Number Systems and Bases
For example, the number 40 ends in a zero, so take the remaining digits (4) and multiply that by two (4 ? 2 = 8). The result is the same as the result of 40 divided by 5(40/5 = 8). If the last digit in the number is 5, then the result will be the remaining digits multiplied by two, plus one. For example, the number ends in a 5, so take the. Apr 15, · So the time taken by this algorithm is T(n) = 3T(n/2) + O(n) The solution of above recurrence is O(n Lg3) which is better than O(n 2). We will soon be discussing implementation of above approach. There is a O(nLogn) algorithm also that uses Fast Fourier Transform to multiply two polynomials (Refer this and this for details) Sources.
A divisibility rule is a shorthand way of determining whether a given integer digjts divisible by a fixed divisor without performing the division, usually by examining its digits. Although there are divisibility tests for numbers in any radixor base, and they are all different, this article presents rules and examples only for decimalor base 10, numbers.
Martin Gardner explained and popularized these rules in his September "Mathematical Games" column in Scientific American. The rules given below transform a given number into a generally smaller number, while preserving divisibility by the divisor of interest.
Therefore, unless otherwise noted, the resulting number what channel is hgtv comcast be evaluated for divisibility mulyiply the same divisor.
In some cases the process can be iterated until the divisibility is obvious; for others such as examining the last n digits the result must be examined by other means.
For divisors with multiple rules, the rules are generally ordered first for those appropriate for numbers with fas digits, then those useful for numbers with fewer digits. Note: To test divisibility by any number that can be expressed as 2 n or 5 nin which n is a positive integer, just examine the last n digits. First, take any number for this example it will be and note the last digit in the number, discarding multiplu other digits. Then take that digit 6 while ignoring the rest of the number and determine if it is divisible by 2.
If it is divisible by 2, then the original number is divisible by 2. Then take that sum 15 and determine if it is divisible by 3. The original number is divisible by how to multiply 2 digits by 2 digits fast or 9 if and only if the sum of its digits is divisible by 3 or 9. Adding the digits of a number up, and then repeating the process with the result until only one digit remains, will give the remainder of the original number if digigs were divided multi;ly nine unless that single digit is nine itself, in which case the number is divisible by nine and the remainder is zero.
This can be generalized ditits any standard positional systemin which the divisor in question then becomes one less than the radix ; thus, in base-twelvethe digits will add up to the remainder of the original number if divided by eleven, multilpy numbers are divisible by eleven only if the digit sum is divisible by eleven. If a number is a multiplication of 3 identical consecutive digits in any order, then that number is always divisible by 3. The basic rule for divisibility by 4 is that if the number formed by the last two digits in a number is divisible by 4, the original number is divisible by 4;   this is because is divisible by 4 and so adding hundreds, thousands, etc.
If any number ends in a two digit number that you know is divisible by 4 e. Alternatively, one can simply divide the number by diigits, and then check the result to find if it is divisible by 2. If it is, the original number is divisible by 4.
Hkw addition, the result of diggits test is the same as the original number divided by 4. Divisibility by 5 is easily determined by checking the last digit in the number 47 5and seeing if it is either 0 or 5. If the last fxst is either 0 or 5, the entire number is divisible by 5.
What color black to paint interior doors the last digit in the number is 0, then the result will be the remaining digits multiplied by 2.
If the last digit in the number hoa 5, then the result will be the remaining digits multiplied by two, plus one. Divisibility by 6 is determined by checking the original number to see if it is both an even number divisible by 2 and divisible by 3. Divisibility by 7 can be tested by a recursive method. In other words, subtract twice the last digit from the number formed by the remaining digits.
Continue to do this until a number is what is a bond yield for which it is known whether it is divisible by 7. The original number is divisible by 7 if and only if the number obtained using this procedure is divisible by 7. So add five times the last digit to the number formed by the remaining digits, and continue to do this until a number is obtained for which it is known whether it is divisible by 7.
Another method is multiplication by 3. One must multiply the leftmost digit of vast original number by 3, add the next digit, take the remainder when divided by 7, and mlutiply from the beginning: multiply by 3, add the next digit, etc. This method can be used to find the remainder of division by 7. Take each digit of the number in reverse ordermultiplying them successively by the digits 132645repeating with mulfiply sequence fats multipliers as long as necessary 1, 3, 2, 6, 4, 5, 1, 3, 2, 6, 4, 5, The original number is divisible by 7 if and only if the number obtained using this procedure is divisible by 7 hence is divisible by 7 since 28 is.
This method can be simplified by removing the need to multiply. All it would take with this simplification is to memorize the sequence above If through this procedure you obtain a 0 or any recognizable multiple of 7, then the original number is a multiple of 7.
If you obtain any number from 1 to 6that will indicate how much what is jj watts phone number should subtract from the original number to get a multiple of 7.
In other words, you will find the remainder of dividing the number by 7. For example, take the number :. I twisted my ankle what should i do minus 4, which ismust be a multiple of 7. So 2 and 9 must have the same reminder when divided by 7. The remainder is 2. Therefore, if a number n is a multiple of 7 i. Divits this procedure does, as explained above for most divisibility rules, is simply subtract little by little multiples of 7 mlutiply the original number until reaching a number that is small enough for us to remember whether it is a multiple of 7.
The digitx reason applies for all the remaining conversions:. Convert the divisor seven to the nines family by multiplying by seven. Start on the right. Multiply by 5, add the product to the next miltiply to digirs left. Set down that result on a line below that digit.
Repeat that method of multiplying the units digit by five and adding that product to the number of tens. Add the rigits to the next digit to the left.
Write down that result below the digit. Continue to the end. If the end result is zero or a multiple of seven, then yes, the number hoq divisible by seven. Otherwise, it is not. This follows the Vedic ideal, one-line notation. Pohlman—Mass method of divisibility by 7 The Pohlman—Mass method provides a quick solution that can determine if most integers digitts divisible by seven in three steps or less. Step A: If the integer is 1, or less, subtract twice the last digit from the number how is chlorophyll related to phytoplankton by the remaining digits.
If the result is a multiple of seven, then so is the original number and vice versa. For example:. Because 1, is divisible by tto, an interesting pattern develops for repeating sets of 1, 2, or 3 digits that form 6-digit numbers leading zeros are allowed in that all such numbers are divisible by seven.
For all of the above examples, subtracting the first three digits from the last three results in a multiple of seven. Notice that leading zeros are permitted to form a 6-digit pattern. Step B: If the integer is between 1, and one million, find a repeating pattern of 1, 2, or 3 digits that forms a 6-digit number that is close to the integer leading zeros are allowed and can help you visualize the pattern.
If the positive difference is less than 1, apply Yow A. This can be done by subtracting the first three digits from the last three digits. The fact thatis a multiple of fash can be used for digiys divisibility of integers larger than one million by reducing multippy integer to a 6-digit number that can be determined using Step B.
This can be done easily by adding digigs digits left of the first six to the last six and follow with Step A. Step C: If the integer is larger than one million, subtract the nearest multiple ofand then apply Step B. For even larger numbers, use larger sets such as digits ,, and so on. Then, break the integer into a smaller number that can be solved using Step B.
This allows adding and subtracting alternating sets of three digits figits determine divisibility by seven. Understanding these patterns allows you to quickly calculate divisibility of seven as seen in the following examples:.
Recurring numbers: 1, digirs, 2, 6, 4, 5 Positive sequence. Multiply the right most digit by the left most digit in the sequence and multiply the second right most digit by the second left most digit in the sequence and so on and so for. Next, compute the sum of all the values and take the modulus of 7. Example: What is the remainder when is divided by 7? That is, the divisibility of any number by seven can be tested by first separating the number into digit pairs, and then applying the algorithm on three digit pairs six digits.
When the number tp larger than six digits, fwst repeat the cycle on the next six digit group and then add the results. How to fix ash blonde highlights that look gray the algorithm until the result is a small number.
The original number is divisible by seven if and only if the number obtained using this algorithm is divisible by seven. This method is especially suitable for large numbers. Example 1: The number to be tested is First we separate the number into three digit pairs: 15, 75 and Example 2: The number to be tested is The method is based on the observation that leaves a remainder of 2 when divided by 7.
And since we are breaking the number into digit pairs we essentially have powers of If you are not comfortable with negative numbers, then use this sequence. Multiply the right most how to open a wii without a tri wing screwdriver of the number with the left most number in the sequence shown above and the second right most digit to the second left most digit of the number in the sequence.
The cycle goes on. Example: What is the remainder when is divided by 13? A number is divisible by diglts given divisor if it is divisible by the highest power of each of its prime factors.
For example, to determine divisibility by 36, check divisibility by bt and by 9. A table of prime factors may be useful. A composite divisor may also have a rule formed using the what does a respiratory nurse do procedure as for a prime divisor, given below, with the caveat that the manipulations involved may not introduce any factor which is present muultiply the divisor.
For instance, one cannot make a rule for 14 that involves multiplying the equation by 7.
Sep 03, · Every possible finite sequence of digits in pi is conjectured to come up eventually. This is equivalent to saying that pi is what is called 'normal' when written in any base. This means that if you go far enough there will be an equal proportion of each of the digits. So statistically a 1 and a 2 are just as likely to come up. Significant digits are certain digits that have significance or meaning and give more precise details about the value of the number. If in our opening scenario, I offered you $2,, the 2 in Sep 22, · The main computation was done using a new version of y-cruncher (version v). The verification was done using the same BBP program from before. Our previous computation of 10 trillion digits required days. This time we were able to achieve trillion digits in "only" 94 days.
October 17, The record has been improved to 10 trillion digits. This is a followup to our previous announcement of our computation of 5 trillion digits of Pi. This article details some of the methods that were used for the computation as well as the hardware and the full timeline of the computation.
Some of you knew this was coming It was just a matter of when. Although we were very quiet about this computation while it was running, anyone who followed my website or my XtremeSystems thread between February and May would have easily guessed that we were attempting a world record.
Here are the full computation statistics. As with all significant computations that are done using y-cruncher - A Multi-Threaded Pi Program , a screenshot and validation is included. Since the computation was done in multiple sessions, there is no single screenshot that captures the entire computation from start to finish. The screenshot provided here is simply the one that shows the greater portion of the computation. The main computation took 90 days on Shigeru Kondo's desktop.
The computer was dedicated for this task. Over the course of the computation, one error was detected and corrected via software ECC. Since the error was corrected, the final results are not affected. The computation error is believed to be caused by a hardware anomaly, or by hardware instability. If the digits were stored in an uncompressed ascii text file, the combined size of the decimal and hexadecimal digits would be 8. Validation File: Validation - Pi - 5,,,, After Fabrice Bellard's announcement of 2.
Shigeru Kondo and I wanted to see how much better we could do if we used some more powerful hardware. Both of us are hardware fanatics. And both of us especially Shigeru Kondo had some very powerful machines at our disposal.
So with that, we decided to see how far we could push the limits of personal computing using personally owned hardware. Unlike Fabrice Bellard's record which focused on efficiency and getting the most out of a small amount of hardware. Our computation focused more on getting the most performance and scalability from a LOT of hardware.
The main challenge for a computation of such a size, is that both software and hardware are pushed beyond their limits. For such a long computation and with so much hardware, failure is not just a probability. It is a given. There are simply too many components that can fail. Verification was done using two separate computers. Both of these computers had other tasks and were not dedicated to the computation.
Click here for a full list of computers that contributed to this computation. The program that was used for the main computation is y-cruncher v0. As of this writing, it also holds the world record for most digits computed for several other famous constants. There are several aspects of y-cruncher that set it apart from most other similar Pi-crunching programs:. This version was a very early private-beta for v0. It was slightly modified to display more detail on the progress of the computation.
All public versions display only a percentage. This program implements the digit-extraction algorithm for Pi using the BBP formulas. It's sole purpose was to verify the main computation.
Both programs are written by me Alexander J. Yee and are available for download from their respective pages. This webpage is the first article I have written that reveals significant details on the inner workings of y-cruncher Although the main computation was done on one computer. There were other computers that were involved. Below is a complete list detailing all the computers that contributed to this computation.
Run the BBP formulas to verify the main computation. This was my primary computer at the time. These runs were done using spare CPU time over the course of the main computation. During development, this was by far the most important computer I had as it was the only machine in my possession that could test memory-intensive code.
Note that there were at least 8 other computers that took part in the development and testing of y-cruncher. But aside from a couple of spectacular laptops, most of these other machines had less than impressive specs and are not worth mentioning. Chudnovsky's Formula was used with Binary Splitting to compute 4,,,, hexadecimal digits of Pi. To verify correctness of the hexadecimal digits, the BBP formulas were used to directly compute hexadecimal digits at various places including the 4,,,,th place.
The two primary verification runs consisted of running both formulas to compute 32 hexadecimal digits ending with the 4,,,,th. These were done simultaneously on two different computers - "Ushio" and "Nagisa". Both computations were started at the same time. The faster formula Bellard was assigned to the slower computer Ushio and the slower formula Plouffe was assigned to the faster computer Nagisa. One radix conversion was done to convert the digits from base 16 to base This produced 5,,,, decimal digits.
The radix conversion was verified using modular hash checks. The method of verification is similar if not identical to the method that Fabrice Bellard used to verify his record of 2.
Unlike most world record-sized computation, only one main computation was performed. To provide sufficient redundancy to ensure that the computed digits are correct, several error-checking steps were added to the computation.
As mentioned in the previous section, the hexadecimal digits were verified by directly computing the digits at various places and comparing them with the main computation. This combined with a Modular Hash Check on the final multiplication is sufficient to verify ALL the digits of the main computation. Depending on the where the error occurs, the last digits of a product may or may not be affected by an error occuring during the product.
This is especially true if the error occurs in the inverse transform or the final carryout of an FFT-based multiply. Large numbers that reside on disk as stored the same way, but with a file handle instead of a pointer to memory. All basic arithmetic uses bit integer-words. Nevertheless, this is not a major drawback because y-cruncher actually spends very little time doing arithmetic in the integer unit. Most of the run-time is spent executing vector SSE and stalling on memory and disk access.
For large computations, the bottleneck is either memory bandwidth or disk bandwidth depending on the number of hard drives and their configuration. Small products are done using the Basecase and Karatsuba multiplication algorithms. It also unnecessarily increases the sizes of the Twiddle Factor tables.
Despite the lack of hand-coded assembly, the bit version of the FFT achieves speeds comparable to that of prime The Hybrid NTT is a currently unpublished algorithm that was first conceived back in and then developed and implemented in It is able to achieve speeds comparable to that of floating-point FFT while requiring only a fraction of the memory and memory bandwidth.
The current implementation used by y-cruncher is multi-threaded and uses both integer and floating-point instructions. This has a side bonus of gaining additional benefit from HyperThreading Technology. The multiplication algorithms described here have been present in y-cruncher since version v0. However, they have since had some fairly significant modifications. The FFT was completely rewritten between v0. And the Hybrid NTT was re-tuned between v0. Disk multiplications are the gigantic multiplications that will not fit in memory.
These are done exclusively using Hybrid NTT. The threshold at which the program will switch between 3-step and 5-step is a carefully tuned value based on the 2TB RPM desktop hard drives that were available at the time when this part of the program that was written. Because of the massive amount of ram that was available 96 GB , only the largest multiplications 2 trillion x 2 trillion digits or more used the 5-step method.
In actuality, we had GB of ram at our disposal which is enough to completely eliminate the need for the 5-step method , but due to restrictions in hardware specifications, GB of ram would have run at a lower clock speed. With respect to the size of our computation 5 trillion digits and our hardware configuration 16 hard drives , it was found that GB of ram is near the point of diminishing return for memory quantity. Therefore, additional memory clock speed provided more benefit than additional memory quantity.
If we used only 48 GB of ram we could have run the memory at an even faster MHz. But 48 GB of ram was found to be less than optimal. Due to the memory intensive nature of the computation, the total amount of memory here is important - which seems to matter more than the speed of the CPU. Although GB is at the point of dimishing return, 48 GB is the exact opposite.
We estimate that clocking the CPU to 3. But it was found out later that, despite what EVGA officials stated, the motherboard is in fact capable of supporting up to 96 GB of ram. The disk multiplication described here has been present in y-cruncher since v0.
No particular optimizations are done. The implementations for Division and Square Root that are used in y-cruncher have been virtually untouched since v0.