Multiplication table

“ In the last analysis, division is only the obverse side of multiplication, and any one who has not learned his multiplication table is reduced to some such expedient as that of the Egyptian. Indeed, whenever we pass beyond the range of our memorized multiplication table-which for most of us ends with the twelves—the experimental character of the trial multiplication through which division is finally effected does not so greatly differ from the experimental efforts which the Egyptian was obliged to apply to smaller numbers. ”

“ In Dodson's Calculator we find a table of multiplication extending as far as 10 times 1000. [2] In 1775, a still more extensive table was published to 10 times 10,000. The Board of Longitude subsequently employed the late Dr Hutton to calculate and print various numerical tables, and among others, a multiplication table extending as far as 100 times 1000; tables of the squares of numbers, as far as 25,400; tables of cubes, and of the first ten powers of numbers, as far as 100. ”

“ For example, a table of logarithms is a table of single entry, the numbers being the arguments and the logarithms the tabular results; an ordinary multiplication table is a table of double entry, giving xy as tabular result for x and y as arguments. The intrinsic value of a table may be estimated by the actual amount of time saved by consulting it; for example, a table of square roots to ten decimals is more Valuable than a table of squares, as the extraction of the root would occupy more time than the multiplication of the number by itself. ”

“ For these reasons it cannot consistently compromise, any more than a mathematician could change the multiplication table to meet the view of some one who should deny that five times five are twenty-five. ”

“ Additions and subtractions could be performed on the abacus quite easily, but in multiplication the abacus could be used only for adding the particular products, and in division for performing the subtractions occurring in the process. Doubtless at this point recourse was made to mental operations and to the multiplication table. Possibly finger-multiplication may also have been used. But the multiplication of large numbers must, by either method, have been beyond the power of the ordinary arithmetician. ”

“ If it is true, it does not need to be inspired. If it is true, it makes no difference whether it was written by a man or a god. The multiplication table is just as useful, just as true as though God had arranged the figures himself. ”

“ When we employ the multiplication-table in multiplying numbers of several places, and so use the results of old operations of counting instead of performing the whole of each operation anew; when we consult our table of logarithms, replacing and saving thus new calculations by old ones already performed; when we employ determinants instead of always beginning afresh the solution of a system of equations ”

“ Teach the Roman notation to C; how to tell the time of day; how to make change with money; and how to solve easy exercises in pt., qt., pk., and bu.,—gi., pt., qt., and gal.—and in., ft., and yd.
17. The teacher, using a pointer, should drill the pupils thoroughly on the following table. (Try to acquire speed and correctness) .
These constitute the multiplication table with the duplicate combinations cut out, leaving but 36 products to learn in the entire field of the common multiplication table. ”

“ I had been to school most all the time and could spell and read and write just a little, and could say the multiplication table up to six times seven is thirty-five, and I don’t reckon I could ever get any further than that if I was to live forever. I don’t take no stock in mathematics, anyway. ”

“ It will be noticed that the rods only give the multiples of the number which is to be multiplied, or of the divisor when they are used for division, and it is evident that they would be of little use to any one who knew the multiplication table as far as 9×9. ”

“ How many of the most important affairs of life can be demonstrated by means of the multiplication table? It would be safe to say not one in ten. Again, mathematics frequently deal with purely ideal figures, which never did or never can exist. ”

“ It requires a multiplication-table for all numbers from one to sixteen. The mental labor of calculating would, therefore, be increased. The number of symbols required would not be quite so large, but the advantage from this source would be slight. It may be noted, in passing, that in the Hindoo notation, the smaller the radix the greater is the number of symbols required to express any number, but the easier the mental work of calculating. ”

“ The diagram was present in my mind from early childhood; I remember that I learnt the multiplication table by reference to it at the age of seven or eight. I need hardly say that the impression is not that of perfectly straight lines, I have therefore used no ruler in drawing it."
J.S. "The figures are about a quarter of an inch in length, and in ordinary type. They are black on a white ground. The numeral 200 generally takes the place of 100 and obliterates it. There is no light or shade, and the picture is invariable." ”

“ In the same way as with multiplication, we can insert a number of coding lines in between the dividend line and the first quotient line.
Both multiplication and division are carried out in the same unit of the machine, the multiply-divide unit. The machine first constructs a table of the multiples of the multiplicand or divisor: 1 times, 2 times, 3 times, ···, 9 times. In multiplication this table is then used by selecting multiples according to the digits of the multiplier one after another. ”

“ If we take several different units, and write down their successive multiples in parallel columns, preceded by the number-series, we obtain a multiple-table such as the following:—
It is to be considered that each column may extend downwards indefinitely.
37. Successive Multiplication.—In multiplication by repetition the unit is itself usually a multiple of some other unit, i.e. it is a product which is taken as a new unit. When this new unit has been multiplied by a number, we can again take the product as a unit for the purpose of another multiplication ”

“ Nor is it true that every one who has once been brought to see it is equally sure of it. A boy who has just learned the multiplication table is pretty sure that twice two are four, but is often extremely doubtful whether or not seven times nine are sixty-three. ”

“ It is, that we may sometimes know a general proposition in cases where we do not know a single instance of it. Take such a case as the following: We know that any two numbers can be multiplied together, and will give a third called their product. We know that all pairs of integers the product of which is less than 100 have been actually multiplied together, and the value of the product recorded in the multiplication table. ”

“ ‘I’m sure I’m not Ada,’ she said, ‘for her hair goes in such long ringlets, and mine doesn’t go in ringlets at all; and I’m sure I can’t be Mabel, for I know all sorts of things, and she, oh! she knows such a very little! Besides, she’s she, and I’m I, and—oh dear, how puzzling it all is! I’ll try if I know all the things I used to know. Let me see: four times five is twelve, and four times six is thirteen, and four times seven is—oh dear! I shall never get to twenty at that rate! However, the Multiplication Table doesn’t signify: let’s try Geography. ”

“ The way to use the Multiplication table. loke qwere þat figure sittes in 23 þe lyft side of þe triangle, & loke qwere þe diget sittes in þe neþer most rewe of þe triangle. & go fro hym vpwarde in þe same rewe, þe quych rewe gose vpwarde til þou come agaynes þe oþer digette þat sittes in þe lyft side of þe triangle. And þat nounbre, þe quych þou leaf 155 b. fyn*des þere is þe nounbre þat comes of the multiplicacioɳ of þe 2 digittes, as yf þou wold wete qwat is 2 tymes 2. loke quere sittes 2 in þe lyft side in þe first rewe, he sittes next 1 in þe lyft side al on hye, as þou may se ”

“ Why should there have been four original multiplication tables? One is enough, and if anybody has got any use for it he can copy that one. The very fact that we have got four gospels shows that it is not an inspired book. ”

“ Knowledge of square root can be regarded somewhat as a separate tool to be sharpened, polished, and used by itself, whereas knowledge of the multiplication tables cannot. Yet even square root is probably best made more closely a part of the total ability, being taught as a special case of dividing where divisor is to be the same as quotient, the process being one of estimating and correcting. ”

“ You subtract thing from ten; there remain ten less thing. Multiply the one by the other, it is ten things less a square. This is the product of the multiplication of one of the two parts by the other. At present you divide this by the difference between the two parts, which is ten less two things. The quotient of this division is, according to the statement, five and a fourth. If, therefore, you muliply five and one-fourth by ten less two things, the product must be equal to the above amount, obtained by multiplication, namely, ten things less one square. ”

“ Within them is embraced the whole of computation; but because every number, every aggregation of units, is also a new unit capable of being added or subtracted, there are also the operations of multiplication and division, which consists in one case of the addition of several equal numbers together, and in the other, of the subtraction of several equal numbers from a greater until that is exhausted. ”

“ It is a useful exercise to multiply and divide numbers represented in other scales of notation than the common or decimal one. The rules are in all respects the same for all systems, the number carried being always the radix of the system. Thus, in the quinary system we carry fives instead of tens. I now give an example of multiplication and division: [Pg 170]
Another way of turning a number from one scale into another is as follows: Multiply the first digit by the old radix in the new scale, and add the next digit ”

“ I want to know the sum of five seventeens, or I ask the following question: There are five heaps of pebbles, and seventeen pebbles in each heap; how many are there in all? Write five seventeens in a column, and make the addition, which gives 85. In this case 85 is called the product of 5 and 17, and the process of finding the product is called multiplication, which gives nothing more than the addition of a number of the same quantities. Here 17 is called the multiplicand, and 5 is called the multiplier. ”

“ As there is 2⁄5 in the denominator, we must take the part of which 2⁄5 will be a unit. That part is larger than a unit; it is 21⁄2 units; 21⁄2 is that of which 2⁄5 is 1. The above fraction then directs us to repeat 21⁄2 units 31⁄4 times. By extending our word ‘multiplication’ to the taking of a part of a time, all multiplications are also divisions, and all divisions multiplications, and all the terms connected with either are subject to be applied to the results of the other.
If 21⁄3 yards cost 31⁄2 shillings, how much does one yard cost? ”

“ To take any quantity a number of times, it will be enough to take every one of its parts the same number of times. Thus, a sack of corn will be increased fifty-fold, if each bushel which it contains be replaced by 50 bushels. A country will be doubled by doubling every acre of land, or every county, which it contains. Simple as this may appear, it is necessary to state it, because it is one of the principles on which the rule of multiplication depends. ”

“ Our present method was known specifically as Italian division, and was not introduced until the beginning of the last century.
One writer on arithmetic, a pious monk, furnishes a good illustration of mediæval logic. He is embarrassed by the usage and meaning of the term "multiplication" in the case of fractions in which the product is less than the multiplicand, and he proposes the question, "Whether the multiplication of fractions is an increasing process?" ”

“ Multiply the numerator by the whole number, and let the denominator remain.
117. If the denominator of the fraction be divisible by the whole number, the rule may be stated thus: Divide the denominator of the fraction by the whole number, and let the numerator remain. For example, multiply 7/36 by 6. This (116) is 42/36, which, since the numerator and denominator are now divisible by 6, is (108) the same as 7/6. It is plain that 7/6 is made from 7/36 in the manner stated in the rule.
118. Multiplication has been defined to be the taking as many of one number as there are units in another. ”

“ You say, therefore, thing multiplied by thing is a square positive; and ten by thing is ten things positive; and minus ten by thing is ten things negative. You now remove the positive by the negative, then there only remains a square. Minus ten multiplied by ten is a hundred, to be subtracted from the square. This, therefore, altogether, is a square less a hundred dirhems.
(19) Whenever a positive and a negative factor concur in a multiplication, such as thing positive and minus thing, the last multiplication gives always the negative product. ”

“ After thousandths, come ten thousandths, hundred thousandths, mil- Konths, &c. but for ordinary uses we seldom come down to so small a fraction. Multiplication is performed as if there were no comma ; but in the sum total, as many figures must be cut off at the right hand of the comma, as are cut off in both the multiplier and sum multiplied. ”

“ The combinations to which algebraical subtraction give rise, are analogous to the preceding. Let us pass on to multiplication. When two numbers to be multiplied are of the same sign, the result is positive; if the signs are different, the product must be negative. In order that the machine may act conformably to this law, we have but to conceive that on the column containing the product of the two given numbers, the digit which indicates the sign of that product, has been formed by the mutual addition of the two digits that respectively indicated the signs of the two given numbers ”