Nicolas Pike.

# A new and complete system of arithmetick. Composed for the use of the citizens of the United States online

. (page 3 of 43)
Font size meeting, or. at the right hand of (he first, and \u\iiev the second,
you will find (he sum â€” as, 6 and 8 is 13.

When you %vould subtract : Find the number to be subtracted in
the left band column, run your eye along to the right hand till you
find the number from %vhich it is* taken, and right over it at (op yoq
uill find the diOfcreuce â€” as 8, taken from 13, leaves 5.

(K:rtirt,and, if the rrmainder be the Tame ai tliat againfl the fumtasU is lu this
cj^ample, the work it prcfumed to l)e right.

An eafier method of cafting out the nines, is to begin as before, and when the
fum txceeJs mine^ to add the (inures tliemfelves of this fum as before, and fo pro*
(ccd, aird this new Aim wilt alwavs be equal to the remainder after nine is ta-
ken from the firft fum. Thus, as before, 3 and 7 arc la, â€” now add the num-
bers of this fum, which, being i and 2, make 3, equal to the remainder after 9
is taken from ta; then 3 and 3 added to 8 mike I4i â€” ^d the i and 4, and the
fum is 5, the fame as the remainder al>ove. In the next row,*â€” omitting the 9,
the fum is i3,the numbers of which, i and 2, make 3, the remainder as abpvc.
The fame will hold true in any cafe.

Note. It ihould be noticed that the method of proof for this rule, and vari-
ous others, depends upon the accuracy of both operations. It does not follow
bcaufe the refult is the fame- by both operations, that there tan ^ no error. For
both operations may (>e incorrectly performe(|Â» aud the refult s, though alike,
4 rroneous. The t^ proof that any refult is riglit, is tkc corral fer/'^rKo/tce of all
fj.t cferathnt.

Digitized by

{Examples.

23

J.

2,

3.

4.

5.

6.

x-

m.

Cirt.

MileÂ«.

Yards.

ÂŁ.

1

n

123

1234

12345

987654321

2

34

466

5678-

67890

123456789

3

66

78&

9098

98765

234567891

4

78

12

7654

43210

345678910

5

90

345

32J0

12345

456789123

6

1

678

-62

67890

567879287

7

2S

901

4713

74100

678900028

a

45

234

131

64786

789400690

9

67

567

9128

19876

548769138

t>oin. 45

40908

' :>

In the first Example, the student adds tog^ether the several num-
bers, and finds the sum to he 45 ; and, as there is hut one column,
he must set down 45 for the answer.

in the 4th Ex. tho student will add the numbers of the column on
the rif^ht hand, which he will find to be 38 ; he will set the 8 un-
der the column, and carry 3 to the next column. The next col-
umn with the 3 to be carried, he will find to be 40 ; he roust set
down the 0, and carry 4 to the next column. This will be foun<i
to be 29; the 9 is to be set under the column, and the 2 carried
to the next column, which makes 40 ; the cypher is to he put un-
der the columoy and the 4 will take the next higher place, for \t
h evident the whole must be set down. The same course must
^ pursued in each example.

7.

8.

9.

10.

1234567

1234567

67

1234567

2345Â«78

723456

123

9876543

5456789

34565 '

4567

2102865

4567890

4566

89093

4321234

5678209

333

654321

5682098

6789098

90

1234667

6543218

1997577

n. What IS the snm of 3406, 7980, 345 and 27 ? Ans. 1175^.

12. A man borrowed of his neighbour, thirty dolhirn atone time^
one hundred and 66 at another, and seventy &ve at another : how
much did he borrow in the whole ? Ans. 271 dolls.

13- Four boys collected chcsnuts ; A. had 4096^ B. 16784, C-
11690, and D. fiv^ hundred and 67 ; how many were there in the
whole ? Ans. { V ; ^

14. Four boys, on counting their apples, found that A. had 67,
B. II niÂ«re than A, CÂ« had 101^ and D. had sixteen naorethan C;.

15. The Deluge*^ happened 2348 years before the birth of oar
Saviour, and America was discovered 1492 years a+terit; bow
iwny years intervened ? 3 \ '

Digitized by

Â»4 SIMPLE SUBTRACTION.

SUBTRACTION.

TEACHES to take a less number from a greater, to find aihrnl,
ftliewingr the inequality or differetice between the given numbers.
'J 'he greater number is cnlled the Minuend. The les\$ number is
r.alleil (he Subtrahend. The diCTerence, or, what is leftaAer tiie
sitbtractiun is rnade^ is called the Remainder.

SIMPLE SUBTRACTlpK

Teaches to find the diflference between any two numbers, which
are of the same kind.

Rule.
Place the larger number up|>ermost, and the less tjnderneatb,so
that units may stand untler units, tens under tens, &c. theoi draw-
ing a line underneath, begin with the units, and subtract the lower
from the upper figutc, and set down the remainder ; bat if the low-
er figure be greater than the upper, add ten, and subtract the low-
er tigure therefrom : To ih\K difference, add the upper figure,
which being set down, you must add one to the ten's place of tlie
lowet* line, for that which you added before ; and thus proceed
through*the whole.*

Proof.
In either simple, of compound Subtraction, add the renminder
and the less line together, whose sum, if the work be right, wilt
be equal to the greater line : Or subtract the remainder from the
greater line, and the difference will be equal to the les\$.

ExxaiPLcs.

e. 3. 4.

jCr Mites. Yards.
305 4670 5^934
103 4020 6182

1.

ÂŁâ€˘

From

25

Take 12

Kern.

13

Proof.

tb

5.

6.

Feet.

Civt.

879C47

9 I 87641

164348

9.1843

2762

68934

â€˘ Dem. When all the figures of the IcCi number arc Icfs th>n their corres-
pondent figures in the greater, the difference of the fi)^ut tt, in ll>c federal like
places, muft, all taken together, make the true diflrrcnce ruu;>ht ; bccaufe, as
the fom of the parts is equal to the whole ; fo muft the fum uf the differences,
uf |,ll the fimiiar parts, be equal to the difference of the whole.

2. When any figure in the greater nunitMrr is krfs than its correfpondent fig-
ure in the lefst.the ten which is added by the Rule, is the vntue of an unit in
tiic next higher pUce, by the nature of notation ; and the one which is added
to the next place of the lefi number, is to diminiih the correfpondent place of
the greater, accordingly ; which is only taking from one pUce and adding aa
much to another, whcM)y the total is never changed : And, by this means, (he
greater is refolved into fuch parts, as are, each, greater than, or equal to, the
fiiniUr part of the lefs ; and Â»be difference of the correfpondent figures, taken
together, will, evidently, make up the difference of the whole.

The truth of the method of proof is evident ; for the diffcrercc of two num-
I'cra itddfil to il;c !(.0, i? manifcflly, equal to the greater.

Digitized by

SIMPLE MULTIPLICATION. S6

The operation oo the firet three Examples is mfficienUy plaio.
In the 4th Ex. 1 beg^in on the right hand, and take 2 from 4, and
set dovm the difference 2, under the column. An 8 is greater than
3Â« \ add 10 to 3, which makes 13, and from it take the C^, and Â§ it
the difference to be set down. As 1 add, 10 to the 3, I now add 1
to the 1 in the next higher place, becaase 10 in one place is equal
only to 1 in the nent higher place, and take the 2 from the 9, and
the difference is 7. The rest of the work is obvious. The same
proofs must be followed in every similar case.

7-

1 00200300400600600700800900
98076054032011023045067089 .

8.

9.

10000

1000000

9999

1

10. What is the difference of 40875 and 38968? Ans. 1907.

11. What number must be added to 6892, so that the sura shall
be 8265? Ans. 1373.

12. America was discovered in 1492 ; how many years have
elapsed since ?

13. If you lend your friend 3646 dollars, and aAerwards are
paid 2998 dollars ; how much is still due ? Ans. 648 dollars.

14. If a man was seventy five years old in the year 1821, in what
year was he born ? Ans. 1746.

15. The Independence of the United States was declared July
4th, 1776 ; how many years have passed since 7 Ans.

16. Sir Isaac Newton died in the year 1727, aged eighty five ;
in what year was he bom ? Ans. 1642.

MULTIPLICATION

TEACtlES to find the amount of one number increased as many
times as there are units in another, and thus performs the work of
many additions in the most compendious manner ; brings numbers
of great denominations into small, as pounds into shillings, pence
or farthingp, he. and, by knowing the value of one thing, we find
the value of many.

The#)umber given to be muhipKed, is called the Multiplicand'
The number given to multiply by, is called the Multiplier,
The multiplicand and multiplier are otten called /aceorx.
The result of the operation, or the number found by multiplying,
is called the Product.

Multiplication is distinguished into Simple and Compound.

SIMPLE MULTIPLICATlok

Is the multiplying of any two numbers together, without having
regard to their signification ; as 7 tiines 8 i^ 56, kr.

D

Digitized by

'J6

SIMPLE MULTIPLICATION.

MutTIPLIClTION IND

DiviMov Table.

1

2

'31

4|

6

6

7

Â«l

9|

10

U

12

2

4

6|

8t

10 1

12

Hi

16 1

18 1

20,

22

24

3

6

9|

.12 1

16

18

21

|24 1

27

30

33

36

4

8

12 1

16 1

20

24

28

|32(

â€˘36

40

44

48

6

10 1

16 1

20 1

25

30

35

1 'io|

45

50

55 1

60

6

12 1

18

24 1

30 1

36

42

48 1

54

60

66 1

72

7

14 1

21

28 i

35 1

42 1

49

56 1

63

70

77 1

84

8

16

24

32

40|

48 I

66

64 1

72

80

88 I

96

9

18

27 1

36

45

54 1

63

72 1

81

90

99 1

108

10

20

30 1

40

50

60

70

80 1

90|

100

110|

120

11

22

33

44

55

66

[77

88 1

99 t

110

121

132

12

24

36 1

48

60

72

84

1 96 1

108 1

120

132 1

144

To learn this Table for Multiplication : Find your muUiplier hi
the left hand column, and joar multiplicand at top, and in the com-
mon angle of meeting, or against your multiplier, along at the
right band, and under your multiplicand, you will find the product,

To leatn it for Division : Find the dirisor in the left hand col-
umn, and run your eye along the row to the right band until you
find the^ dividend ; then, directly over the dividend, at top, you will
find the quotient, shewing how often the divisor is contained in the
dividend.

Rdlc.
Having placed the multiplier under the multiplicand so that unit?
stand under units, tens under tens, &c. and drawn a line undei
them, then,

1. When tlic muitiplief does not exceed 12; begin at the right band
of the multiplicand and multiply each figure by the multiplier, set-
ting down the unit figure under units, and so on, and carrying for
the tens to the next place, as in addition, and the work is done.*^

2. When the multiplier exceeds 12; multiply each figure of the
multiplicand by every figure in the multiplier as before, placing
the first figure of each product exactly under its multiplier : then

* Dem, When the muUiplier it a fingle digit, it is plain that we find the pro-
dtt<5l; for, by multiplying every figure, that it, every part of the multiplicand,
we multiply the whole ; and, the writing down of the products, which arc lefs
than ten, or theexcefs of tens, in the places of the figures multiplied, and car-
rying the number of tens to the product of the next place, is only gathering to*
^ether the fimilar parts of the refpective products, and is, therefore, the fame,
in effect, as though we wrote down the multiplicand as often as the multiplier
cxprclTcsi and added them together; for the fum of every column. is the pro*
duct of the figures in the place of that column ; and the products, collected to-"
gether, are evidently equal to the whole required product.

Digitized by

SIMPLE MULTtPIf^ATION. 27

Â«dd together these several {>roduct8 as they stand, an^ Iheir suq
will be the total product.*

* If the multiplier bc a number, made up of more than one figure; after we
have found the product of the multiplicand by the firft figure of the multiplier,
aÂ« above, we fuppofe the multiplier divided into parts, and, after the fame
oianoer, find the product of the multiplicand by the fecond figure of the mul-
tiplier; but at the figure, by which we are multiplying, dindt in the place of
tens, the product muft be ten times its fimple value ; and, therefore, the firfl fig-
ure in this product muH be nored in the place of tens, or, which is the fame,
directly under the figure we are multiplying by. And, proceeding in the fame
maaner with all the figures of the multiplier, feparatdy, it is evident we (halt
amiltlply' all the parts of the multiplicand by all the parts of the multiplier;
ther^ore, thefc (eireral products, being added together, will be equal to the
whole required product.

The reifon of the method of proof, depends upon this propolition, that if
two numbers are to He multiplied together, either of them may bc made the
multiplier or multiplicand, and the product will be the fame.

A (mall attention to the nature of numbers will make this truth evident ; for
5X9 = 45 =9X5; and,ingeneral,2x3x4X6X6,&c. = 3X2x6X5X4,lcc.
ivithout any regard to the order of the terms ; and this is true of any number

The following examples are fubjoined, to make the teafon of the rule appear
as clearly as po0ibleÂ»

^4763
â– : 3728

1903648 =^ 8 times eke multiplicand.
475912 =s 20 times ditto.
1665692 rs 700 times ditto.
30 =60000X5 7 13868 as 3000 tiipcs ditto.

1

5=

3X5

ss

=

50X5

35

=

700X5

20

=

4000X5

323765=64753 X 5 88709906H=3728 times ditto.

Another method of proving the rult is as follows. Let the factors be 6475'.^
and 5. Now ;>47o3=t>0000-f 4000+700+o0-f u\ The fum of the products
of thefe quantities (evcrallf multiplied by 5, is the true product. Then
600004-4UOO-f-700-|-50-f3 is one faaor. 5 the multiplier the other fa^or.
DOOOOOH-20000-|-3500-f 250-f- 15=32.^65=04753 X 5.
Or let the favors bc 45 and 24. Then 45= 104-5, and 243=20-f 4* and â€˘
40+5 multiplicand. JLct the favors he 24 and 24. Then,

20-i-4 mu ltiplier. 20-|-4

8Â«J0-f 100=46X20 J21i__

160 + 20=4 5X4 400+80 = 24X^

800+260+20 = 1000=15X21. , _80+l6 = 24x4.

400+160+16 = 576=24X2^1.

Multiplication may al(b be proved, by c^ing out the nines ; but is liable to
the iiico!Â«venieaee before mentioned.

Umay likewife be, very naturally, proved by divifion ; for the produd, be-
ing divided by either of the fadlors, wiU, evidently, give the other ; and it might
not be amifi for the pupil to be taught diviiion,at the fame time with multipH-
cation ; as it not only fcrves f^r proof ; but alfo gives him a readier knowledge
of them both. But it would have been contrary to good method to have^iv-
eo this rule in the text, bccaufc the pupil is foppofcd, as yet, to be unacquaint-
ed with divificn.

Digitized by

28 SIMPLE MULTIPLICATION.

Proof.

Multiply the maltiplier ^y the multiplicand.

Multiply 3851 by 3. By additioo.

3851 Multiplicand. 3851

aMultiplierÂ« ^ 3851

3851

11553 Product. â€” â€”

11553 SuQi.

Having placed the numbers according to the rule,-Â«^then say, 3
times 1 is 3, and place 3 directly under units ; tben3 times 5 is 15,
set down 5 and carry the one to the next product. Then, 3 times
8 is 24, to which the 1 is to h{ added, making 25 ; set down 5 and
carry 2. Then 3 times 3 is 9, and the 2 to be carried, make !1,
which^set down, and the work is done. The result is the^same as

Multiply 6053 by It.
11

Prod. 66583

Procee<ling as before, multiply 3 by 11, and of the product, 33^
set down 3 under units, and carry 3 ; then 5 by 11, and to the pro-
duct, 55, add the 3 to be carried, set down 8, and carry 5 ; then O
by U, aod as the product is 0, set down the 5Â» which was to be
carried ; Iben 6 by U, and, as there is none to carry ,^ set down the
product, 66j and the operation is finished.

Multiply 67013 by 2?.
67013 Multiplicand.
29 Multiplier.

603117 Product by 9, the units of the multipHer.
134026 Product by 2, the tens of the multiplier.

In this example, the multiplLcand is first multiplied by 9, the naitÂ»
of the multiplier, and the product set down, as in the preceding ex-
amples. The multiplicand is theo multiplied by 2, the tens of the
multiplier, as before, the first figure of the product is placed under
the 2, in the place of tens. The two products are then added, and
their sum is the whole product or answer.

Examples.

I.

2.

3.

4.

37934

76930a

4980076

763896

2

3

4

5

Prod. 7586a

Digitized by

SIMPLE HDLTIM^lCATKkN.

29

6.
67689
6

6.
603764 :

7

7. 8. *
3018296 9164785

8 9

â€˘

Prod. 405534

9.

4879567
10

10.
5864794
11

11.

e583478646
12

Prod.

64512734

12.

6357534
47

13.

8324629
59

14.

46293845
106

44602738
25430136

r

277763070
46293845

Prod. 298804098

4907147570

15.

647906
4873

16.
760483
9152

17.
- 91867584
6875

3157245938

18. Maltiplj 103 by sixty seven. Aos. 6901.

19. Said Jack to tifarry, yoa hare only 77 cbesniits, bat I have
seven times as maoy ; how many have I ? Ads. 539.

20. If four boshiels of wheat make a barrel of floor, and the
price of wheat be one dollar a boshel, what will 225 barrels of
floor cost ? Aos. 900 dolls.

21. Eighty nine men shared equally in a prize, and recei? ed 17
dolls, each ; how much was the prize ? Ans. 1513 dolb.

22. Multiply 308879 by twenty thousand five hundred and three.
Ans. 6332946137.

In some cases the operations of multiplication are shortened by
particular rules. Several Cases follow.

Note. A composite number is the product of two or more num-
bers, as 27, which 3x9, and, as 316, which =? 5X7X9.

CASE I,
When the multiplier is a composite number, multiply th^ multi-
plicand by one of those figures, first, and that product by the other,
^ and the last product will be the total required.*

* The resdbn of this method is obvious : For any number, multiplied by the
compocent partt of another namber, muft give the fame produ^ at though it
were maltipKed by that number at once t Thus, in example firft, \$ times the
produce of 7, multiplied into the given number, makes 35 timet that given
number, at pUioly as 5 times 7 mtket 35.

Digitized by

SIMPLE MULTIPLICATION.

1.
Mult. 59375 by 35.

7X5 = 35

415625
5

Examples.

2.
39187 by 48.

3.

91632 by 56.

2078125

4.
3065 by 68.
6.
14567 by 144

5.
6061 by 121.

CASE If.

When there are Ofphers on the right hand of either the multiplieand^
0T multiplier^ or both : Neglect those cyphers ; then place the m*
oificaot figures under one another, and multiply by them only ; add
them together, as before directed, and place to the right hand at
many cyphers as there are in both the factors.

Examples.
K 2. 3.

67910 956700 930137000

5600 320 9500

Prod. 380296000 306144000 8836301500000

CASE III.

To multiply by 10, 100, 1000, 4^c. : Set down the multiplicand un-
derneath, and join the cyphers in yonr multiplier to the right hand
of them.*

EsfAMPLES.

1. 2. 3. 4.
57935 8493S 613975 8473965

10 100 1000 lOOOO

Prod. 579350

CASE IV.
To multiply by 99, 999, ^c. in one line : Place as many dots at
the right hand of the multiplicand, as there are figures of 9 in your
multiplier, which dots suppose to be cyphers ; then, beginning with
the right hand dot, subtract the given multiplicand from the new oneÂ»
and the remainder Will be the product.f

* Thiiif evidcBt fron^ the nature of numbcra, fince every cypher annexed to
iht right of a number increafet the value of that number in a tenfold proportion.

f Here it may eafily he feen that. If yon mnlttply any fum by 9, the prodwfl
will be but 9 tcntha of the producSk of the fame fum, multiplied by .10 : and as
the anncziog oft doc or evpber, to the right hand of the multiplicand, fnppof-
es it to be iocreafed tenfold : therefore, fubtradling the given multiplicand
from the tenfold multiplicand, it is evident that the remainder will be nincibld
the faid given multiplicand, equal to the produA ot the fame by 9; and the
Tame will hold true of any number of sines. -

Digitized by

SIMPLE MULTIPLICATION.

31

Examples.

1.

2.

3.

6473..

857389...

5384976

99

999

9999

64082T

'\

53844375024

That these exampTei maj appear as clear aa posflibU, I wilt il*
lustrate them by giving another.

Mult. 371967.
by 999

371595033

1

According to the rule,
it will stand thus.

371967... Minaend.
371967 Sabtrab.

371595033 Rem. or
â–  ' total Pro.

CASE V.

Tq mmWply by 13, 14, 1^ f-c to 19 ; alto from 101 io 109, from
1001 to 1009, ^c. : Multiply with the unit figure only, of the mul-
tiplier, remoTing the product one place to the right hand of the
multiplicand, and so many places further as there may be cyphers
betwÂ«eii the significant figures ; then add all together, and their
snm will be the product.^

EXAJCPJLES.

1. 2. 3*

75964X13 7593x104 6735x1005

227892. ' 30392 33675

Prod. 987532

CASE VI.

To multiply 5y 21, 31, 41, 4-c. t& 91, also by the tame figures with
any number of cyphers between them : Multiply by the left hand fig-
ure, only, of the multiplier, and set the unit figure of the product
one place to the left, and as many places farther as there are cy-
phers between the significant figures ; and add the numbers togeth
er for the product.

EXJIMPLES.

1. 2. 3,

73918 X21 66934 X301 45936 X4001
147836 170802

Prod. 1552278

17137134

4.
3167X500001.

* The resioD of thtt Rule, and of ibc following one %\Co, will bÂ« cvideot om
lofpcdHag an eaniple undtr each riHc.

Digitized by

32 DIVISION. ,

CASE. VIL

To multiply any number^ by any i|tifii6er, giving only the Product i
Pat dowD the product figare of the first figure of the multiBlicaDd
by the first of the multiplier. To the product of the second of the
multiplicand by the first of the multiplier, add the nomber to be
carried, and tbi^ product ofih^fifst of tbemultiplicapd by the second
of the multiplier ; theo, carnring for the tens in the sam, put down
the rest. To the product of the third of the multiplicand by the
first of the multiplier, add the number to be carried, and the pro*
duct of the second of the multiplicand by the s^coÂ»ulof the multipli*
erÂ« also the product of the first of the multiplicand by the third of
the multiplier, carry the tens, and put down the rest, and so pro*
ceed till. you have multiplied the highest of the multiplicand by the
lowest of the multiplier. Then multiply the highest of the multi-
plicand by the second of the multiplier : Add the number to be
carried, and the product of the last but one of the multiplicand by
the third of the multiplier, and the product of the last but two of
the multiplicand by the fourth of the multiplier, &c. Then to the
product of the laH but one of the mnltiplioand by the fourth of the
multiplier; and so proceed till you have multiplied the last of the
multiplicand^ by the last of the multiplier, which finishes the work.

Example, Explanation, , Â« .

Mult 5321415

By 2364 .6x4=20

Prod. 12626610910 1X4+2+6x6=31

4x4+3+1x6+5x3=39

1X4+3+4 X 5+lx3+5x2t=:40

2X4+4+1x6+4x3+1x2=31

3x4+3+2x6+1x3+4x2=36

"6X4+3+3x6+2x3+ 1x2=46

6X 6+4+3X 3+2X 2=42

5>r3+4+3x 2=26

â€˘â€˘ 5x2+2=12

DIVISION

TEACHES to separate any number or quantity given, into any
tttimber of parts assigned ; or to find how often one number is con-
tained in another ; and is a concise way of performing several Sub-
traction;.

Digitized by VjOOQ IC

SIMPLE DIVISION. 38

There are four parts to be noticed in Division, vi2.
â–  The Dividends id the number given to be divided.

The Divisor^ is the number given to divide bj.

The Qtfo/tenC, or ansfver to the question, sho^va how many
times th^ divisor is contained in the dividend.

If there be any thing left after the operation is performed, it is
called the Renuiinder ; sometimes there is a remainder and some-
limes there is not. The remainder, when there is any, is of tlie
same denomination as the dividend.

Diviaion is both Simple and Compound.