"084 numbers as ""local rate"" or ""lo-call"" or 087 numbers as ""national rate"". A small number of landline providers and a smaller number of mobile providers have allowed 0870 (and a few landline providers have also allowed 0845) numbers to be used in inclusive minutes bundles. However, 0844 and 0871 numbers are never inclusive. From landlines, 084 and 087 numbers cost anything up to 20 pence per minute. From mobile phones, 084 and 087 numbers cost anything up to 45 pence per minute. The price includes a revenue-share ""premium"" or ""service charge"" of up to 5 pence per minute for"
"a number between 1 and 10 using a two-out-of-five code with the following numeric equivalence: 1, 2, 4, 7, 0. The numbers are summed together. For example, the number 6 is encoded NWWNN, with wide bars occupying the positions for 2 and 4 (2+4=6). In the case of NNWWN which is it is assigned to 0 for digits (+0), and 10 for the letter columns (+10 – +30). When encoding the (+10 to +30) Letters the equation needs a ""−1"" added so 'A' is WNNNW → 1 + 10 − 1 → 10 as shown in the table. The last"
"the full range of numbers is −9.999999×10 through 9.999999×10. The numbers −""b"" and ""b"" (here, −1×10 and 1×10) are the smallest (in magnitude) ""normal numbers""; non-zero numbers between these smallest numbers are called subnormal numbers. Some numbers may have several possible exponential format representations. For instance, if ""b""=10 and ""p""=7, −12.345 can be represented by −12345×10, −123450×10, and −1234500×10. However, for most operations, such as arithmetic operations, the result (value) does not depend on the representation of the inputs. For the decimal formats, any representation is valid, and the set of these representations is called a ""cohort"". When a result"
"10 values are valid (0000 through 1001). So adding 6 to the invalid entries results in the following: Thus the result of the subtraction is 1001 1001 0010 0101 (−925). To confirm the result, note that the first digit is 9, which means negative. This seems to be correct, since 357 − 432 should result in a negative number. The remaining nibbles are BCD, so 1001 0010 0101 is 925. The ten's complement of 925 is 1000 − 925 = 75, so the calculated answer is −75. If there are a different number of nibbles being added together (such as"
"digits are summed This method works for divisors that are factors of 10 − 1 = 9. Using 3 as an example, 3 divides 9 = 10 − 1. That means formula_2 (see modular arithmetic). The same for all the higher powers of 10: formula_3 They are all congruent to 1 modulo 3. Since two things that are congruent modulo 3 are either both divisible by 3 or both not, we can interchange values that are congruent modulo 3. So, in a number such as the following, we can replace all the powers of 10 by 1: which is exactly"
"10 = 0 (where the 2 is the tens digit of 24 from the sum for the final digit). This set of numbers ending ...404 may therefore contain one or more missing-digit sums. Similar reasoning can be applied to sums in which two, three and more digits are deleted from the original number. 14358846, 14400000, 15000000, 28758846, 28800000, 29358846, 29400000, 1107488889, 1107489042, 1111088889, 1111089042, 3277800000, 3281400000, 4388888889, 4388889042, 4392488889, 4392489042, 4500000000, 5607488889, 5607489042, 5611088889, 5611089042, 7777800000, 7781400000, 8888888889, 8888889042, 8892488889, 8892489042, 10000000000, 20000000000, 30000000000, 40000000000, 50000000000, 60000000000, 70000000000, 80000000000, 90000000000 Missing-digit sum Missing-digit sums are integer numbers that are equal"
"3 left to right. These weighted digits are added together. The check digit is the integer between 0 and 9 that makes the sum a multiple of 10. For instance, for the item with ISMN beginning 979-0-060-11561: As 85 mod 10 = 5, the check digit is 10 - 5 = 5 and the full number is 979-0-060-11561-5. For another example, Robert Fripp's collection of Guitar Craft scores has the ISMN 979-0-9016791-7-7. Given first 12 digits 979-0-9016791-7, the ISMN algorithm evaluates which implies that the check digit is indeed 7 (because 123+7=130=13x10). International Standard Music Number The International Standard Music"
"1, 0), (0, 8, 3, 8, 3, 0) Multiply the group containing the rightmost character (which is the FIRST group) by 2: Add up the individual digits: (6 + 4 + 0 + (1 + 4) + 6 + 2 + 0) + (0 + 8 + 3 + 8 + 3 + 0) = 45 Take the 10s modulus of the sum: 45 mod 10 = 5 Subtract from 10: 10 - 5 = 5 Take the 10s modulus of the result (this final step is important in the instance where the modulus of the sum is 0, as"
"Summation In mathematics, summation (denoted with an enlarged capital Greek sigma symbol formula_1) is the addition of a sequence of numbers; the result is their ""sum"" or ""total"". If numbers are added sequentially from left to right, any intermediate result is a partial sum, prefix sum, or running total of the summation. The numbers to be summed (called ""addends"", or sometimes ""summands"") may be integers, rational numbers, real numbers, or complex numbers. Besides numbers, other types of values can be added as well: vectors, matrices, polynomials and, in general, elements of any additive group (or even monoid). For finite sequences"
"numerical ratio of the first four integer numbers which derive respectively from the relations of string length: the eighth (1/2), the fifth (2/3) and the fourth (3/4). The sum of those numbers 1 + 2 + 3 + 4 = 10 was for Pythagoreans the perfect number, because it contained in itself ""the whole essential nature of numbers"". Werner Heisenberg has called this formulation of musical arithmetical as ""among the most powerful advances of human science"" because it enables the measurement of sound in space. Pythagorean tuning is a system of musical tuning in which the frequency ratios of all"
"is simply the ones digit of the sum of the cards, which is why it does not matter whether the 10, jack, queen, and king count as zero or ten. For example, a hand consisting of a 3 and a 2 has a value of five ""taem"". Likewise, a hand consisting of a 7 and an 8 has a value of five ""taem"" because the sum, fifteen, has a one's digit of five. A hand with a jack and a 5 also has a value of five ""taem"" because the sum, either five or fifteen, has a one's digit of"
"one, since the general solution may naturally have a simpler algorithm and cleaner design, and typically can take less time to solve in comparison with a particular problem. The sum of numbers sequentially from 1-99: This process, although not impossible to do in one's head, can prove to be difficult for most. However, the ability to generalize the problem exists, in this case by reordering the sequence to: In this form, the example can be solved by most without the use of a calculator. If one notices the problem's lowest and highest numbers (1 + 99) sum to 100, and"
"Sum-product number A sum-product number is an integer that in a given base is equal to the sum of its digits times the product of its digits. Or, to put it algebraically, given an integer ""n"" that is ""l"" digits long in base ""b"" (with ""d"" representing the ""x""th digit), if: formula_1 then ""n"" is a sum-product number in base ""b"". In base 10, the only sum-product numbers are 0, 1, 135, 144 . Thus, for example, 144 is a sum-product number because 1 + 4 + 4 = 9, and 1 × 4 × 4 = 16, and 9"
"that the sum of all the ten digits, each multiplied by its weight in ""ascending"" order from 1 to 10, is a multiple of 11. For this example: Formally, we can say: The two most common errors in handling an ISBN (e.g., typing or writing it) are a single altered digit or the transposition of adjacent digits. It can be proved that all possible valid <nowiki>ISBN-10</nowiki>'s have at least two digits different from each other. It can also be proved that there are no pairs of valid <nowiki>ISBN-10</nowiki>'s with eight identical digits and two transposed digits. (These are true only"
"9 and must be such that the sum of all the thirteen digits, each multiplied by its (integer) weight, alternating between 1 and 3, is a multiple of 10. Formally, using modular arithmetic, we can say: The calculation of an <nowiki>ISBN-13</nowiki> check digit begins with the first 12 digits of the thirteen-digit ISBN (thus excluding the check digit itself). Each digit, from left to right, is alternately multiplied by 1 or 3, then those products are summed modulo 10 to give a value ranging from 0 to 9. Subtracted from 10, that leaves a result from 1 to 10. A"
"dropping two, three and four digits, respectively. Here is the delete-2 sum, containing 12! / (2!(12 - 2)!) = 66 integers: In any base ""b"", there will be a set of delete-1 missing-digit sums with ""b""+1 digits and first digits in the range 1...""b""-1, followed by ""b"" zeroes. Consider base-2 and base-3: Accordingly, the eleven-digit numbers 10,000,000,000 through 90,000,000,000 are trivial missing-digit sums in base ten, because their sums take this form: And in hexadecimal or base sixteen, the seventeen-digit numbers 10,000,000,000,000,000 through F0,000,000,000,000,000 are missing-digit sums of the same form. Searching for delete-1 missing-digit sums is simplified when one"
"the sums adding to ""n"" + 1 in a different way, this time by the location of the first 2. Specifically, the first group consists of those sums that start with 2, the second group those that start 1 + 2, the third 1 + 1 + 2, and so on, until the last group, which consists of the single sum where only 1's are used. The number of sums in the first group is ""F""(""n""), ""F""(""n"" − 1) in the second group, and so on, with 1 sum in the last group. So the total number of sums is"
"that the Fibonacci numbers can be defined by a Diophantine equation, which led to his solving Hilbert's tenth problem. The Fibonacci numbers are also an example of a complete sequence. This means that every positive integer can be written as a sum of Fibonacci numbers, where any one number is used once at most. Moreover, every positive integer can be written in a unique way as the sum of ""one or more"" distinct Fibonacci numbers in such a way that the sum does not include any two consecutive Fibonacci numbers. This is known as Zeckendorf's theorem, and a sum of"
"Fibonacci numbers in popular culture The Fibonacci numbers are a sequence of integers, starting with 0, 1 and continuing 1, 2, 3, 5, 8, 13, ..., each new number being the sum of the previous two. The Fibonacci numbers, and in conjunction the golden ratio, are a popular theme in culture. They have been mentioned in novels, films, television shows, and songs. The numbers have also been used in the creation of music, visual art, and architecture. John Waskom postulated that stages of human development followed the Fibonacci sequence, and that the unfolding psychology of human life would ideally be"
"10 10 (ten) is an even natural number following 9 and preceding 11. Ten is the base of the decimal numeral system, by far the most common system of denoting numbers in both spoken and written language. The reason for the choice of ten is assumed to be that humans have ten fingers (digits). As is the case for any base in its system, ten is the first two-digit number in decimal and thus the lowest number where the position of a numeral affects its value. Any integer written in the decimal system can be multiplied by ten by adding"
"The following example is decimal, which simply means the base is 10. This is nothing other than converting to scientific notation. In detail: This is the true result, the exact sum of the operands. It will be rounded to 7 digits and then normalized if necessary. The final result is: Note that the low 3 digits of the second operand (654) are essentially lost. This is round-off error. In extreme cases, the sum of two non-zero numbers may be equal to one of them: Another problem of loss of significance occurs when two close numbers are subtracted. e=5; s=1.234571 and"
"A − C ≤ 9, both the first and third digits are guaranteed to be single digits.) Its reversal is 100 × ['] + 10 × ' + 1 × [']. The sum is thus 101 × ['] + 20 × ' + 101 × ['] = 101 × [' + '] + 20 × = 101 × [−1 + 10] + 180 = 1089. Multiplying the number 1089 by the integers from 1 to 9 produces a pattern: multipliers adding up to 10 give products that are the digit reversals of each other: Also note the patterns within each"
"is equal to its own missing-digit-sum in base ten. Now consider the set of seven-digit numbers ending in ...04. For these numbers, the final digit of the sum is equal to (0 x 1 + 4 x 6) modulo 10 = (0 + 24) mod 10 = 24 mod 10 = 4. This set may therefore contain one or more missing-digit sums. Next consider seven-digit numbers ending ...404. The penultimate (last-but-one) digit of the sum is equal to (2 + 4 x 2 + 0 x 4) modulo 10 = (2 + 8 + 0) mod 10 = 10 mod"
"a zero to the end (e.g. 855 × 10 = 8550). The Roman numeral for ten is X (which looks like two Vs [the Roman numeral for 5] put together); it is thought that the V for five is derived from an open hand (five digits displayed), and X for ten from both hands. Incidentally, the Chinese word numeral for ten, is also a cross: . The digit '1' followed by '0' is how the value of ""p"" is written in base ""p"". (E.g. 16 in hexadecimal is 10.) The SI prefix for 10 is ""deca-"". The meaning ""10"" is"
"Power of 10 In mathematics, a power of 10 is any of the integer powers of the number ten; in other words, ten multiplied by itself a certain number of times (when the power is a positive integer). By definition, the number one is a power (the zeroth power) of ten. The first few non-negative powers of ten are: In decimal notation the ""n""th power of ten is written as '1' followed by ""n"" zeroes. It can also be written as 10 or as 1E""n"" in E notation. See order of magnitude and orders of magnitude (numbers) for named powers"
"Aliquot sum In number theory, the aliquot sum ""s""(""n"") of a positive integer ""n"" is the sum of all proper divisors of ""n"", that is, all divisors of ""n"" other than ""n"" itself. It can be used to characterize the prime numbers, perfect numbers, deficient numbers, abundant numbers, and untouchable numbers, and to define the aliquot sequence of a number. For example, the proper divisors of 15 (that is, the positive divisors of 15 that are not equal to 15) are 1, 3 and 5, so the aliquot sum of 15 is 9 i.e. (1 + 3 + 5). The"
"an exhaustive search of [1,2916] one finds that for summing the cubes of digits base 10 there are happy numbers and eight different types of unhappy number: those that eventually reach formula_30 or formula_31, which perpetually produce themselves. those that eventually reach the loops: formula_32 formula_33, as well as those that alternate between formula_34 and formula_35 or between formula_36 and formula_37. All multiples of three end in 153. This fact can be proved by the exhaustive search up and noting that a number is a multiple of three if and only if the sum of digits is a multiple of"
"− see Dime (United States coin)#Denomination history and etymology. 10 10 (ten) is an even natural number following 9 and preceding 11. Ten is the base of the decimal numeral system, by far the most common system of denoting numbers in both spoken and written language. The reason for the choice of ten is assumed to be that humans have ten fingers (digits). As is the case for any base in its system, ten is the first two-digit number in decimal and thus the lowest number where the position of a numeral affects its value. Any integer written in the"
"Kempner series The Kempner series is a modification of the harmonic series, formed by omitting all terms whose denominator expressed in base 10 contains the digit 9. That is, it is the sum where the prime indicates that ""n"" takes only values whose decimal expansion has no nines. The series was first studied by A. J. Kempner in 1914. The series is counter-intuitive because, unlike the harmonic series, it converges. Kempner showed the sum of this series is less than 80. Baillie showed that, rounded to 20 decimals, the actual sum is 22.92067 66192 64150 34816 Heuristically, this series converges"
"a maximum of four digits each. When two numbers are added together, the result is called a ""sum"". The two numbers being added together are called ""addends"". Suppose you have two bags, one bag holding five apples and a second bag holding three apples. Grabbing a third, empty bag, move all the apples from the first and second bags into the third bag. The third bag now holds eight apples. This illustrates the combination of three apples and five apples is eight apples; or more generally: ""three plus five is eight"" or ""three plus five equals eight"" or ""eight is"
"be C1 C2 C3 C4 C5 C6 C7 C8 C9 C10 C11. (C1 is the first digit, C2 is the second digit, and so on) <br>""Step 1 "": Starting from the left, add all the Characters in the even position. S1 = C2 + C4 + C6 + C8 + C10 <br>""Step 2 "": Multiply the sum by 3 to get 3×S1 <br>""Step 3 "": Starting from the left.. add all the Characters in the odd position. S2 = C1 + C3 + C5 + C7 + C9 <br>""Step 4 "": Add the sum of step 2 to the sum"
"can deduce the 10-adic expansion of −1 thus an expansion which clearly is a ten's complement representation. In this notation, 10-adic expansions can be extended indefinitely to the left, in contrast to decimal expansions, which can be extended indefinitely to the right. Note that this is not the only way to write -adic numbers – for alternatives see the ""Notation"" section below. More formally, a 10-adic number can be defined as where each of the is a digit taken from the set {0, 1, … , 9} and the initial index may be positive, negative or 0, but must be"
"in all bases): When the convention formula_27 is used the following numbers are also PDDIs (as well as 0, in all bases): Perfect digit-to-digit invariant A perfect digit-to-digit invariant (PDDI) (also known as a Munchausen number) is a natural number that is equal to the sum of its digits each raised to the power of itself. In any base, there are only a finite number of Munchausen numbers. For example, in base 10 (decimal) there are only four: 0, 1, 3435 and 438579088. A number formula_1 in base formula_2 with digits formula_3 (where each formula_4 is an integer such that"