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From Wikipedia

Multiplication table

In mathematics, a multiplication table (sometimes, less formally, a times table) is a mathematical table used to define a multiplication operation for an algebraic system.

The decimal multiplication table was traditionally taught as an essential part of elementary arithmetic around the world, as it lays the foundation for arithmetic operations with our base-ten numbers. Many educators believe it is necessary to memorize the table up to 9 × 9.

In his 1820 book The Philosophy of Arithmetic, mathematician John Leslie published a multiplication table up to 99 × 99, which allows numbers to be multiplied in pairs of digits at a time. Leslie also recommended that young pupils memorize the multiplication table up to 25 × 25.

Traditional use

In 493 A.D., Victorius of Aquitaine wrote a 98-column multiplication table which gave (in Roman numerals) the product of every number from 2 to 50 times and the rows were "a list of numbers starting with one thousand, descending by hundreds to one hundred, then descending by tens to ten, then by ones to one, and then the fractions down to 1/144" (Maher & Makowski 2001, p.383)

The traditional rote learning of multiplication was based on memorization of columns in the table, in a form like

1 × 10 = 10 2 × 10 = 20 3 × 10 = 30 4 × 10 = 40 5 × 10 = 50 6 × 10 = 60 7 × 10 = 70 8 × 10 = 80 9 × 10 = 90

10 x 10 = 100 11 x 10 = 110 12 x 10 = 120 13 x 10 = 130 14 x 10 = 140 15 x 10 = 150 16 x 10 = 160 17 x 10 = 170 18 x 10 = 180 19 x 10 = 190 100 x 10 = 1000

This form of writing the multiplication table in columns with complete number sentences is still used in some countries instead of the modern grid above.

Patterns in the tables

There is a pattern in the multiplication table that can help people to memorize the table more easily. It uses the figures below:

→ → 1 2 3 2 4 ↑ 4 5 6 ↓ ↑ ↓ 7 8 9 6 8 � � 0 0 Fig. 1 Fig. 2

For example, to memorize all the multiples of 7:

  1. Look at the 7 in the first picture and follow the arrow.
  2. The next number in the direction of the arrow is 4. So think of the next number after 7 that ends with 4, which is 14.
  3. The next number in the direction of the arrow is 1. So think of the next number after 14 that ends with 1, which is 21.
  4. After coming to the top of this column, start with the bottom of the next column, and travel in the same direction. The number is 8. So think of the next number after 21 that ends with 8, which is 28.
  5. Proceed in the same way until the last number, 3, which corresponds to 63.
  6. Next, use the 0 at the bottom. It corresponds to 70.
  7. Then, start again with the 7. This time it will correspond to 77.
  8. Continue like this.

Figure 1 is used for multiples of 1, 3, 7, and 9. Figure 2 is used for the multiples of 2, 4, 6, and 8. These patterns can be used to memorize the multiples of any number from 1 to 9, except 5.

In abstract algebra

Multiplication tables can also define binary operations on groups, fields, rings, and other algebraic systems. In such contexts they can be called Cayley tables. For an example, see octonion.

Standards-based mathematics reform in the USA

In 1989, the National Council of Teachers of Mathematics (NCTM) developed new standards which were based on the belief that all students should learn higher-order thinking skills, and which recommended reduced emphasis on the teaching of traditional methods that relied on rote memorization, such as multiplication tables. Widely adopted texts such as Investigations in Numbers, Data, and Space (widely known as TERC after its producer, Technical Education Research Centers) omitted aids such as multiplication tables in early editions. It is thought by many that electronic calculators have made it unnecessary or counter-productive to invest time in memorizing the multiplication table. NCTM made it clear in their 2006 Focal Points that basic mathematics facts must be learned, though there is no consensus on whether rote memorization is the best method.



From Encyclopedia

Binary Number System Binary Number System

The binary number system, also called the base-2 number system, is a method of representing numbers that counts by using combinations of only two numerals: zero (0) and one (1). Computers use the binary number system to manipulate and store all of their data including numbers, words, videos, graphics, and music. The term bit, the smallest unit of digital technology, stands for "BInary digiT." A byte is a group of eight bits. A kilobyte is 1,024 bytes or 8,192 bits. Using binary numbers, 1 + 1 = 10 because "2" does not exist in this system. A different number system, the commonly used decimal or base-10 number system, counts by using 10 digits (0,1,2,3,4,5,6,7,8,9) so 1 + 1 = 2 and 7 + 7 = 14. Another number system used by computer programmers is the hexadecimal system, base-16 , which uses 16 symbols (0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F), so 1 + 1 = 2 and 7 + 7 = E. Base-10 and base-16 number systems are more compact than the binary system. Programmers use the hexadecimal number system as a convenient, more compact way to represent binary numbers because it is very easy to convert from binary to hexadecimal and vice versa. It is more difficult to convert from binary to decimal and from decimal to binary. The advantage of the binary system is its simplicity. A computing device can be created out of anything that has a series of switches, each of which can alternate between an "on" position and an "off" position. These switches can be electronic, biological, or mechanical, as long as they can be moved on command from one position to the other. Most computers have electronic switches. When a switch is "on" it represents the value of one, and when the switch is "off" it represents the value of zero. Digital devices perform mathematical operations by turning binary switches on and off. The faster the computer can turn the switches on and off, the faster it can perform its calculations. Each numeral in a binary number takes a value that depends on its position in the number. This is called positional notation. It is a concept that also applies to decimal numbers. For example, the decimal number 123 represents the decimal value 100 + 20 + 3. The number one represents hundreds, the number two represents tens, and the number three represents units. A mathematical formula for generating the number 123 can be created by multiplying the number in the hundreds column (1) by 100, or 102; multiplying the number in the tens column (2) by 10, or 101; multiplying the number in the units column (3) by 1, or 100; and then adding the products together. The formula is: 1 × 102 + 2 × 101 + 3 × 100 = 123. This shows that each value is multiplied by the base (10) raised to increasing powers. The value of the power starts at zero and is incremented by one at each new position in the formula. This concept of positional notation also applies to binary numbers with the difference being that the base is 2. For example, to find the decimal value of the binary number 1101, the formula is 1 × 23 + 1 × 22 + 0 × 21 + 1 × 20 = 13. Binary numbers can be manipulated with the same familiar operations used to calculate decimal numbers, but using only zeros and ones. To add two numbers, there are only four rules to remember: Therefore, to solve the following addition problem, start in the rightmost column and add 1 + 1 = 10; write down the 0 and carry the 1. Working with each column to the left, continue adding until the problem is solved. To convert a binary number to a decimal number, each digit is multiplied by a power of two. The products are then added together. For example, to translate the binary number 11010 to decimal, the formula would be as follows: To convert a binary number to a hexadecimal number, separate the binary number into groups of four starting from the right and then translate each group into its hexadecimal equivalent. Zeros may be added to the left of the binary number to complete a group of four. For example, to translate the number 11010 to hexadecimal, the formula would be as follows: Bits are a fundamental element of digital computing. The term "digitize" means to turn an analog signal—a range of voltages—into a digital signal, or a series of numbers representing voltages. A piece of music can be digitized by taking very frequent samples of it, called sampling, and translating it into discrete numbers, which are then translated into zeros and ones. If the samples are taken very frequently, the music sounds like a continuous tone when it is played back. A black and white photograph can be digitized by laying a fine grid over the image and calculating the amount of gray at each intersection of the grid, called a pixel . For example, using an 8-bit code, the part of the image that is purely white can be digitized as 11111111. Likewise, the part that is purely black can be digitized as 00000000. Each of the 254 numbers that fall between those two extremes (numbers from 00000001 to 11111110) represents a shade of gray. When it is time to reconstruct the photograph using its collection of binary digits, the computer decodes the image, assigns the correct shade of gray to each pixel, and the picture appears. To improve resolution, a finer grid can be used so the image can be expanded to larger sizes without losing detail. A color photograph is digitized in a similar fashion but requires many more bits to store the color of the pixel. For example, an 8-bit system uses eight bits to define which of 256 colors is represented by each pixel (28 equals 256). Likewise, a 16-bit system uses sixteen bits to define each of 65,536 colors (216 equals 65,536). Therefore, color images require much more storage space than those in black and white. see also Early Computers; Memory. Ann McIver McHoes Blissmer, Robert H. Introducing Computer Concepts, Systems, and Applications. New York: John Wiley & Sons, Inc., 1989. Dilligan, Robert J. Computing in the Web Age: A Web-interactive Introduction. New York: Plenum Press, 1998. White, Ron. How Computers Work: Millennium Edition. Indianapolis: Que Corporation, 1999.


From Yahoo Answers

Question:sup! I'm alexis, I'm 17. It's funny how none of this stuff actually sticks for me, but my little brother has 3 pages of math homework and neither one of us can figure it out. lol. sad. but true. and some of it i swear i never did in his grade. or like ever. _______________________________________________________________ ok the first one says "fill in the boxes to make equivalent fractions" 2/4 = 4/__ __/12 = 4/6 ____________________________________________________________ "evaluate the given fractions. Do the students have units that are equivalent in size? explain. Prove your answer with a sketch or with an example of an equivalent fraction" 10. arra has 2/3 of a candy bar. Brenda has 5/6 of a candy bar. __________________________________________________________ "If two numbers are adjusted equally, the answer to the problem stays the same. The subtrahend, 57, is 3 away from 60; add 3 to both the subtrahend and the minuend." ex. 71 +3 ------ 74 - 57 + 3 ------ - 60 ---------- ------- ----------- 14 ------- 14 "look at the problem. make an adjustment. rewrite the problem and subtract" 51 - 26 ______ ___________________________________________ "write the decimal that is equal to ech word phrase on the first line. write the equivalent fraction or mixed fraction on the second line." 1. four hundredths _______ _______ 13. two thousand, five hundred forty two and six hundredths ______ _______ ____________________________________ "If the grid is worth one, shade the following decimals" 1. 0.05 (just describe what i'd do) ___________________________________ thanks guys. and to answer any possible questions.. YES, i am serious. and NO im not retarded, i have an A in pre-cal i simply don't remember this stuff k? so don't be rude just answer the questions!

Answers:I think I can help with a few... 2/4 = 4/8 2/3 = ?/6 and 5/6 2/3 = 4/6 and 5/6 Brenda has the bigger piece. do a pie chart divided into 6...... 51 + 3 = 54 -26 + 3 = 29 ___________ 25 + 3 = 25 (subtract each set of numbers before and after the 3s) 4/100 and 25 2,542 6/100 and 2542.16

Question:1: What is the number of degrees of arc in a circle? 45 90 180 360 2: What is the measure in degrees of the angles of a triangle? 45 90 180 360 3: Which of the following is true? More difficult math questions are worth more toward your score, so you should make sure you spend a lot of time answering these questions. More difficult math questions are worth less toward your score, so they are not worth taking the time needed to answer them. All math questions are worth the same, so you answer those questions that are easier first, and then return to the more difficult questions later. You are not allowed to got back to a question once you have skipped it, so you should try to come up with the answer to each question as you come to it. 4: A/B: what is the term used to refer to "A" in this fraction? Numerator Denominator Terminator Numberator 5: Define "set": It is another word for a fraction. It is a number that had at least one fractional representation. It is a collection of distinct things not considered as a whole. It is a collection of distinct things considered as a whole. 6: A/B: what is the term used to refer to "B" in this fraction? Numerator Denominator Terminator Numberator 7: How would you read the following ratio: 1:2? One out of two Two out of one One to two Two to one 8: Which of the following is an example of a geometric sequence? 2, 4, 5, 7 2, 3, 4, 5 2, 4, 6, 8 2, 4, 8, 16 9: Which of the following is an example of an arithmetic sequence? 1, 3, 7, 15 1, 3, 5, 7 1, 3, 9, 27 1, 3, 4, 7 10: Which of the following is FALSE of grid-in questions? There are no negative answers. There are sometimes two correct answers for one question. The grid contains decimal points and a fraction bar if your answer is not a whole number. There is only one correct answer for each question. 11: What is a scalene triangle? Two sides have equal length with two equal internal angles. It has one 90 internal angle. All sides are different lengths and the internal angles are all different. All sides of equal length 12: What is the range of a data set? The difference between the smallest and largest element. The number within a set that appears most frequently. The number that divides the higher half of a data set, number grouping, or a probability distribution from the lower half. A collection of measurements or quantities. 13: On the SAT, what does the word "average" refer to? Arithmetic mode Arithmetic mean Arithmetic median Arithmetic range 14: The average velocity v of an object moving a distance d in a straight line during a time interval t is described by what formula? v = d x t v = d/t v = d + t v = t/d 15: What are coefficients? Fixed values, like 3 or 8 that can be represented by letters Unknown or generic real numbers represented by letters Constant factors that multiply a variable or powers of a variable Numbers or variables separated by + or -

Answers:Most of these are standard definition and test taking techniques. If you need help here, there's nothing we can do to help you with your SAT.

Question:Ohkay. I need some MAJOR help on math. The title is "relateing rational numbers" and there is a box that says: FRANKS BUDGET: clothing - 1/8 entertainment - 1/20 food - 27.5% other - 0.4% transportation: 15% ----- then it says: "The grid below represtents 100% of Frank's monthly salary after taxes. Shade the grid to represent the information in the table above. [there is a grid with 100 little blocks] then there is questions: 12. What fraction of Frank's budget is used for TRANSPORTATION? explain the strategy you used to convert the perfect to a fraction. 13. hat percent of Frank's budget is used for CLOTHING? explain the strategy you used to convert the fraction to percent. 14. Use the shaded model to verify the percent given in problem 7 is equivalent to 1/8. 15. What decimal of Frank's budget is allotted for ENTERTAINMENT? explain the strategy you used to convert the fraction to decimal. 16. [same question, but with OTHER] 17. [same question but with FOOD] 18. What fraction of Frank's budget does he spend the least amount? Justify your response. 19. What catefory of Frank's budget does he spend the least ammount? justify your response. 20. Describe how to verify the given amounts in the table represent 100% of Frank's budget after taxes. What form of rational numbers did you choose to verify the total of the amounts represent 100% and why? 21. Without doing any calculations, estimate a resonable solution for the amount Frank spends on clothing each month if his salary is $3,500 after taxes. Describe the process you use. 22. Calculate the amount Frank spends on clothing each month if his salary after taxes is $3,500. Justify your answer. OKAY! :] i just need help, because i'm not that smart, i'd ask my dad but he is comeing back from Austin and is getting back late tonight. I NEED HELP right now, or i'll get into some trouble at school.. i'd do this earlier but i've been busy! pleaseee! -mandy

Answers:Clothing= 1/8 (12.5%) Entertainment= 1/20 (5%) Food= 27.5% Other= 0.4% Transportation= 15% *I got the percentages by dividing the numerators by denominators and then multiplying the result by 100. I will explain that below. But for now, shade in 60% of the grid, because all of those percentages add up to about 60. 12. Put the fraction over 100, meaning have 100 has the denominator, because the denominator is expressing the grid and 15 as the numerator because the percentage expresses the part that is shaded. 13. Divide the numerator by the denominator and multiply the result by 100. (1/8)100 *Whenever a number is right next to something, like in this case, parentheses, multiply it, so that means to multiply by 100. 14. What's problem 7? 15. Do the same as you did for percent and then divide the percent by 100. 16. I think you divide 0.4 by 100 and leave the percent sign out, Not totally sure about this one. 17. Divide the percent by 100. 18. I don''t think the question is written correctly. Sorry. 19. Frank spends the least on Other because he spends 12.5% on clothing, 5% on entertainment, 27.5% on food, and 15% on transportation. 20. On the chart, fill in 60 boxes because the percentages add up to 60. To answer the second half, percentages because they are easiest to add and calculate with. 21. About 60 dollars because you divide 35 by 6 and get around 6. Add the zeros to the 6's and multiply. You get 3600, which is close to the amount in problem 21. 22. 35/6= 5.83. Add the zeros and get 58.30 dollars because you divide 35 by 6 and get 5.83. Add a 0.. *If I am wrong, excuse me. I tried my best!

Question:According to the Russian Meteorological Center, their country is in the midst of a heat wave which "nothing similar to this on the territory of Russia during the last one thousand years". http://en.rian.ru/russia/20100809/160128496.html The heat wave has pushed maximum temperatures more than 20 F above the August Moscow average, and the Russian death toll is probably at least 15,000 (not to mention decimating the country's wheat crop). http://www.wunderground.com/blog/JeffMasters/comment.html?entrynum=1568 This comes on the heels of a flood in Tennessee earlier this year which in much of the state was a one-in-1,000-year event. http://www.srh.noaa.gov/ohx/?n=may2010epicfloodevent Then of course there are the floods in Pakistan, record high temperatures across Africa and Asia, along the US East Coast, etc. And of course there was the 2003 European heat wave which killed 30,000 people and resulted in 13 billion Euros' worth of losses. http://www.grid.unep.ch/product/publication/download/ew_heat_wave.en.pdf Sometimes it's suggested that rather than reduce greenhouse gas emissions now, we should simply adapt to the consequences of global warming. Is there any way to adapt to the increase in these extreme weather events? I suppose one way would be to start deploying air conditioning units in regions which previously didn't necessitate them, in order to reduce the number of heat-related deaths. Any other suggestions?

Answers:Some Numbers As you may have gathered from my previous answers, I like to mess around with numbers. And that s what I ve been doing in respect of the extreme weather events over these last few months. There have been numerous instances of unusual weather stretching from the Arctic to the Antarctic and pretty much most places in-between. At the moment we have India, Pakistan, Western Europe, China, Russia, the Arctic, Sub Saharan Africa, the Western Pacific and southern South America that are all concurrently experiencing very unusual weather. Each event in itself wouldn t be that remarkable, not when looked at on a global scale. For example, there are 195 countries in the world, statistically it would be normal, in any one year, for one country to experience a specific event with a 200 year return period [1], or for 2 countries to experience events with 100 year RP s. There are numerous specific events that could occur including extreme temperatures, droughts, cyclonic energy, sustained wind speeds etc. All told there are perhaps 300 such records but live feeds only provide data on about 100. So what we re looking at is 200 countries (rounded up) each capable of registering 100 extremes 20,000 possibilities in total. Thus, in any given year we would expect to see the sum total of the RP s for all unusual weather events totalling around 20,000 (the equivalent for example of 1,000 events with a 20 year RP, 200 events with a 100 RP s or any combination that adds up to 20,000). Since the 1st January this year the RP Index , instead of being around 12,100 is in fact somewhere between 40,000 and 50,000. In simple terms this means there has been 3 to 4 times as many weather extremes this year as would ordinarily be the case. I haven t yet worked out the probability of this happening but it s going to be many thousands to one against. In respect of the flooding in the UK, we were commissioned by two organisations to conduct some research, once in 2006 and again in 2007 following the devastating floods. Prior to the record breaking floods, the probability of the number of floods occurring, without some external force affecting the climate, was just under a million to one, in the wake of the 2007 floods this rose to nearly 5 million to 1 and following the 2008, 2009 and 2010 floods it now stands at 8 million to 1. [1] Return period (AKA return event or recurrence interval) is the expected length of time between successive occurrences of a natural disaster such as a flood, drought or earthquake; i.e. an event with an RP of 100 would be expected to occur once every 100 years. An Answer to Your Question I don t think there can be any doubt that the climate is changing and that this year the weather has gone haywire. There have been somewhere in the region of 70 contributory factors to all the different events but perhaps the most significant have been the abnormalities with the jet stream, the strong El Nino effect, changes to the North Atlantic Oscillation earlier this year, the Arctic dipole anomaly and the unusual poleward transition of Equatorial air masses. All the effects are interactive, some amplify and some attenuate each other and combined they provide some pretty crazy weather. Quite how big a role global warming has had in this is very difficult to say. What we do know from longer term observations is that the climate is changing and consequent to this are the increases in the number of adverse weather events. These are not always detrimental, sometimes for example, they bring rain to normally dry areas or they warm up areas that are usually cold. On the whole, the negative effects far exceed the positive ones and the impact on humans is significant, particularly so in the less economically developed countries. In terms of mitigating against the effects of a changing climate, a lot depends on the availability of resources. In the developed world we can often adapt, we can change agricultural practices, build flood defences and storm shelters and just as importantly, we can do the smaller things as well. Buying an extra bottle of water when it s hot, turning up the air-con, or paying a little extra for a loaf of bread because the wheat harvest has failed. The third world countries don t have these luxuries. When they get hit by a drought or a flood or other extreme weather event the consequences hit hard. Their homes are washed away, epidemics break out, crops are destroyed with no reserves available to replace them, water supplies become contaminated, the infrastructure collapses and the economies take a serious pounding. It s not just the obvious effects, there s the more insidious ones as well, such as the spread of diseases like malaria. The World Health Organisation has calculated that 150,000 people are dying each year because of climate change, a number expected to double in the next 20 years. Almost all of these deaths are occurring in the third world. There are of course, many more consequences of a changing climate but it tends to be the same picture not so bad in the developed world, disastrous in the developing world. - - - - - - - - - - - - Mark s comment is very interesting considering that his account was only created yesterday. No prizes for guessing which denier has created another account so that s/he can abuse the system. Things are looking pretty grim for the deniers, not only do they have to rely on imaginary evidence but they ve now made some imaginary friends as well.

From Youtube

Converting Fractions to Decimals Using the 100-Grid :This video shows one way to convert a fraction into a decimal - by using the 100-grid.