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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 Ã— 9.
In his 1820 book The Philosophy of Arithmetic, mathematician John Leslie published a multiplication table up to 99 Ã— 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 Ã— 25.
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 Ã— 10 = 10 2 Ã— 10 = 20 3 Ã— 10 = 30 4 Ã— 10 = 40 5 Ã— 10 = 50 6 Ã— 10 = 60 7 Ã— 10 = 70 8 Ã— 10 = 80 9 Ã— 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:
â†’ â†’ 1 2 3 2 4 â†‘ 4 5 6 â†“ â†‘ â†“ 7 8 9 6 8 â†� â†� 0 0 Fig. 1 Fig. 2
For example, to memorize all the multiples of 7:
- Look at the 7 in the first picture and follow the arrow.
- 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.
- 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.
- 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.
- Proceed in the same way until the last number, 3, which corresponds to 63.
- Next, use the 0 at the bottom. It corresponds to 70.
- Then, start again with the 7. This time it will correspond to 77.
- 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
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.
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 Ã— 102 + 2 Ã— 101 + 3 Ã— 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 Ã— 23 + 1 Ã— 22 + 0 Ã— 21 + 1 Ã— 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 signalâ€”a range of voltagesâ€”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.
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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
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.
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!
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 , 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.  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.