the most amazing UFO event so far, as it is very different from other UFO events that it showed a complete process of “landing”, “docking” or hovering above a building, and taking off in a blasting speed which could be thousands of times more powerful than any vehicle we have on the Earth including the rocket. With that kind of speed, it only needs less than an hour to reach the Moon. The UFO then really showed off its super intelligence that we have never understand before, and could not be explained with our present physics theories, which may be set aside in years to come soon.

If the UFO visit is true, it will be a very puzzling miracle. Why the UFO chooses the Dome of the Rock this time? Are they investigating something in the shrine? Are there something beneath or nearby they are looking for? So many questions will arise if you are trying to think about it.