Transcriber’s Notes:
Underscores “_” before and after a word or phrase indicate _italics_
in the original text.
Equal signs “=” before and after a word or phrase indicate =bold=
in the original text.
Carat symbol “^” designates a superscript.
Small capitals have been converted to SOLID capitals.
Illustrations have been moved so they do not break up paragraphs.
Old or antiquated spellings have been preserved.
Typographical and punctuation errors have been silently corrected.
ELEMENTS OF ARITHMETIC.
BY AUGUSTUS DE MORGAN,
OF TRINITY COLLEGE, CAMBRIDGE;
FELLOW OF THE ROYAL ASTRONOMICAL SOCIETY,
AND OF THE CAMBRIDGE PHILOSOPHICAL SOCIETY;
PROFESSOR OF MATHEMATICS IN UNIVERSITY COLLEGE, LONDON.
“Hominis studiosi est intelligere, quas utilitates
proprie afferat arithmetica his, qui solidam et
perfectam doctrinam in cæteris philosophiæ partibus
explicant. Quod enim vulgo dicunt, principium esse
dimidium totius, id vel maxime in philosophiæ partibus
conspicitur.”--MELANCTHON.
“Ce n’est point par la routine qu’on e’instruit, c’est par
sa propre réflexion; et il est essentiel de contracter
l’habitude de se rendre raison de ce qu’on fait: cette
habitude s’acquiert plus facilement qu’on ne pense; et une
fois acquise, elle ne se perd plus.”--CONDILLAC.
_SEVENTEENTH THOUSAND._
LONDON:
WALTON AND MABERLY,
UPPER GOWER STREET, AND IVY LANE, PATERNOSTER ROW.
M.DCCC.LVIII.
LONDON:
PRINTED BY J. WERTHEIMER AND CO.,
CIRCUS-PLACE, FINSBURY-CIRCUS.
PREFACE.
The preceding editions of this work were published in 1830, 1832,
1835, and 1840. This fifth edition differs from the three preceding,
as to the body of the work, in nothing which need prevent the four,
or any two of them, from being used together in a class. But it is
considerably augmented by the addition of eleven new Appendixes,[1]
relating to matters on which it is most desirable that the advanced
student should possess information. The first Appendix, on
_Computation_, and the sixth, on _Decimal Money_, should be read and
practised by every student with as much attention as any part of the
work. The mastery of the rules for instantaneous conversion of the
usual fractions of a pound sterling into decimal fractions, gives the
possessor the greater part of the advantage which he would derive from
the introduction of a decimal coinage.
At the time when this work was first published, the importance
of establishing arithmetic in the young mind upon reason and
demonstration, was not admitted by many. The case is now altered:
schools exist in which rational arithmetic is taught, and mere rules
are made to do no more than their proper duty. There is no necessity
to advocate a change which is actually in progress, as the works which
are published every day sufficiently shew. And my principal reason for
alluding to the subject here, is merely to warn those who want nothing
but routine, that this is not the book for their purpose.
A. DE MORGAN.
_London, May 1, 1846._
[1] Some separate copies of these Appendixes are printed, for those who
may desire to add them to the former editions.
TABLE OF CONTENTS.
BOOK I.
SECTION PAGE
I. Numeration 1
II. Addition and Subtraction 14
III. Multiplication 24
IV. Division 34
V. Fractions 51
VI. Decimal Fractions 65
VII. Square Root 89
VIII. Proportion 100
IX. Permutations and Combinations 118
BOOK II.
I. Weights and Measures, &c. 124
II. Rule of Three 144
III. Interest, &c. 150
APPENDIX.
I. On the mode of computing 161
II. On verification by casting out nines and elevens 166
III. On scales of notation 168
IV. On the definition of fractions 171
V. On characteristics 174
VI. On decimal money 176
VII. On the main principle of book-keeping 180
VIII. On the reduction of fractions to others of nearly
equal value 190
IX. On some general properties of numbers 193
X. On combinations 201
XI. On Horner’s method of solving equations 210
XII. Rules for the application of arithmetic to geometry 217
ELEMENTS OF ARITHMETIC.
BOOK I.
PRINCIPLES OF ARITHMETIC.
SECTION I.
NUMERATION.
1. Imagine a multitude of objects of the same kind assembled together;
for example, a company of horsemen. One of the first things that must
strike a spectator, although unused to counting, is, that to each man
there is a horse. Now, though men and horses are things perfectly
unlike, yet, because there is one of the first kind to every one of the
second, one man to every horse, a new notion will be formed in the mind
of the observer, which we express in words by saying that there is the
same _number_ of men as of horses. A savage, who had no other way of
counting, might remember this number by taking a pebble for each man.
Out of a method as rude as this has sprung our system of calculation,
by the steps which are pointed out in the following articles. Suppose
that there are two companies of horsemen, and a person wishes to
know in which of them is the greater number, and also to be able to
recollect how many there are in each.
2. Suppose that while the first company passes by, he drops a pebble
into a basket for each man whom he sees. There is no connexion between
the pebbles and the horsemen but this, that for every horseman there
is a pebble; that is, in common language, the _number_ of pebbles and
of horsemen is the same. Suppose that while the second company passes,
he drops a pebble for each man into a second basket: he will then have
two baskets of pebbles, by which he will be able to convey to any
other person a notion of how many horsemen there were in each company.
When he wishes to know which company was the larger, or contained most
horsemen, he will take a pebble out of each basket, and put them aside.
He will go on doing this as often as he can, that is, until one of the
baskets is emptied. Then, if he also find the other basket empty, he
says that both companies contained the same number of horsemen; if the
second basket still contain some pebbles, he can tell by them how many
more were in the second than in the first.
3. In this way a savage could keep an account of any numbers in which
he was interested. He could thus register his children, his cattle,
or the number of summers and winters which he had seen, by means
of pebbles, or any other small objects which could be got in large
numbers. Something of this sort is the practice of savage nations at
this day, and it has in some places lasted even after the invention
of better methods of reckoning. At Rome, in the time of the republic,
the prætor, one of the magistrates, used to go every year in great
pomp, and drive a nail into the door of the temple of Jupiter; a way of
remembering the number of years which the city had been built, which
probably took its rise before the introduction of writing.
4. In process of time, names would be given to those collections of
pebbles which are met with most frequently. But as long as small
numbers only were required, the most convenient way of reckoning them
would be by means of the fingers. Any person could make with his
two hands the little calculations which would be necessary for his
purposes, and would name all the different collections of the fingers.
He would thus get words in his own language answering to one, two,
three, four, five, six, seven, eight, nine, and ten. As his wants
increased, he would find it necessary to give names to larger numbers;
but here he would be stopped by the immense quantity of words which
he must have, in order to express all the numbers which he would be
obliged to make use of. He must, then, after giving a separate name
to a few of the first numbers, manage to express all other numbers by
means of those names.
5. I now shew how this has been done in our own language. The English
names of numbers have been formed from the Saxon: and in the following
table each number after ten is written down in one column, while
another shews its connexion with those which have preceded it.
One eleven ten and one[2]
two twelve ten and two
three thirteen ten and three
four fourteen ten and four
five fifteen ten and five
six sixteen ten and six
seven seventeen ten and seven
eight eighteen ten and eight
nine nineteen ten and nine
ten twenty two tens
twenty-one two tens and one fifty five tens
twenty-two two tens and two sixty six tens
&c. &c. &c. &c. seventy seven tens
thirty three tens eighty eight tens
&c. &c. ninety nine tens
forty four tens a hundred ten tens
&c. &c.
a hundred and one ten tens and one
&c. &c.
a thousand ten hundreds
ten thousand
a hundred thousand
a million ten hundred thousand
or one thousand thousand
ten millions
a hundred millions
&c.
[2] It has been supposed that _eleven_ and _twelve_ are derived
from the Saxon for _one left_ and _two left_ (meaning, after ten is
removed); but there seems better reason to think that _leven_ is a word
meaning ten, and connected with _decem_.
6. Words, written down in ordinary language, would very soon be too
long for such continual repetition as takes place in calculation. Short
signs would then be substituted for words; but it would be impossible
to have a distinct sign for every number: so that when some few signs
had been chosen, it would be convenient to invent others for the rest
out of those already made. The signs which we use areas follow:
0 1 2 3 4 5 6 7 8 9
nought one two three four five six seven eight nine
I now proceed to explain the way in which these signs are made to
represent other numbers.
7. Suppose a man first to hold up one finger, then two, and so on,
until he has held up every finger, and suppose a number of men to do
the same thing. It is plain that we may thus distinguish one number
from another, by causing two different sets of persons to hold up each
a certain number of fingers, and that we may do this in many different
ways. For example, the number fifteen might be indicated either by
fifteen men each holding up one finger, or by four men each holding up
two fingers and a fifth holding up seven, and so on. The question is,
of all these contrivances for expressing the number, which is the most
convenient? In the choice which is made for this purpose consists what
is called the method of _numeration_.
8. I have used the foregoing explanation because it is very probable
that our system of numeration, and almost every other which is used
in the world, sprung from the practice of reckoning on the fingers,
which children usually follow when first they begin to count. The
method which I have described is the rudest possible; but, by a little
alteration, a system may be formed which will enable us to express
enormous numbers with great ease.
9. Suppose that you are going to count some large number, for example,
to measure a number of yards of cloth. Opposite to yourself suppose a
man to be placed, who keeps his eye upon you, and holds up a finger for
every yard which he sees you measure. When ten yards have been measured
he will have held up ten fingers, and will not be able to count any
further unless he begin again, holding up one finger at the eleventh
yard, two at the twelfth, and so on. But to know how many have been
counted, you must know, not only how many fingers he holds up, but also
how many times he has begun again. You may keep this in view by placing
another man on the right of the former, who directs his eye towards his
companion, and holds up one finger the moment he perceives him ready
to begin again, that is, as soon as ten yards have been measured. Each
finger of the first man stands only for one yard, but each finger of
the second stands for as many as all the fingers of the first together,
that is, for ten. In this way a hundred may be counted, because the
first may now reckon his ten fingers once for each finger of the second
man, that is, ten times in all, and ten tens is one hundred (5).[3]
Now place a third man at the right of the second, who shall hold up
a finger whenever he perceives the second ready to begin again. One
finger of the third man counts as many as all the ten fingers of the
second, that is, counts one hundred. In this way we may proceed until
the third has all his fingers extended, which will signify that ten
hundred or one thousand have been counted (5). A fourth man would
enable us to count as far as ten thousand, a fifth as far as one
hundred thousand, a sixth as far as a million, and so on.
[3] The references are to the preceding articles.
10. Each new person placed himself towards your left in the rank
opposite to you. Now rule columns as in the next page, and to the right
of them all place in words the number which you wish to represent;
in the first column on the right, place the number of fingers which
the first man will be holding up when that number of yards has been
measured. In the next column, place the fingers which the second man
will then be holding up; and so on.
|7th.|6th.|5th.|4th.|3rd.|2nd.|1st.|
I.| | | | | | 5 | 7 | fifty-seven.
II.| | | | | 1 | 0 | 4 | one hundred and four.
III.| | | | | 1 | 1 | 0 | one hundred and ten.
IV.| | | | 2 | 3 | 4 | 8 | two thousand three hundred
| | | | | | | | and forty-eight.
V.| | | 1 | 5 | 9 | 0 | 6 | fifteen thousand nine
| | | | | | | | hundred and six.
VI.| | 1 | 8 | 7 | 0 | 0 | 4 | one hundred and
| | | | | | | | eighty-seven thousand
| | | | | | | | and four.
VII.| 3 | 6 | 9 | 7 | 2 | 8 | 5 | three million, six hundred
| | | | | | | | and ninety-seven
| | | | | | | | thousand, two hundred and
| | | | | | | | eighty-five.
11. In I. the number fifty-seven is expressed. This means (5) five tens
and seven. The first has therefore counted all his fingers five times,
and has counted seven fingers more. This is shewn by five fingers of
the second man being held up, and seven of the first. In II. the number
one hundred and four is represented. This number is (5) ten tens and
four. The second person has therefore just reckoned all his fingers
once, which is denoted by the third person holding up one finger;
but he has not yet begun again, because he does not hold up a finger
until the first has counted ten, of which ten only four are completed.
When all the last-mentioned ten have been counted, he then holds up
one finger, and the first being ready to begin again, has no fingers
extended, and the number obtained is eleven tens, or ten tens and one
ten, or one hundred and ten. This is the case in III. You will now find
no difficulty with the other numbers in the table.
12. In all these numbers a figure in the first column stands for only
as many yards as are written under that figure in (6). A figure in
the second column stands, not for as many yards, but for as many tens
of yards; a figure in the third column stands for as many hundreds of
yards; in the fourth column for as many thousands of yards; and so on:
that is, if we suppose a figure to move from any column to the one on
its left, it stands for ten times as many yards as before. Recollect
this, and you may cease to draw the lines between the columns, because
each figure will be sufficiently well known by the _place_ in which it
is; that is, by the number of figures which come upon the right hand of
it.
13. It is important to recollect that this way of writing numbers,
which has become so familiar as to seem the _natural_ method, is not
more natural than any other. For example, we might agree to signify one
ten by the figure of one with an accent, thus, 1′; twenty or two tens
by 2′; and so on: one hundred or ten tens by 1″; two hundred by 2″; one
thousand by 1‴; and so on: putting Roman figures for accents when they
become too many to write with convenience. The fourth number in the
table would then be written 2‴ 3′ 4′ 8, which might also be expressed
by 8 4′ 3″ 2‴, 4′ 8 3″ 2‴; or the order of the figures might be changed
in any way, because their meaning depends upon the accents which are
attached to them, and not upon the place in which they stand. Hence,
a cipher would never be necessary; for 104 would be distinguished
from 14 by writing for the first 1″ 4, and for the second 1′ 4. The
common method is preferred, not because it is more exact than this, but
because it is more simple.
14. The distinction between our method of numeration and that of the
ancients, is in the meaning of each figure depending partly upon the
place in which it stands. Thus, in 44444 each four stands for four of
_something_; but in the first column on the right it signifies only
four of the pebbles which are counted; in the second, it means four
collections of ten pebbles each; in the third, four of one hundred
each; and so on.
15. The things measured in (11) were yards of cloth. In this case one
yard of cloth is called the _unit_. The first figure on the right is
said to be in the _units’ place_, because it only stands for so many
units as are in the number that is written under it in (6). The second
figure is said to be in the _tens’_ place, because it stands for a
number of tens of units. The third, fourth, and fifth figures are in
the places of the _hundreds_, _thousands_, and _tens of thousands_, for
a similar reason.
16. If the quantity measured had been acres of land, an acre of land
would have been called the _unit_, for the unit is _one_ of the things
which are measured. Quantities are of two sorts; those which contain an
exact number of units, as 47 yards, and those which do not, as 47 yards
and a half. Of these, for the present, we only consider the first.
17. In most parts of arithmetic, all quantities must have the same
unit. You cannot say that 2 yards and 3 feet make 5 _yards_ or 5
_feet_, because 2 and 3 make 5; yet you may say that 2 _yards_ and 3
_yards_ make 5 _yards_, and that 2 _feet_ and 3 _feet_ make 5 _feet_.
It would be absurd to try to measure a quantity of one kind with a unit
which is a quantity of another kind; for example, to attempt to tell
how many yards there are in a gallon, or how many bushels of corn there
are in a barrel of wine.
18. All things which are true of some numbers of one unit are true of
the same numbers of any other unit. Thus, 15 pebbles and 7 pebbles
together make 22 pebbles; 15 acres and 7 acres together make 22 acres,
and so on. From this we come to say that 15 and 7 make 22, meaning that
15 things of the same kind, and 7 more of the same kind as the first,
together make 22 of that kind, whether the kind mentioned be pebbles,
horsemen, acres of land, or any other. For these it is but necessary to
say, once for all, that 15 and 7 make 22. Therefore, in future, on this
part of the subject I shall cease to talk of any particular units, such
as pebbles or acres, and speak of numbers only. A number, considered
without intending to allude to any particular things, is called an
_abstract_ number: and it then merely signifies repetitions of a unit,
or the _number of times_ a unit is repeated.
19. I will now repeat the principal things which have been mentioned in
this chapter.
I. Ten signs are used, one to stand for nothing, the rest for the first
nine numbers. They are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. The first of these
is called a _cipher_.
II. Higher numbers have not signs for themselves, but are signified
by placing the signs already mentioned by the side of each other, and
agreeing that the first figure on the right hand shall keep the value
which it has when it stands alone; that the second on the right hand
shall mean ten times as many as it does when it stands alone; that the
third figure shall mean one hundred times as many as it does when it
stands alone; the fourth, one thousand times as many; and so on.
III. The right hand figure is said to be in the _units’ place_, the
next to that in the _tens’ place_, the third in the _hundreds’ place_,
and so on.
IV. When a number is itself an exact number of tens, hundreds, or
thousands, &c., as many ciphers must be placed on the right of it as
will bring the number into the place which is intended for it. The
following are examples:
Fifty, or five tens, 50: seven hundred, 700.
Five hundred and twenty-eight thousand, 528000.
If it were not for the ciphers, these numbers would be mistaken for 5,
7, and 528.
V. A cipher in the middle of a number becomes necessary when any one of
the denominations, units, tens, &c. is wanting. Thus, twenty thousand
and six is 20006, two hundred and six is 206. Ciphers might be placed
at the beginning of a number, but they would have no meaning. Thus 026
is the same as 26, since the cipher merely shews that there are no
hundreds, which is evident from the number itself.
20. If we take out of a number, as 16785, any of those figures which
come together, as 67, and ask, what does this sixty-seven mean? of what
is it sixty-seven? the answer is, sixty-seven of the same collections
as the 7, when it was in the number; that is, 67 hundreds. For the 6
is 6 thousands, or 6 ten hundreds, or sixty hundreds; which, with the
7, or 7 hundreds, is 67 hundreds: similarly, the 678 is 678 tens. This
number may then be expressed either as
1 ten thousand 6 thousands 7 hundreds 8 tens and 5;
or 16 thousands 78 tens and 5; or 1 ten thousand 678 tens and 5;
or 167 hundreds 8 tens and 5; or 1678 tens and 5, and so on.
21. EXERCISES.
I. Write down the signs for--four hundred and seventy-six; two thousand
and ninety-seven; sixty-four thousand three hundred and fifty; two
millions seven hundred and four; five hundred and seventy-eight
millions of millions.
II. Write at full length 53, 1805, 1830, 66707, 180917324, 66713721,
90976390, 25000000.
III. What alteration takes place in a number made up entirely of nines,
such as 99999, by adding one to it?
IV. Shew that a number which has five figures in it must be greater
than one which has four, though the first have none but small figures
in it, and the second none but large ones. For example, that 10111 is
greater than 9879.
22. You now see that the convenience of our method of numeration arises
from a few simple signs being made to change their value as they
change the column in which they are placed. The same advantage arises
from counting in a similar way all the articles which are used in
every-day life. For example, we count money by dividing it into pounds,
shillings, and pence, of which a shilling is 12 pence, and a pound 20
shillings, or 240 pence. We write a number of pounds, shillings, and
pence in three columns, generally placing points between the columns.
Thus, 263 pence would not be written as 263, but as £1. 1. 11, where £
shews that the 1 in the first column is a pound. Here is a _system of
numeration_ in which a number in the second column on the right means
12 times as much as the same number in the first; and one in the third
column is twenty times as great as the same in the second, or 240 times
as great as the same in the first. In each of the tables of measures
which you will hereafter meet with, you will see a separate system of
numeration, but the methods of calculation for all will be the same.
23. In order to make the language of arithmetic shorter, some other
signs are used. They are as follow:
I. 15 + 38 means that 38 is to be added to 15, and is the same thing
as 53. This is the _sum_ of 15 and 38, and is read fifteen _plus_
thirty-eight (_plus_ is the Latin for _more_).
II. 64-12 means that 12 is to be taken away from 64, and is the same
thing as 52. This is the _difference_ of 64 and 12, and is read
sixty-four _minus_ twelve (_minus_ is the Latin for _less_).
III. 9 × 8 means that 8 is to be taken 9 times, and is the same thing
as 72. This is the _product_ of 9 and 8, and is read nine _into_ eight.
IV. 108/6 means that 108 is to be divided by 6, or that you must find
out how many sixes there are in 108; and is the same thing as 18. This
is the _quotient_ of 108 and 6; and is read a hundred and eight _by_
six.
V. When two numbers, or collections of numbers, with the foregoing
signs, are the same, the sign = is put between them. Thus, that 7
and 5 make 12, is written in this way, 7 + 5 = 12. This is called an
_equation_, and is read, seven _plus_ five _equals_ twelve. It is plain
that we may construct as many equations as we please. Thus:
12
7 + 9 - 3 = 12 + 1; --- - 1 + 3 × 2 = 11,
2
and so on.
24. It often becomes necessary to speak of something which is true not
of any one number only, but of all numbers. For example, take 10 and 7;
their sum[4] is 17, their difference is 3. If this sum and difference
be added together, we get 20, which is twice the greater of the two
numbers first chosen. If from 17 we take 3, we get 14, which is twice
the less of the two numbers. The same thing will be found to hold good
of any two numbers, which gives this general proposition,--If the sum
and difference of two numbers be added together, the result is twice
the greater of the two; if the difference be taken from the sum, the
result is twice the lesser of the two. If, then, we take _any_ numbers,
and call them the first number and the second number, and let the first
number be the greater; we have
[4] Any little computations which occur in the rest of this section may
be made on the fingers, or with counters.
(1st No. + 2d No.) + (1st No. - 2d No.) = twice 1st No.
(1st No. + 2d No.) - (1st No. - 2d No.) = twice 2d No.
The brackets here enclose the things which must be first done, before
the signs which join the brackets are made use of. Thus, 8-(2 + 1) × (1
+ 1) signifies that 2 + 1 must be taken 1 + 1 times, and the product
must be subtracted from 8. In the same manner, any result made from
two or more numbers, which is true whatever numbers are taken, may be
represented by using first No., second No., &c., to stand for them, and
by the signs in (23). But this may be much shortened; for as first No.,
second No., &c., may mean any numbers, the letters _a_ and _b_ may be
used instead of these words; and it must now be recollected that _a_
and _b_ stand for two numbers, provided only that _a_ is greater than
_b_. Let twice _a_ be represented by 2_a_, and twice _b_ by 2_b_. The
equations then become
(_a_ + _b_) + (_a_ - _b_) = 2_a_,
and (_a_ + _b_) - (_a_ - _b_) = 2_b_.
This may be explained still further, as follows:
25. Suppose a number of sealed packets, marked _a_, _b_, _c_, _d_, &c.,
on the outside, each of which contains a distinct but unknown number of
counters. As long as we do not know how many counters each contains, we
can make the letter which belongs to each stand for its number, so as
to talk of _the number a_, instead of the number in the packet marked
_a_. And because we do not know the numbers, it does not therefore
follow that we know nothing whatever about them; for there are some
connexions which exist between all numbers, which we call _general
properties_ of numbers. For example, take any number, multiply it by
itself, and subtract one from the result; and then subtract one from
the number itself. The first of these will always contain the second
exactly as many times as the original number increased by one. Take
the number 6; this multiplied by itself is 36, which diminished by one
is 35; again, 6 diminished by 1 is 5; and 35 contains 5, 7 times, that
is, 6 + 1 times. This will be found to be true of any number, and, when
proved, may be said to be true of the number contained in the packet
marked _a_, or of the number _a_. If we represent a multiplied by
itself by _aa_,[5] we have, by (23)
_aa_ - 1
------------- = _a_ + 1.
_a_ - 1
[5] This should be (23) _a_ × _a_, but the sign × is unnecessary here.
It is used with numbers, as in 2 × 7, to prevent confounding this,
which is 14, with 27.
26. When, therefore, we wish to talk of a number without specifying
any one in particular, we use a letter to represent it. Thus: Suppose
we wish to reason upon what will follow from dividing a number into
three parts, without considering what the number is, or what are the
parts into which it is divided. Let _a_ stand for the number, and _b_,
_c_, and _d_, for the parts into which it is divided. Then, by our
supposition,
_a_ = _b_ + _c_ + _d_.
On this we can reason, and produce results which do not belong to any
particular number, but are true of all. Thus, if one part be taken away
from the number, the other two will remain, or
_a_ - _b_ = _c_ + _d_.
If each part be doubled, the whole number will be doubled, or
2_a_ = 2_b_ + 2_c_ + 2_d_.
If we diminish one of the parts, as _d_, by a number _x_, we diminish
the whole number just as much, or
_a_ - _x_ = _b_ + _c_ + (_d_ - _x_).
27. EXERCISES.
What is _a_ + 2_b_ - _c_,
where _a_ = 12,
_b_ = 18,
_c_ = 7?--_Answer_, 41.
_aa_ - _bb_
What is ----------- ,
_a_ - _b_
where _a_ = 6 and _b_ = 2?--_Ans._ 8.
What is the difference between (_a_ + _b_)(_c_ + _d_)
and _a_ + _bc_ + _d_, for the following values of
_a_, _b_, _c_, and _d_?
_a_ | _b_ | _c_ | _d_ | _Ans._
1 | 2 | 3 | 4 | 10
2 | 12 | 7 | 1 | 25
1 | 1 | 1 | 1 | 1