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•VYitLIAM 'Q. pECK, pH.g, pp.p.,
2*vof, of Jtalhematics and Astronomy in Columbia
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F I I E F A . C E
D A V I E S A N D P E C K ' S
SHORT COURSE IN MATHEMATICS
I N F O U R B O O K S .
E L E M E N T A R Y A R I T H M E T I C ,
C O M P L E T E A R I T H M E T I C .
M A N U A L O F A L G E B R A .
M A N U A L O F G E O M E T R Y .
Copyrighted, 1878, by William G. Peck.
^ovk is designed as the Introductory Volume
X of the Two Book Course of Daties and Peck. It
is especially adapted to beginners. It is believed that the
subjects are treated in sucli a manner as to interest and
awaken the attention of the young.
In preparing the work, three objects have been con
stantly kept in view.1. To make it educational.
3. To make it practical.
3. To adapt it to the capacity of any child whose mind is sufRciently mature to commence the study of arith
m e t i c .
To attain these objects, every new subject has been
introduced by an inductive process, and the idea thus
developed has been expressed in the form of a definition.
The methods and rules have been deduced from prac tical operations and enforced by familiar illustrations. To direct the attention to important principles, leading test questions have been freely introduced.
In determining the subjects to be included, and the space to be assigned to each, the author has been guided by a consideration of the natural development of the
N j f
W P B B F A C E .
mental facnities. The book may be said to consist of
five parts. The first part contains simple, familiar
Lessons in Numbers. The second paçt contains theFundamental Operations followed by General Pnnci[)lcs
and Properties of Numbers. The third contains Frac
tions, in which great pains liave been taken to render
the work intelligible to young students. Currency and
the Metric System follow, because of their intimate rela
tion to Decimal Fractions. The fourth contains Compound Numbers and Heduction. The fiftli, Percentage
and its applications.The logical development of principles, the systematic
arrangement of the subjects, the copiousness and variety
of exercises will, it is believed, greatly aid the teacher
in exciting the interest of the pupil.
Teachers who desire to give a more extended drill in
the simplest operations, are referred to "Peck's First
Lessons in Nuiibees."
To facilitate references, a complete Index to the Sub
jects and Definitions is inserted at the end of the volume.
The author takes great pleasure in acknowledging his
obligations to many teachers who have favored him ^rith
suggestions and criticisms. But more than a passing
acknowledgment is due to Prof. John Dunlar,- whose
long experience and superior ability as a Teacher have
enabled him to render much valuable assistance in the
preparation of this work.O O n S T T E - T S T T S . F o r m a t i o n o f N u m b e r s . P A C E N u m b e r s f r o m i t o l o 8 " " t o t o 2 0 9 " " 2 0 t o 3 0 1 0 " " 3 0 t o 1 0 0 I I
Increasing and Diminishing by i... \i
" " b y 2 . . . 1 3 " " b y 3 . . . 1 4 • * " b y 4 . . . I S " " b y 5 . - . 1 6 E x e r c i s e s i n N u m b e r s 1 7 H i g h e r N u m b e r s b y F i g u r e s 2 0 N u m b e r s b y L e t t e r s 2 1 N o t a t i o n a n d N i i i i i c r n t i o n . T h i i e e M e t h o d s o f W r i t i n g N u m b e r s 3 3 O r d e r s o f U n i t s 2 3 S i m p l e a n d L o c a l V a l u e s 2 6 P e r i o d s o f F i g u r e s 2 9 C l a s s i fi c a t i o n o f N u m b e r s 3 1 F u n d n m c n t n i O i i c r n t i o n s . A d d i t i o n 3 3 E x p l a n a t i o n o f S i g n s 3 5 O p e r a t i o n s 3 ^ S u b t r a c t i o n 4 5 E x p l a n a t i o n o f S i g n s 4 7 Figure in Subtrahend less than 1
F i g u r e i n M i n u e n d ' Figure in Subtrahend greater than
Figure in Minuend
M u l t i p l i c a t i o n 6 0
E x p l a n a t i o n o f S i g n 6 0
E l e m e n t s o f M u l t i p l i c a t i o n 6 3
M u l t i p l i e r b u t O n e F i g u r e 6 5 Multiplier any Number of Figures. 67 C o m p o s i t e N u m b e r s 7 0 S 3 P A G E D i v i s i o n 7 5 E x p l a n a t i o n o f S i g n s 7 7 T h e D i v i d e n d l e s s t h a n D i v i s o r 7 9
Methods of Performing Operations. 81
S h o r t D i v i s i o n 8 2 F r a c t i o n i n t h e Q u o t i e n t 8 2 L o n g D i v i s i o n 8 4 G e n e r a l P r i n c i p l e s . N o t a t i o n 9 3 A d d i t i o n 9 3 S u b t r a c t i o n 9 3 M u l t i p l i c a t i o n 9 3 D i v i s i o n 9 3 P r o p e r t i e s o f N u m b e r s . D e fi n i t i o n o f P r o p e r t i e s 9 5 E x a c t D i v i s o r 9 5 P r i m e N u m b e r 9 ® E v e n N u m b e r 9 ^ O d d N u m b e r 9 ^ C a n c e l l a t i o n 9 8 G r e a t e s t C o m m o n D i v i s o r J o o L e a s t C o m m o n M u l t i p l e ' 0 3 F o r m a t i o n o f F r a c t i o u s . D e n o m i n a t o r ' ® 5 N u m e r a t o r . V a l u e o f a F r a c t i o n T e r m s o f a F r a c t i o n K i n d s o f F r a c t i o n s " 7 Principles of Fractions R e d u c t i o n o f F r a c t i o n s ' o g A d d i t i o n o f F r a c t i o n s S u b t r a c t i o n o f F r a c t i o n s i ' 9 M u l t i p l i c a t i o n o f F r a c t i o n s r s i D i v i s i o n o f F r a c t i o n s * 3 7
V I
{J
C O N T E N T S . F o r m a t i o n o f D e c i m a l s . P A G B Notation of Decimals 136 Numeration of Decimals 137 P r i n c i p l e s o f D c d m a l s 1 3 8 A d d i t i o n o f D e c i m a l s 1 3 9 Subtraction of Decimals 141 Multiplication of Decimals 143 Division of Decimals 146 » S « 152 155 •55 i S S C u r r e n c y . Definitions U. S. Currency Canada Currency French Currency ! English Money Metric System. Measure of Length ,57To write numbers in the Metric 1
S y s t e m . j ' S 8 To read numbers in the Metric 1
S y s t e m ( ■ » 5 8 Measure of Surface.! ! !!." ! ! ! ! " " Measure of Volume .!.■ Measure of Capacity Measure Of Weights !! ,60 Business Opcratlniia.
Terms used in Business Transac} ,
l i o n s f 1 6 3 . . . I t ' o n s . . . M e t h o d s Aliquot Paru !/!."!!! 162 166 B i l l s a n d A c c o u n t s . P A G E
Definitions and Abbreviations 169
O p e r a t i o n s 1 7 0 C o m p o i i i t d N n i n b c r s . T a b l e s o f W e i g h t s 1 7 1 T a b l e s o f T i m e 1 7 3 M e a s u r e s o f L e n g t h 1 7 3 M e a s u r e o f S u r f a c e 1 7 5 M e a s u r e o f V o l u m e 1 7 6 Angular Measure and Longitude.. 178
R e d u c t i o n i g o
Addition of Compound Numbers.. 185
Subtraction of Compound Numbers 187 Multiplication of Compound Num
b e r s 1 8 9
Division of Compound Numbers.. 191 P c r c c n t n f f c a n d A p p l i c a t i o n s . Explanations and Definitions 195
I'rinciples of J'ercentagc 198
O p e r a t i o n s 1 9 8
C o m i n i s s i n n 2 0 1
P r o fi t a m i L i i S S . , 0 4
S i m p l e I n t e r e s t 2 0 5
Aliquot Parts of a Year 307
M l q u o t P a r t s o f . 1 M o n t h 2 0 8 N o t e s P a r t i a l P a y m e n t s 2 1 3 D i s c o u n t 8 1 4 H a n k s n i u l B u n k D i s c o u n t 2 1 6 M i s c e i . l a n e u u s E x a m p l e s 3 1 8 A N ' s w b r s 2 1 9 I . v o c x 2 2 6
FORMATION OF NUMBERS.
L E S S O N I . C O U N T I N G .Look at tlie picture and count the objects named below.
f
How many houses? How many horses ? How many sail-boats ?
How many high trees? How many boats?
How many boys at play ?
How many windows in front?
How many small trees ?How many birds ?
8 Î ^ U i l B E R S .
X U i l B E E S . 9
L E S S O N I I .
W R I T I N G N U M B E R S . 1 T O 1 0 . *
"Write the word that telJs Jiow many liouses there are in
tlie picture. (Pne. One Is a Unit.
Write the word that tells Jiow many liorses.
How many ones, or units, in two ?
Write the word that tells liow many persons there are
in tlie carriage. S^iee. How many units in tliree 'i
How many units, or ones, in four? In five ? In six ?
In seven ? In eight ? In nine ? How many in ten ?
One, two, three, four, five, etc., are called numhors. ^ A Number is one or more things of the same kind.What number tells how many girls tlicrc are on the grounds ? What is the number of boys ?
Thus far we have used words to express numbers ; we
may also use Figures.The number of houses may bc written one or I; the
number of horses, two or 2; the number of sail-boats,
three or 3j tlie number of girls, four or A; number of
boats on the lake,/ye or 5; number of boys, six or 0;
number of windows, sevew or 7; number of small trees,
eight or 8; number of birds, nine or .9.
We use one more figure, 9. it jg called nauqhf, end
standing alone expresses no number, })ufc jg used witliother figures to express niimbers.
These are all the figures in use. JEIow many are tiiere ?
Write the ten figures ; thus,
1, 2, S, 4, 6', 7, 8, 9, 0.
Read the following figures • 3. 2, 1 ; 4^ 5 q . ç)^ 7^ 0*
Which is the least number? The greatest^
• See Picture, page 7.
L E S S O N I I I .
N U M B E R S F R O M 1 0 T O 2 0 .
How many figures do we use to express numbers?
The number ten is written by means of figures, thus, 10.
This is one ten. How many units in one ten ? Writet e n .
How many bo3's arc snow-bulling?
We write the number by means of figures, thus, 11.
The right-hand figure is 1 unit. The second figure
from the right is 1 ten. Eleven is one ten and one unit.
How many units in ten ? How many units in eleven ?
We will Avrite, by means of figures, all the numbers
I r o m 1 0 t o 3 0 :
1 0 , 1 1 , 1 2 , 1 3 , U , I S '
ten, eleven, tmlve, thirteen, fourteen, ffteen,
1 6 ,
1 7 ,
1 8 ,
1 9 ,
2 0 .
eixteen, seventeen, eighteen, nineteen, ticenty.
10 X U i T B E R S . n u m b e r s . * 1 1 L E S S O N I V .
numbers from 20 TO 30.
Tvto tens and 1 unit are twenty-one, 21.
Two tens and 2 unit-s are twenty-two, 22!
Two tens and 3 units are twenty-three, 23!
Two tens and 1 units are twenty-four, 21.
Two tens and 5 units are twenty-live,' 25
Two tens and 0 units are twenty-six, 20!
wo ns and 7 units are twenty-seven, 27.
wo tens and 8 units are twenty-eiglit, 28
Tw-o tens and 9 units are twentî-nino, 29.
Mow man} are one ten and i ? a i.
ten and 3 ? One ton and 4 ? À ^ ^
andC.^ One ten "0.10
9? Two tens are how manv'^T /'
tens and 3 ? 3 tens and ,8T' ,
5? 2 tens and G ? 2 tens and 7 s ,!
and 9 ? 7-2 tens and 8 ? 2 tens
Write, in fi^ires, eiffht ^
one twenty fii-o fwo ^ ' ®®^®^teen, nineteen,
twenty-on^twent)-hvc,twenty-s,x,tn.enty-nine
Write as one number i fn« ,and 8, 2 tens and 5, 2 tens and 7"" t!' ^ ^
Bead the following numbel 99 1Ô
21, 11, 12, 15, 21, 28? '
One ten and one unit are how many? Two tens and
two units are how manvî' T\i'^ +
How many are one tL and si i
six units ? Wiite two tens and five"u ^^ns aiic
Write three tens and six one nninber.
units; two tens and eight units- two
6 umcs, two tens and nine units.L E S S O N V .
N U M B E R S F R O M 3 0 T O 1 0 0 .
Three tens are thirty, 30. Four tens are forty, 40.
Five tens are fifty, 50. Six tens are sixty, 60. Seven
tens are seventy, 70. Eight tens are eighty, 80. Nine
tens are ninety, 90. Ten tens are one-hundred, 100.
In writing tens we use two figures, and the second figure
fi-om the right tells how many tens we have written.
In writing 100 we use three figures, and the third figure
from the right shows how many hundreds we have written.
If we use 2 instead of 1 in the third place, we have 200,
{two hundred). If we use 3, we have 300; if 4, 400, and
s o o n .
Write the numbers between 30 and 40:
Thus, SI, 32, 33, 34, 35, 36, 37, 33, 39.
AVrite the numbers between 40 and 50 ; between 50 and
60 ; between 60 and 70 ; between 70 and 80 ; between 80
and 90; between 90 and 100.Read the following numbers, 11, 14, 29, 23, 28, 31, 40,
37, 36, 42, 45, 49, 51, 53, 57, 62, 65, 69, 70, 75, 78, 82, 90,
87, 93, 71, 98, 86, 99, 100.
Four tens and 1 are how many ? 4 tens and 3 ?
Five tens and 6 are how many? 5 tens and 7? Six tens and 9 ? 6 tens and 5 ? 6 tens and 8 ?Seven tens and 4 are how many ? 7 tens and 8 ? 7 tens
and 9 ?
Eight tens and 5 are how many? 8 tens and 6? 8 tens
and 7 ?
Nine tens are how many ? 9 tens and 1 ? 9 tens and
2 ? 9 tens and 3 ? 9 tens and 9 ? 9 tens and 7 ?n n u m b e r s .
I ' E S S O N V I .
increasing and diminishing by I.
■?. How many eggs are two
J eggs and one egg:- 2 and 1,
l i o i n i m u y y - e W y
eggs ^rill be left?'"2
3, Tliree shoej) and one
slieep are liow manv sheejj ?
d and 1, are ]iow many 1
3, are how many?
J/, 't we take 1 sheen r , ,
eliet'i) will ij6 fcfti 3 iv f '• «lieop, liow iiiuuy
t loaves how many ?
3. Four bird.s and
one bird are liow many birds? 4 and 1 are how
how many:- One sheep and fnn ^
^ I f W P f T r 1 - 1 ^ m a n y ?
0. It we take one bird from g i • ,
will be left? 4 from 5 leaves i
7. How many boys are five "
many are 5 and 1 ? 1 and 5
<?. If we take one apple' fT'
apples will be left l from p apples, how many
6 leaves how many? oaves how many ? 5 from
9. Six chairs and one chair arp l
10. If we take one book from chairs?
books are left ? 1 fi-om 7 leavp i
weaves bow many?
>• UMBERS. 1 3
L E S S O N V I I .
i n c r e a s i n g a n d D I M I N I S H I N G B Y 2 .
1. Two apples and two apples are how
many apides ?
If we take 'd apples from i apples, how
many apples will be left?
S. Three sheep
p a n d t w o s h e e p a r e
liow many sheep?3 and 2 are how many ? 2 and 3 are how many ^
Jf. If we take 2 sheep from 5 sheep, how many sheep
will be left? 3 from 5 leaves how many? 3 from 5
IcuvcB how muiiy ?
/). How inanj' clieiTies
— / ( ^ ^ c h e r r i e s a n d 2 c h e r -
2 a r e h o wmany ? 2 and 4 are bow
many ?
6. If we take 2 cherries from 0 cherries, how many
cherries will be left ? 2 from 6 leaves how many ? 4 from
6 leaves how many ?
7. How many birds are five birds and two birds? 5 and
2 are how many? 2 and 5 are how many ?
S. If we take 2 birds from 7 birds, how many birds will be left? 2 from 7 leaves how many? 5 from 7 leaves how many?
1 4
N U 3 I B E B 8 . N U M B E B S . 1 5
LESSOK VIII,
increasing and diminishing by 3.
Ihree balls and throe balls are
bowmauy balls ?
ma. bles ,v.U be left ? 3 f.-om G leaves how ...a.,y ?
3. How many pears are
4 pears and 3 pears ? 4
and 3 are how nianv ? 3
, 0 1 , . h o w n i a n v ?
I 3 apples from 7 apples leaves how n.ai.y appl'es ? 3
from 7 leaves how many ? 4 fom 7 leaves how ...any ?
3. Five cberi'ics and (lirec
clieiTies are how many
ehcr-ries ? 3 units and 5 units
/ ? T f w - A f ^ i 1 1 . m a n y u n i t s ?
may units? ^ from 8 leaves howlaiiyf'
7. How many roses
are G roses and 3 roses :
0 and 3 are how many?
r n i . , • ^ 6 a r e h o w m a n y ?
o, Ihree trees from nine troAo i i j. ^ ^
Q 0 1 l e a v e s h o w m a n v t r e e s .
0 i r o m J l e a s e s h o w m a n v 9 r r , , « i ' ' j
0 „ 1 ^ ^ ' e a v e s h o w m a n y f
How many a77TnÎ3 '"C"
m Tf wl ioir n ^ are 3 and 7 ?
l^leaves how many? ^
L E S S O N I S .
I N C R E A S I N G A N D D I M I N I S H I N G B V 4 .
1. How many balls are 4 balls
and 4 balls ?
If we take away 4 marbles from 8 marbles, how many
niarbles will be left?
3. 5 sheep and
4 sheep are how many sheep? 5
and 4 are how many ? 4 and 5 are how many ?
Jf. If we take away 4 horses from 9 horses, how many horses will Ixî left ? 4 from 9 leaves how many ? 5 from
9 leaves how many ?
5. How many flowers
arc 6 flowers and 4 flow ers ? C and 4 arc how
many ? How many are
4 and 6 ?
6. 4 roses from 10 roses leaves how many roses ? 6 from 10 leaves how many ? 4 from 10 leaves how many ?
7. How many balls are
7 balls and 4 balls? 7 and
4 are how many? 4 and
7 are how many ?
S. If we take 4 marbles from 11 marbles, how many
marbles will be left ? 4 from 11 leaves how many ? 7
fi-om 11 leaves how many?
9. Jane has eight pears, and Julia has four pears ; liow
many pears have both ? 8 and 4 are how many ? 4 and 8
X T T M B E R S . 1 7
1 6 N U i l B E R S .
L E S S O N X .
I N C R E A S I N G A N D D I M I N I S H I N G B Y 5 .
1. How many slieep are 5 sheep and 5 slice]) ?
2. If we take 5 sheep from 10 sheep, Iiow many sliccp
will be left ? n from 10 leaves how many ?
J
/
J
^
p e a r s
braneli and 6 p e a r s o n ( l i e o i l i e r ;
' ' " l i o w m a n y p e a r s e n
both branches?
If we take 5 pears from 11 pears, how many pears
IVill be left ? G from 11 leaves how many?
5. How many are 5 and 7? How many are 7 and 5 ?
e. If we take 5 lilies from 12 lilies, how many lilies will
be left ? 5 from 12 leaves how many ? 7 from 13 leaves
how many ?
7. How many acorns are 5 acorns and 8 acorns ? 5 and
S are how many ? How many are 8 and 5 ?
8. 5 from 13 leaves how many ? 8 from" 13 leaves hoW
many?
9. Five and nine, how many ? 5 from 14, how many ?
L E S S O N X I ,
E X E R C I S E S .
1. How many are nine cherries and three cherries? 9
and 3 are liow many ? 3 and 0 are how many ?
2 . I f w e t a k e 9 b a l l s
from 11 balls, how many
balls are left? 9 from 11 leaves how many? 3 from 11
leaves how many ? 9 books from 11 books liow many?
3. Here is a flock
of swans; 4 are on
land, and 5 on the water: how many in all? How many
are five and four ?
Jf. 4 from 9 leaves
§ h o w m a n y ? 5
from 9 leaves how
many?
5. How many birds
are on the roof of the
bird-house ? How many
are flying in the air ?
How many are on the shelf? How many are there in all? 2, 4 and
3 are how many ? If the
birds on the shelf fly
N U M B E E S . 1 9 1 8
!
N U 3 I B E K S . L E S S O N " X I I , E X E R C I S E S .1. Ho^v' many boys are skating towards tJie riglit ? IIoW
many towards the left ? How many in all ?
2. If five leave tiie ice, how many will be left skating .
6 from 11 leaves how many ? 5 fi-om 11^ how many ?
Here is a book-rack con
taining books; some arc
stand-np, and some are lyingdoM'n. How many are standing
tlio loAvcr shelf? How many
are lying on tlie lower shelf?
^ow many hooks are tliere
alto-gether on tiie lower shelf?
H o w m a n v b o o k s a r c
standing on the upper shelf?
How many are lying down?
How many books are there
alto-„ gother on the up2ier shelf?
J. How many more are lying down on both shelves
than there are standing ?
L E S S O N X I I L
E X E R C I S E S .
1. Two acorns and two acorns arc
how many acorns ? How many lemons
arc 2 lemons and 2 lemons? How
many acorns are 2 times 2 acorns ? How many lemons arc 2 times 2 lemons ? IIow many are 2 times 2 ?
2, Two ai)ples from four apjiles, leaves how
^ many apiilcs ? 2 pears from 1 pears, leaves
liow many jiears ? How many times 2 apples are 1 apples ? How many times 2 pears in 1 pears? 2 in 4 how many times?
S. How many
sheep are 3 sheep and3 sheep? How
many sheep are 2 times 3 sheep? 3 times 2 sheep ?
there i
n 6 eg^s?
How many times 2 boys, in G boys ? How many times
2 in G ? 3 times 2 are bow many ?
B. How many marbles are four
marbles and four marbles? How
many marbles are 2 times 4 marbles ? How many are
2 times 4 ?
0. How many times 2 boats are there in 8 boats? How many times is 2 contained in 8 ?
7. If there arc 2 bunches of acorns, and each bunch con tains 5 acoras; how many acorns are there in both ? How
2 0
l î t n i B E E S . N U i l B E E S . 2 1
L E S S O N X I V.
WHITING HIGHER NUMBERS BY MEANS OF FIGURES.
Numbers from ninety-nine to onc-tlionsand are written
by three figures. The figure on the right, a.s we have
a ready learned stands for units, the second figure from
hundrrfs. '
are no units, the fignre on the right is 0.
i, " second fignre from tlio right
one hundred"^ numbers from one liundred to
one hundred and twenty hy means of ligures •
1 1 3 u l o s . 1 » 0 , 1 1 0 , i n , 1 1 2 ,
113, in, iio UG, m, 118, U9, m.
liundlr'n Fi" V"""
units; thus, 200.'° °
two hundrlLt'd ÎlV'',iite't' *^7
Write flirûû 1 1 ' imiidrcil and twelve.
sixhuudrS ; eT h:^,
"■^•^'^"i-Tsi^hÎdr'dÏdforr'
O n e 1 0 0 0 .
one number, a'^eCr'itteu 7'^°''' """
ber is read one thon ^
Write in fi7I ~ and eleven,
one thousand Tnd fi "'°"®and two hundred fifteen i
and five; and one thousand and ten.
L E S S O N X V .
W R I T I N G N U M B E R S B Y L E T T E R S .
We have learned two methods of writing numbers, one
by words, and another figures.
We will now learn a third method; the lessons in this
book are numbered by this method.In this method we use seven letters, I, V, X, L, C, D, M.
The following table shows how numbers are expressed
by these seven letters.
1 . I . 2 1 . . . X X L 2 . . I I . 2 2 . . . X X I L 3 . . I I I . 2 3 . . . X X I I L 4 . . I V . 2 4 . . . X X I V . 5 , . V . 2 5 . . . X X V . G . . V I . 2 6 . . . X X V L 7 . . . v n . 2 7 . . . X X V I L 8 . . V I I I . 2 8 . . . X X V I I I . 9 . . I X . 2 9 . . . X X I X . 1 0 . . X . 3 0 . . . X X X . 1 1 . . X I . 4 0 . . . X L . 1 2 . . x n . 5 0 . . L . 1 3 . , . x i n . 6 0 . . L X . 1 4 . . . X I V . 7 0 . . . L X X . 1 5 . . . X V . 8 0 . . . L X X X . I G . . . X V L 9 0 . . . X C . 1 7 . . . X V I I . 1 0 0 . . C . 1 8 . . . X V I I I . 5 0 0 . . D . 19 - . . X I X . 1 0 0 0 . . M . 2 0 . . . X X . 1 0 0 0 0 0 0 . . M .
K O T A T I O X A X D X U i l E R A T I O X . 2 3
"We liave three metliods of expressing
n u m b e r s .
i. By words—called the Word Method ; as one, two, three, four, etc.
i 2, By figures—called the Arabic Method ;
as 1, 2, 3, 4, etc.
3, By letters—called the Roman Method ; as I, Hj
III, IV, etc.Write all the numbers to fifteen, by each of the three
methods.
Ihe method of wnling numbers is called Notation.
The method of reading written numbers is called
N u m e r a t i o n .
RECAPITULATION AND DEFINITIONS.
1. A Unit is a single thing.
2. A Number is one or more things of the same
kind-3. A Figure is a character used to denote a number.
4. Notation is the method of writing numbers.
5. Numeration is the method of reading i^ ritteo
numbers.
6. The Word Notation is the method of writio^
numbers by means of words.
7. The Arabic Notation is the method of writing
numbers by means of ligures.
S. The Roman Notation is the method of writlne:
numbers by means of letters.
9, The ten figures iukeii separately are called digits.
Tlie naught is also called cipher or zero, and when con
sidered by itself, has no valitc. The other ligures arc called significant, because each has a value.
10. Arithmetic is the seicucc of numbers, and the art
of computing by them.
T E S T Q U E S T I O N S .
What is a unit? What is a number? Write five numbers;
write seven numbers. How many units in each? In how many
ways can wo express numbers ? What is the first method ? What
i s i t c a l l e d ? W h a t i s t h e s e c o n d ? W h a t i s i t c a l l e d ? W h a t i s t h e t h i r d ? i s i t c a l l e d ? W h a t i s n o t a t i o n ? W h a t i s n u m e r
a t i o n ? W h a t i s A r i t h m e t i c ?
O R D E R S O F U N I T S .
11. How many fingers and thumbs have you? Write the number by a word.
Will any one of the ten figures express this number? What is the greatest number that one figure will ex
press ?
One book is a single thing—a unit. If we make a bundle of ten books this handle is a single thing, and is, therefore, a unit. But these units arc not alike. One
unit is a single booh; the other is a single handle of
ten books. The single book is called a a nit of the first
order ; ihe single bundle of ten books is called a unit of
34
Mil'Jùïî IWI) NUMKUA'riojj.
iio,r mmj «n^fio ftooKe miilfo llie siiitflP lniiidlo?
one „.,it of (1.
ir WC n.ako ten bundles witli ten books in oaeli bundle
b l l L t ' t l T " - k i n g l - ' - g e
bundle, the large bundle trill also be 0,>e thmi-x imit.
H^V many small bundles in the large bundle?
TIow ma 's a unit of the third order.
Ï
0 -
- i t
O f
fig™ onr^efl T' of it write another
bgure one. Jibe first one is a unit of the first order Tlie
second one is a unit of the second orde.
s e c o n l o T r V ' " " ' ^ h e
TlItsVoneÏnr?®"''' '"^'t of the second.
iiiis last one e.xpresses a unit of 11,e third order
H o w i n a D v u i i i l : s o f r , - . o u i l i .
t h e t h i r d o r d e r ? o " "
a uTuInre'Sorfer "hd
order make one unit of the fourt^Z?
and runiUnlfrtL'4Ï"°'' "
TV^iito with ouc fiffiiro i-mrt , -i ^
of the second; ttro of the th'"! *''®
of the fifth; aiidtum oftl Si;.
me place, màtlfcllrgcTZl^r oUn7^''' fionrcs occupies t"'
T T 7 « î f ^ • n ' ^ o r d e r i s n i n e .
Write m figures, as one number three -i r tbe first
order, four units of the second ;der: tr units of the
J ^ O ' t . \ T I o r f A H O I f t O i l E R A T r n w . s &
H, of flift J l i î r t l o r c l e t - , f i v e n u l l s f » r H i e f u u r H i o r d e r , o n e m i l
lîi'fli ortloi', smit nine iuiiIh oP fllO OPilOl'.
'IMieso iiuinl)(>rs ni'o înirgerM.
1*2. An Integferifi a whole lUimhGl'.
Units of the First Order either stand alonCj or
occupy the right-hand place.
14. Units of the Second Order occupy the second
place from the right.
15. Units of the Third Order occupy the third
place.
Yon may now tell what place units of the foiirtli order
occnpy; nuits of the tiftli order, sixth order, seventh
order, eighth order, ninth order, tenth order.Units of the second order may be expressed without
a unit of the first order, by jnitting a cipher in the place
of the unit of the first order. Thus, 10.
The orders of units are indicated by the relative positions
of the figures.
Units of any order may he written without express
ing the units of other orders by putting ciphers in the
place of the other units. Thus, two units of the third
order are written 200; two units of the first order and
four units of the fifth order are written thus, 40002,
ciphers taking the place of the absent units.
AVrite two tens. What number have j^ou written ?
Write three tens, and at the right of it two units.
What number have you now ?
Write three units of the fourth order, and in the same line one unit of the third order, five units of the second
order, and nine units of the first. What number do they
express ?
2 6 NOTATION AND NUMERATION.
Units of the First Order express single things^ and
are called simply units.
Units of the Second Order express collections of
ten single things, and are therefore called tens.
Units of the Third Order express coUeclions of (en
tens or one hundred, and are therefore called hundreds.
Units of the Fourth Order express collections of
en mn reds oi one thousand, and are therefore called
thousands, as shown in the following
n u m e r a t i o n t a b l e . m •a £ a s W • c e -o n a n •a a a 2 œ 2 1 N o t e rnouJd. ready to Eh 3 1 3 S a W P o P 1 3 a 5 4 p 6 9
= I
a s ^ S o . a E l « M O V Î • a ■13 a c 3 C O S o o I t o T ) S g 3 O 'S < u b t 3 a 3 - " 7 8 0 7 a 4 O ' ^The tSchef^omied ni the sai»0
varv hT.t/n variety, and must be
™py hiB methods of teaching accordingly.
SIMPLE AND LOCAL VALUES.
lowing Presenting this suhject is the
fol-fail to com tlie wants of tliosc wlio
fa. to comprehend the first method.
andaYo^alVle.'™ Simple Value
standinfaTone^oIwhe^ "''of
O EH S P 4 5 3 N O TAT I O N A N D N U M E R AT I O N . 2 7The Local Value of a Fi^re is its value arising
from the place in which it stands. When 2 stands alone
or at the right hand, it denotes 2 units ; when it stands in
the second place from the right, it denotes 2 tens, Jis in
24; wliGii it stands in the third place from the right, as in
234, it denotes 2 hundreds. The local values of figuresincrease from the right to the left by a scale of tens.
T E S T Q U E S T I O N S .
What do units of the first order express? Units of the second order? Third? Fourth? How nuuiy values have figures? Wliat
is the aimple value of a figure ? What is the local value? Write
2 so na to show its simple value. Write 2 so as to denote 2 tens ; to denote 2 hundreds. How do the local values increase?
Places 0Î figures and orders of units ai'C counted from
right to left, but numbers are read from left to right.
E X A M P L E S I N N O T A T I O N A N D N U M E R A T I O N .
IT. J. Numbers from one to nine inclusive are
eollec-tiona of simple units, and are expressed by a single figure.
Numbers from ten to ninety-nine inclusive are composed of tens and units. Thus, twenty-seven is composed
of 2 tens and 7 units; forty-eight is composed of 4 tens
and 8 units.
"Write the following numbei*s by means of figures:
1. Twenty-five. 2. Forty-one. 3. Fifty-seven. 4. Eighty. 6. Ninety-one. 6. Thirt5'-two. 7. Fifty-eight. 8 . N i n e t e e n . 9 . T w e l v e . 10. Seventy-six. 11. Forty-two. 12. Ninety-four. IS. Eighty-two. llf. Sixty-five. 15. Ninety-nine. Read the following numbers : .
8 4 . 6 2 . 9 4 . 3 8 . 2 7 . 7 9 . 4 1 . 8 1 . 5 4 . 4 8 . 3 3 . 8 7 . 7 6 . 78.
2 Î 0 T A T I 0 N A N D N U M E R A T I O N . 2 9
2 8 N O T A T I O N A N D N U M E R A T I O N .
3» Numbers from one huDdred to nine liundred and
ninety-nine inclusive are composed of liimdreds, tens, and
units. Thus, the number four liundred and sixty-live is
composed of 4 hundreds, 6 fens, and 5 units; two hun
dred and three is composed of 2 hundreds, 0 tens, and 3
u n i t s .
E X A M P L E S .
Write the following numbers by means of figures:
7. Two hundred and sixty-five.
Three hundred and ninety.
3. Seven hundred and eight.
Eight liundred and fifty-seven.
5. Nine hundred and eiglity.
3. Four hundred and thirty-two.
7. Two hundred and six.3. One hundred and ninety-nine.
3. Five hundred and seventy-three.
iO. Six hundred and sixty-six.
4, To read a number of three figures, we name
number of hundreds and then read the tens and units, as
though they were by themselves. Thus, 512 is read,
nundred and twelve ; 874 is read, eight hundred au
seventy-four ; 209 is read, two hundred and nine.
Head the following numbers :
384. 7C3. 914. 571-994. 999. t h e 7. 713. 7 , 4 9 5 . 1 3 . 6 4 2 . 19. 2 . 3 . 806. 8 . 8 8 8 . U - 4 0 4 . 2 0 . 2 0 0 . 9 . 2 3 2 , 1 5 . 5 4 6 . 21. I 5 . 817. 10. 4 5 0 . 1 6 . 6 3 4 . 22. 728. 11 . 527. 1 7 . 9 7 8 . 23. 6. 827. 1 2 . 9 3 2 . 1 8 . 7 6 9 . 2Jh
18. Numbers are written by putting each oi'der in its
own place, and if any order is not mentioned, naught
must occupy its place.Write the following numbers by means of figures: 1. Six thousand four liundred and twenty-one.
2. One thousand three hundred and five.
3. Two thousand and ninety-six.
h. Eight thousand and one.
5. Four thousand and nine hundred.
G. Sixty-one thousand three liundred and seven. 7. Twenty-three thousand seven hundred and five. 8. Two hundred and fifty-nine thousand.
9. Three hundred, forty-eight thousand and thirteen. Read the following numbers, and name the nnniber of
units in each order:
(!■) (^.) (.5.) (4.) (d.)
1 4 2 3 2 0 1 0 2 3 7 0 5 6 7 8 3 2 2 5 8 0 1 3
(6.) (7.) (5.) (3.) (70.)
2 5 6 7 1 3 6 5 6 1 3 0 7 5 7 2 4 3 7 0 0 2 3 0
P E R I O D S O F F I G U R E S .
19. Numbers containing more than three figures are
separated into periods of throe figures each, beginning at the right. The loft-hand period may contain less than
three figures.
The first period, counting from the right, is called the jjeriod of units, the second is called the period of
thoii-sands ; the third is the period of millions, and so on, as shown in the following table :
NOTATION AND NUMERATION. ' t l . o i i s n i u l fl , u n i t a . 3 0 P e r i o d s
f « i II ?
N , .
. 1 !
s
Ï S B
,,,, » 1. 8 7 4 I ,J p ^ 2 10,
u m l " I T ' " ; ™ " l ' ï ' ' ! ! 8 7 0
Inllwlis, .llo vnlhons, 300 lltoiimmls, 210.
tbo '■° plûi'surc; Ihe units of
unions etc quadrillions, quinliUions,
sex-Every period except the left-hand one must he complete,
d "its m u" • T"'"" ^■■8"=' ''"t ">■
digits may be ciphers.
1^^74208436173 into periods. TcII iiow many
figures m each period, and read the number.
2 n A D . ° f ^ ' N " T I O N .
u e is a brief direction for performing work.
^ r u l e f o r n o t a t i o n
ihc fi,gi,ves 'of each
Places
T p ' O R N u m e r a t i o n .
into ^dght and separate the uiimher
the leftll ^ ^ figures each, until you reach
three figures. '"-"V leave one, two, or
it stood "rûn read each period as if
last figurZ' "ach period as you read its
NOTATION AND NUMERATION.
Write, point off, and read the following numbers:
2 3 i r » - 1 2 1 1 2 1 8 f t . ' ) S 7 3 4 0 i 4 fi ( i ( i i u ; i ( i . t ; i 6 i 2 fl i . 7 c n i ; i a i f i î i i i i o T ^ i i W i i a ? 7 4 ; ! 2 U ! I 2 8 4 . ' ) 1 7 1 2 7 0 0 8 0 1 ) 1 1 : 4 8 3 0 : 1 ( 1 4 l I K i a i K I 8 7 4 2 i a 0 5 1 4 M 3 1 T E S T Q U E S T 1 0 r < S .
How many periods in the lusi number 1 Name the poriotls in this number. TIow many figures in each period? CJive the rule for notation. Give the rule for numeration. Which period may have less than three figures?
C L A S S I F I C A T I O N O F N U M B E R S .
Connfc five. In this manner of counting, you mention
the numbers without a thought of any object. Num bers used ill this ivay are called abstract numbers.
Count tbe number of scholars in this class j the num
ber of maps on tbe wall; the number of books on my
desk ; the number of panes of glass in the window. Num bers used in connection with the objects counted, indi cating the numtcr of ohjects, are called denominate or
c o n c r e t e .
D E F I N I T I O N S .
21. An Abstract Number is one whose unit is not named ; as three, fwo, seven.
2 2 . A D e n o m i n a t e o r C o n c r e t e N u m b e r i s o n e
whoso unit is named ; as three f/irh, five ponnds seven
pennies, eifjhl pcnciU.
N a m e fi v e a b s t r a c t n u m b e r s . N a m e fi v e d e n o m i n a t e n u m b e r s .
W r i t e f o u r a b s t r a c t n u m b e r s . W r i t e fi v e d e n o m i n a t e n u m b e r s . What is ail abstract number? What is a denominate number
3 2 NOTATION AND NUilEUATION.
23. An Integer or Integral Number is one -n-Iiich
expresses one or more entire tilings.
24:. The Unit of any number is one of the eollecf
wliich constitutes the number.25. Similar or Like Numbers are those tliat lia.
tlie same kind of unit; as eiff/il dai/s and ten (la}fs,
yards seven feet, and six yards eleven feet.
26. Unlike Numbers are those that have different
kinds of units; as eight horses and five coivs, six pencils
and four knives, five feet and three days.
R E V I E W Q U E S T I O N S .
W]iat is a unit? Write a unit. What is a number? Write tw
numbers. Wliat is arithmetic? What is notation? What is minie°
ation ? Name the three methods of notation. The Arabic notation
employs how many figures ? Write tliem. What are these figures
called taken separately? Wlint is an abstract number? Give two examples. What is a concrete, or denominate number? Give three examples. What is an integer? What arc like numbers ? Give two examples. What are unlike mnnhers? Give two exanipieaWhat are the first four orders of units? Write four units of the third order. How many units of any order make one unit of the next higher order? What is the greatest number that can be ox-pressed by one figure? What is the greatest number that can bo expressed by two figures? Wliat by three figures? When there are four figures in a number, of wliat orders is it composed? Give an example and name the order of each figure. Wiat do units of
the first order denote? What do units of the second order denote?
What do units of the third order denote? What is the simple
value of a figure? \^niat is the local value of a figure? Give all
the general principles of notation and numeration. Give the rule
for notation. Give the rule for minicration. How are numbers
expressed in the Roman notation ? Wliat are figure.s ? Name all
the orders to trillions, beginning with units. >inmo the first four periods. Why is the second order called tens ? Why third called
h u n d r e d s ? W h y f o u r t h c a l l e d t h o u s a n d s ? ^ I /■ > -/ ■ 'j: -■ V A d d i t i o n . ii >
• , " a.;"
her mother
another doll., how
(lolls has slic then y
^V'ritc the mnuljop of
dolls tiiat Mary
H o w
m a n y
a r e
i
^
^
,„dl? "-'»'^">yarelandn„Kllandl?
I f a f a r m e r h a s p a s - ncigb-4 and 1 ? 9 and 1 ?Ô and 1 ? G and 1 ?
lOiiûdl? 12 ami 1?
Write the «umber of horses in the picture.
2 h o r s e s i n t h e t n r c a n d h i s
bor puts in 1 more;
liow many horses will
there be in the pas ture? IIow many are
2 and 1 ? 3 and 1 ?
T and 1 ? 8 and 1 ?
3 4 A D D I T I O N . A D D I T I O N . 3 6 ■'f 'V rx*' " IfLt.^onn'^ià S. How many
flîîiifs' arc 3 diwes ami 2 di/iics? How
many arc and 1, and 1 ? How many arc 3, 1. and 1?
How many arc 4,
J . a n d 1 V H o w
many arc 6, 1, and
1 ? How many are
0, l,and 1?
I How maDj ^ cantsi, 2 ccfUf, and 3 cciif?
How mimy are 3. 2, and 1 ? How many are 2, 3, and 1 ?
5. How mall}' «'I'c tlircc we/fj two me», and two
mM? How miiny are 3, 2 and 2 ?
5. How many murMrs arc i mnrbles, 3 marbles, and 2
marbles ? HoiV many are 3. 4 and 2 ?
6 The word (\ kftern, tlie word man lias 3
letters, and tlie ii'oi'tl 2 letters; liow many in
the tliree words? How many in tlic first two words?
The operation of finding how many dolls JIary
how many horses tiiere are in the pasture, etc., is eallcd
Addition, and the number tlius found is called tlic Snni
o r A m o u n t .
D E F I N I T t O N S .
The sum of two or more numbers is a number
which contains as many units as all the numbers taken
together.
38. Addition is the operation of finding the snm of
two or more niiiui^ers.
The numbers to be added must be similar, that is, tliey
must have the same unit. Three days and two days can be added, because they have the same uuit, one day ; but three daj's and two yards cannot Ijc added, because they
have not the same unit.
What is the sum or amount of two or more numbers? What is addition ? What are similar or like numbers 'i
S I G N S .
Jn the examples given a])ove, the word and is used to
denote the addition ; we generally denote it by this sign +, whicii is called plus, and when it is used between num
bers it shows that they arc to be added; thus G-l-3 + 2
are 11, means that the sum of six and three and tiro is
equal to eleven.
In place of the word arc, tlie sign = is used. It is
called tlie sign of cqvalHy, and is road equals or equal to ;
thus, 0 -p 4 -p 8 = 18, is read, six plus four plus eightecpials eighteen.
What is the sign of addition ? « Make it; What does it denote ?
What is the sign of equality ? Make it.
39. The sign of equality placed between numbers
or combinations of numbers, sliows that those at the left
hand are equal to those at the right.
The entire expression is called an equation ; thus,
G-P3 = 9, 7 — 2 = 5, 8x3-^2 = 14 — 6 + 4,
are equations.
30. An Arithmetical Equation is the expression of
equality between numbers or combinations of numbers. What is an equation ? Write an equation.
A D D I T I O N . E X E R C I S E S F O R O R A L V V O R J < . 2 + 2 = ? 3 + 2 = ? 4 + 2 = ? 5 + 2 = ? G + 2 = ? 3 + 3 = 4 + 3 = 5 + 3 = G + 3 = 7 + 3 = 4 + •} = ? 5 + 4 = y G + 4 = ? 7 + 4 = 't 8 + 4 = ? 31. Make and learn the following
A D D r T I O X T A n L 1 5 , 2 + 0 = 2 2 + 1 = 3 2 + 2 = 4 2 + 3 = 5 2 + 4 = G 2 + 5 = 7 2 + G = 8 2 + 7 = 0 2 + 8 = 1 0 2 + 9 = 1 1 2 + 10 = 12 3 + 0 = 3 4 + 0 = 4 5 + 0 = 5 3 + 1 = 4 4 + 1 = 5 + 1 = G 3 + 2 — 5 4 + 2 — n 5 + 2 — 3 + 3 = G 3 + 4 = 7 3 + 5 = 8 3 + G = 0 3 + 7 = 1 0 3+ 8 = 11 3+ 0 = 12 3 + 1 0 = 1 3 4 + 3 = 5 + 3 = S 4 + 4 = 8 5 + 4 = 0 4 + 5 = 0 5 + 5 = 1 0 4+ 0 = 10 4+ 7 = 11 4+ 8 = 12 4+ 9 = 13 4 + 10 = 14 5+ 0 = 11 5 + 7 = 1 2 5+ 8 = 13 5+ 9 = 14 5 + 10=15 G + G + G + G + G + G + G + G + 0 + 8 0 + 9 G + 10 = 7 + 7 + 7 + 7 + 7 + 7 + 7 + 7 + 7 + 7 + = 0 : 7 : 8 : 0 : 1 0 1 1 1 2 1 3 1 4 1 5 = 1 0 7 + ) 0 = 0 : 1 : O . / V -3 : 4 : 5 = 0 = : 9 : 1 0 1 1 1 2 1 3 1 4 1 5 1 0 1 7 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + = 8 = 0 = 10 = 11 : 1 2 : 1 3 : 1 4 : 1 5 ; 1 0 1 7 8 + 10 = 18 = 9 = 10 = 11 = 12 : 1 3 : 1 4 : 1 5 r l G : 1 7 1 8 0 + 10 = 19 0 + 9 + 9 + 9 + 0 + 0 + 9 + 0 + 9 + 0 +
Note.—The teaclier %vil] l)e atnply repaid for thorough drill in
all poJï-ible coinbinations of the digits. The pupil should be
thorui.ghly master of the table. Frequent exercises are required
to secure this and to break up tlio habit of couuting, which is
fatal to rapidity in addition.
A D D I T I O N . E X A M P L E S F O R O R A L W O R K . 3 7 7 . 2 + 7 = ? Z 4 + 9 = ? â. 8 + 0 = ? + 3 + 5 = ? 3 + 4 + 2 + 5 = ? jy,. 4 + 5 + 3 + 4 = ? 75. 7 + 3 + 2 + 5 = ? 0 . G. 7 . 8 . 0 + 5 = ? 8 + 4= ? 8 + 9 = ? 3 + 8 = ? IG. 0 + 3 + 2 + 4 = ? 17. 1 + 3 + 4 + 3 + 2 = ? 2S. 3 + 2 + 3 + 4 + 2 + 3 = ? 9. 9 + 2 = ? 10. 7 + 8 = ? 77. 0 + 9 = ? 72. 8 + 7 = ? 79.4 + 3 + 2 + 3 + 1 + 0=? 3 + 8 + 5 + 4 + G = ? 30. 5 + 2 + 3 + 1+2 + 4 + 5=? 7 + 9 + 8 + 5 = ? ^7. 8 + 5+4 + 2 + 3 + 1 + 7 = ? 9 + 3 + G+4 = ? 33. 9 + C + 3 + 4+G + 2 + 1 =? 4 + 9+8 + 7 = ?
Note.—The signs of interrogation in these and all similar exam ples indicate that the second members are to be supplied by the
pupil.
33. Count by twos from 2 to 50 ; thus, 2, 4, G, 8,10, etc. 34. Count by threes tVoiii 3 to 99; from 27 to 120.
25. Couut by fours from 0 to 100; from 105 to 161. 3G. Couut by Hyes from 1 to 106 ; from 200 to 250. 27. Count by sixes from 0 to 108 ; from 212 to 242. 38. Couut by sevens from 0 to 98; from 100 to 135.
39. Couut by eights from 0 to 104; from 150 to 174.
30. Couut by threes from 4 to 46 ; from 50 to 77. 31. Couut by fives from 0 to 100; from 200 to 250. S3. Couut by nines from 0 to 99; from 100 to 154.
(3^.^ Instead of writing numbers in a liorizontal line
with the sign between them, it is more convenient towrite them in columns witliout any sign, the sum being
written beneath. In the following examples, add fi-om the bottom upwai'd.
3 8 A D D I T I O N " . A D D I T I O N " . 3 9
e x a m p l e s f o r w r i t t e n w o r k .
S.) (.?.) 8 (d.) {5.) 6 8 9 8 3 8 3 2 8 4 9 9 7 7 5 (à.) {'•) C • i 3 5 1 0 (0.) f) 9 •i (-?.) 4 5 7J
_
_
—
—
1 9lu adding, name only the results of each
tlius. Example {IjJ/nreJen, Ji/hr/i,
( ' 2 ) , e l e v e n , 7 î i ? i e f e c n ^ . - p
Prove tlio ivork liy adclin.g from Hie top
the same sum is obtained, the \vork i» thong i
( M ) 1 4 C 9 4 5 3 5 3 3 4 9 2 2 9 (10.) (20.) (31.) (22.) (23.) 8 8 4 4 1 4 2 C 5 3 3 8 5 9 7 G 3 3 2 2 2 7 8 1 4 1 0 9 3 9 S 1 h 9 9 ] S f j 3 2 pies:
(IS.)
8 (16.) 5 5(17.)
1 G 3 3 5 2 9 9 4 5 7(20.)
(20.)
s 4 G 0 9 .5 7 i G 4 0 2 8 a 3 4 7 1 8 /. o i Q 0 7 4. andL'8. Find the sum of d, 4, 8, -h ^ p,
30. Pintl the sum of 0, ti, i. h '^" ,| iT.
TauclS.
!, 7, 9, 0, d, 0,
SO. Find the sum of 0, 8, 6. 7,
SJ. Find the sum of 8
Simple numbers may be added by the following
R U L E .
/. 11'rite the mnn hei'S so that ivniis of the same
o r d e r s h a l l s t a n d i n t h e s a m e c o l u m n .
II. Brain nt the right, add. each coJuDin, and write the sum, if less than ten, under the column.
III. IVhen the sum of any column exceeds
nine, set down the right-hand figure of the sumu n d e r t h e c o J u . m n . a n d . a d d . t h e n u m b e r i n d i
cated by the left-hand figure or figures to the
n e . v t c o l u m n .
II . Continue this operation till all the col
u m n s h a v e b e e . n a d d e d ; w r i t e t h e e n t i r e s u m o f t h e l a s t c o l u m n .
Phooc.—Add, the iin.mhers p'om the fop down
ward; if the result, is the same as the fii'st sum,
the worJc is presumed to he right.
E X A M P L E S . (^•) (2.) (3.) (d.) (5.) (6.) 2 4 3 0 4 5 8 1 1 6 4 2 3 2 2 3 3 3 7 2 1 2 3 3 5 1 G4 G 1 0 3 2 7 2 1 7 0 _72 7 0 5 4 3 8 4 8 Sum, 177 (7.) (S.) (9.) (10.) (77.). (12.) 3 0 1 8 8 4 4 4 8 8 8 0 4 7 4 1 0 2 3 7 GG CD 5 8 8 2 3 1 1 8 2 3 4 5 7 9 C l 1 4 2 9 3 2 5 7
4 0 A 1 > D I T 1 0 2 < . A P D I T I O X . 4 1 (13.) ( U - ) 2 1 3 127 days 4 1 0 213 days 7 2 9 418 days 6'uni, 1358 758 d<fys {1Ô.) 23G quarls 72 (J liar I.H 801 quaria {IG.) 8 1 2 1 7 4 7 0 2
111 the last four examples, which answers are abstract?
are concrete ?
Can 137 days be added to 330 quarts ? Wliy not ?
E X A M P L E S .
17. Find tlie sum of 135, 718, 04, 370, and 715.
18. Find the sum 7'^ years, 173 years, 00 years, 813
years, 43 years, and 197 years.
10. Add 345 quarts, 117 quarts, 123 qiiarts, 885 quarts,
04 quarts, and 543 quarts.
20. Add pounds,pounds,d,lh%poicnds,^,VtG
pounds, and 304 pounds.
21. Find the sum of 77, 213, 315, 421, and 007. Add tlie following gi-oups of numbers:
22. 818, 838, 40, 071, 304, 484, and 793.
23. 15, 812, 75, 717, 045, 720, and 347.
21^. 412 days, 817 days, 510 days, and 893 days.
AnnitE^nATiOxs.—In wlint follows, ft. stands for feet ; yds. for yard» ; lbs. tor jyoands ; in. for inches ; qts. for quarts ; and tbd sign s placed before a number stands for dollars.
When dollars and cents are expressed, we first write tlie
sign, then the number of dollars, then a point or period,
and then the number of cents: thus, 825.75, read
twenfy-five dollars and seventy-twenfy-five cents. If llie number of cents
to be written is less than ten, a cipher must be put in
the tons place ; thus. 810.0.5, read sixteen dollars and five
cents. If cents alone are to he written, we first make the
sign of dollars, then a 0, tlien the point, and then write the
miinlicr of cents; thus, 80.10. read 10 cents. Wlion dol
lars. cents, and mills are to be expressed, write the dollars
and cents as above, and the luills at the right of the cents.
Seven dollars, tweiit^'-five cents and eight mills are
t\rit-ten 87.258.
E X A M P L E S .
F i n d t h e s u m o f
35. 513//., 893//., 911//., 745//., and 14//.
20. 483 yds., 880 yds., 934 yds., 87 yds., and 994 yds.
27. 804 ttfs., 343 th\, 183 lbs., 94 lbs., and 14 lbs.
2S. 0(Ï7 in., 843 in., 918 in., 445 in., and SS7 in.
20. 8818, 8435, 888, 8413, 8807, 8983, and 871.
30. 2,314, 13,107, 310,030, 78,784, and 08,547.
SI. 83,308. 813,135, 841,410, and 818,876.32. 1,280 yd.s., 71,413 yds., 47,489 yds., and 9,297 yds.
33. 1,703/5.S., 4,389 tbs., \20,000 Uks., and 173,794/ôs.
34. 843//., 8,848//., 37,790//.. 153,407//.. and IS//.
35. $64, 8040, $0,833, 89,040, 8118,930, and 88,734.
SG. 614 in., 3,300 in., 89,705 in., 8,884 in., and 286 in.
D E F I N I T I O N S .
;î3. a Problem is a cpiestion requiring a solution.
34. A Solution is the operation of finding tlie re
quired answer.
P R A C T I C A L P R O B L E M S F O R O R A L W O R K .
35. 1. Jane's father gave her G peaches and her mother
gave her 4 more ; how many peaches did both give her?
4 2
A D D I T I O N .
_ Cliaries gave 10 cents for a Faber pencil No 2
0 cents for an Eagle pencil No. 2, an<l 8 cents fc." a
Stoddard penc. ; vvbat did the three pencils cost him ?
The head of a fish caught in Nmvark ]3av was
1 inches long; ])ow long was Hie li.sli ?
to 10«, from 9 fo 51 ; from 11 to 74.
George solved 10 i.rohleins in (l,e morning and 8 in
Hie evening; how mauv did lie solve in all ?
TOR Wniri-KN Work-.
A grocer has 3 liogslieads of sugar, of wliicli tiie
first weiglis 957 Ids., the second 1,0:^3 Ids., and Hie third
l,li9/^.y.; wiiat is Hie weight of them all ?
IvVPK.\XATION-.— Tlir' u-ciglit of iLLUsTit.vTroN.
tlio Wliole is Ofjnal to tlie .sum of the 957 Ids.
V fights of all ihc jiarts. ilenoe, we 1023 Ids
set down the separate weights aud intvo ??a d d t h e m .
Ans. 3150 Ids.
7. A mereiiant bougiit 4 ])ieces of cloth for $129,
G pieces of silk for $312, and 97 pieces of muslin for
$873; what did lie pay for them all? Ans. $1,314,
A gentleman bought a jiair of horses foi* $050, a set'
of harness for $100, and a carriage for $955: what did
they all cost ?0. A merchant bon""]it a liorsc for 112 dollars; after
O
keeping Iiim a short time, he sold him, and gained 25
dollars ; liow mnch did he receive for tiie horse?70. The mail route fi'om Albany to N^ew \ork is 144
miles, from Xew York to Philadelphia 90 miles, from
A D D I T I O N . 4 3
h
-I '
l^hihulelphia to Baltimore 08 miles, aud from Baltimore
to Washington City 38 miles; what is the distance from
iVlbany to Wasliiugton?
F O R O R A L W O R K .
BG, 1. A man bought a lirkin of butter for $9, a keg
of syrup for $(i, and 5 bushels of wheat for $7 ; how
much did he give for the wliole ?2. A boy gave to one of liis companions 7 apples, to
anoHior G, and to a third 8; bow many apples did bo
give away ? 7 + 0-1-8=:?3. A farmer bought a cow for $30, a sheep for $20, and
a calf for $10 ; bow mucli did he give for the whole ?Jf. Ill a young peach orchard Jane found 27 peaches on
one tree, on another 10, on another 8, aud on another 5;
how many peaches did she find ?
5, A lady bought a muff for $25, a boa for $15, and a
pair of gaiters for $10 ; how much did she pay for the
whole ? $25 + $15 + $10 = ?
F O R W R I T T E N W O R K .
1. A grocer sold 289 pounds, of sugar for $28, ten
barrels of flour for $108, and a quantity of pork for $879;what did he get for the whole ?
2. A person pays $950 for a lot of ground, on which ho
builds a house costing $5,430, a barn costing $980, and
then sells the whole so as to gain $914; what was his selling price ?
S. A farmer raises 673 hnshels of wlieat, 1,489 bushels
of corn, 67 bushels of barley, aud 1,682 bushels of oats;
how many bushels of grain does he raise in all ?
u A D D I T I O N .
4. A farmer sells his stock of cattle as follows: for his
oxen lie gets -^88:3. for his cows 81,:^Tî', lor iiis calves $413, and for his horses 8980; wiiat does lie get for I hem alii'
A gentleman builds a lioiise: his lot costs him
81,;3Ô4, the carpenter work costs 84,fhe masonry 8;^,II0, the painting and ]»apering 81.187. and (he niis-ccdlanooiis expenses amount to 81,v77 ; wliat is (he cost
of the whole ?
0. The distance from I3oston to Springlield is 0!) miles,
from Springlield to Albany 102 miles, from Albany to
Ivocliester 220 miles, from lioc-hestcr to liulfalo O.') miles,
and from Ilnlfalo to Chicago 018 miles; how far is it from
Boston to Chicago by (his route?
7. A manufacturer jiaid 88,820 for rent, 817,780 for mulerial, 847,880 for laboi", and then sold his goods so as to clear 811,827 ; what was (ho amount ol his sales?
<V. A speculator hongh( a house aiul lot for 1,904 dollars,
expended 384 dollars in rejiairing ami relltting the
proj)-erty, jiaid taxes and insurance amounling to 50 dollars,
and then sold them so as to gain 300 dollars ; wliat did he
get for the property?T E S T Q U E S T I O N S .
Wliut is addition? What is tlie answer in addition ailled?
is tlie sii^n «>f addition? What docs it mean? Make the sign of
c(jimlitv. Wliy Is it called sign of eijiiaiity? rbo an example in
wliich tljero is the- sign of eijiiality, antl .show linw it is used, (tive (he rule for writing miinbcr.s in suldition. (live the rule foradd-ingr and writing tlic rosnlts. Howdoyoii prove addition ?, Wliat
i.sTin equation? What an-, the mcmhers of an equation? What
niinihors can be added together V Make the sign for dollars. How nianv orders of units or places do cents riccupy? \Mint signis used between dollars and cents? Ho"' dollars, cents, and
mills written for adding? How many places do cents and mills
occupy ?
S U B T R A C T I O N .
1. One of the boys in the picture has 4 npplesy the other hoy has 3 apples ; how maay more apples has the
first hoy than the second ?
2. One of the girls has 4 roses, the other has 2 roses; how many more roses has one girl than the other?
S. On one side of the walk there are 5 frees, on the other side 2 frees ; how many more trees on one side than
o n t h e o t h e r ?
4. On one side of the house yon can see G ir>7}(lo?rs, on the other side 2 windotes ; how many more can you see
4 6
SUBTRACTION.
ô. A man had 13 coivs, and sold 3 of (liem; lion' many
had he left ?
In these live exanijdes we are required to tind/^e/f ?)iuc/i
greater one number is than another. Tlie niimber tlius
lound is the Difference between the two number.^, and
the process of linding it is called Subtraction.
dilïerence is also called a Remainder.
O E F I N I T I O N S .
Si. The Difference,, or Remainder, is a niimher
nhich shows how much gi-eater one of two numbers is
than the other.
•58. Subtraction is the operation of finding the dif
ference between two numbers.
In tliese examples, and in all examples of sulitraction
in Aritlimetic, the greater number is called the
end. The less iinmber is called the Subtrahend.
39. The Minuend is one of two numbers from vhmh
tlie other is to be suldraeted.
40. The Subtrahend is the number to be sulitracted.
^^Tiat is meant by tlie difference between two niunbersV
is subtraction ? What is the minuend ? What is the subtralicnci^
In each of the fo]lo\ving examples tell which is the minuend n"
whicli the subtrahend.
Read and work the following
e x a m p l• e s . E S . ^ p
A 5 less 4 = 1. G, G less 1 = hotv many ^^
i?. 7 less 6 = how many? 7. 6 less 4 = lio"' mmO ^
^ 8 . o l e s s 1 = I m f f
0. 4]GSs2 = ho^vrnnnr^
10. 3 less 1 = bo\r ma"l •
3, 4 less 1 = how many; Jt. 5 less 3 = how many?
5. 4 less 3 = Jiow many ?
11. 8 less 3 are how many ? 8 less 4 ? 8 less 5 ?
S U B T R A C T I O N . 4 7
Instead of the word /ess between two numbers whose
difference is required, this sign — is used. It is called
the Minus Sign, or Sign of Subtraction.
41. Minus denotes aud wheu placed between two
numbers it sliows that the second is to be subtracted
from the first. Thus, 5 — 3 shows that 3 is to be taken
from 5.
Tlie Parenthesis, ( h "s used to show that the
expression (Miclosed by it is to he treated as a single
number. Thus, S - (3 + 2) shows that the sum of 3 and
2 is to tie subtracted from 8.
Keud and work the following
E X A M P L E S .
1, 0-1=5. 7. 5-3 = ? IS. 4-3 = ?
11 0—3 = ? i J . 5 — 1 = ? 1 6 . G — 4 = ? J 7 . 0 — 5 = ? J 8 . 9 — 0 = ? î 4 _ 1 — 9 8 . 4 — 3 = ?I
. - o l y
2 _ i = ?
. 5 - 3 = ? 1 0 . r , - 2 = ? Ô. 3 — 1=? 1^- 5—4 = ? 6 . 8 - 4 = ? 1 3 . 7 - 3 = ?AO. 10—3 = how many? 10—5? 10—7? 10 — 8?
13—1 = how many? 13—3? 13—3? 13 — 0.-'SA How many are 13 —1 ? 13—3? 13—3? 13—4?
SS. How many are 14—1 ? 14—3? 14—3? 14—4."'
Write the sign of subtraction. Wbat is it called ? What doesit mean ? When used between two numbers what does it show?
Read 8 — 3 = 5, and tell which is the miimend, which the subtrahend, and which the remainder ?
Can you always tell, if the sign of subtraction is
used, which is the minuend, and which is the sub
wo iu iC {XTJ. îo uo § ST q i ' u0 0A^ 9q sn nim gn q i -oin ozi jo q 1Î III u oip LMi jq us .lo j s. io qu iu u Sa pi. iA V jo p i3 0:}su i • f-f ' 6 8 o ; Z ZI «1 0.T J ^9 X q x wq Bu niu ip SI ipuo uoqAi. '2 04 98 luo.ij s-ioquinii oq'^ o.iu '09 •Si o ; 0 01 Ï 0^ 8i «1 0.I J Î g Âq q oiî o Sn iq siu uu ip 0; gs U1 0.1 J si oq iu iu i 9 1(4 o uiin i j omu ua o racs oi[ !^ iq ^ c A q po qs in uu ip Q i S po qs iu iu iip
OX
i e
p
aq
sm
ira
ip
e
x ^
= e
iC
q p
aq
su
utn
ip
o
g '1
9
i = (8 + g + l) —8T '8 Z 9 -9 81 + -g 9) 8- =( 6 i = 8 - 8 - 9 - OK '9 9 ri = 8 — t — e — 81 t9 ^ = j; — Ç — g — eg '9 9 've ex — — g g — = f ■ ^ = g - g - g - gx 'T 96 =
(X +
9 +
i)-G
X '0
9
^=(o
+
f+s)
—Qg
'O
o
(i = (s + 8) — e x 'i z é = (f + e) — o g '9 ^ d= e— 9 + i + 6 '9 ^ i = f -8 + g + gX 6 = 9 — 8 + 0 '9 v i t — 8 + i 't o <J = 8 - 9 + 9 'ÏZ *6 — — 9X 'OZ *8 — é — 81 '0 1 '9 = ( i — gx '6 1 •9 = gx •G '8 = 6 -fl '8 T •8 ' — • ^ — 01 '8 + 6 - 81 '1 1 — — e ■I • O X = 6 — ex '9 T •g — 6 9 '9 i = 8 - ex '9 1 — t -XX '9 i = I - 91 t f I = G — gx t i = 9 — ex '9 1 i — 9 — 6 •9 i = 0 — x-x '9 1 i — 8 — 8 V i = 8 — gx 'J I 6 — t — I ' I •M HO AS iv ao s a T d W v x a 6 t •K Oi io YH xa ns d^' ji JO ' uio in iiis 9q jsu iu -Jo pu iu uia j o qx pn u 'p iio qu i^c iu s o qx 'p iio nu iu i a ifx •0 }0 I \ soAi rai g mojj no qu j jo qiu uu ^ uq AV = smj ^ oSi mqo 'u iu Sv ^ -o jo *1 =1 —g : Kiu p lU Jo j oi[ ) aS um f.) -ji q;! A\ au tp iu iq saiu oo oq j uio qjs oq ^ iq im po }«a do a aq p pio iii^ a iq wi siq x—•5 IJ0 >; 6 » 61 6 } > 81 : G » » i l 6 » ? 9 1 6 ? ? 91 8 » 81 :8 J ) i l 8 » ? 91 8 » ? 91 8 »? fl I ) ) i t : i » 91 I-?? 91 i ? ? Ï-T i ? » 81 9 } } 91 9 » ? 9T 9 ? ? fl 9 ? ? 81 9 ? ? gx 9 S ) 9X j j fl 0 ? ? 81 Ç ? ? gx 0 )» IX f fl f » ? SX f ?» gx X -? » n f ? » O X 8 } ) £X £ } } gx, £ ? ? IX 8 ?? 01 . £ »> G g } } g l g ? » XX g *» 01 A / 6 ? . G I g ? » 8 X SM A lU j X X X S .)A l ja[ O X I S , )A l !J( 0 ! I S J Al ! .tI 8 ' I W AI 1 3| lU 10
. 1J
O X
I U
T O
J J;
G
lU
O Jf
g
i iio
. ij i
! m
o .ij
9
G ? » t l G ? » 8X G ? ? g l G -•? XI 6 ? ? 01 8 ? » 81 8 • ? g l 8 î ? XI 8 ? » 01 8 ?? G i ?» gx i ? » XX i ? ? 01 f ? » G i ? ? 8 9 j » I I 9 ? ' OX 9 ?• G 9 ? ? 8 ' 9 •» i ç ? • 01 0 ? » G 9 ? » 8 e ? » g ? ? 9 t î ' G f ». 8 X -?? i f ? ? 9 t ? ? ç 8 ? » S 8 ? » £ ? ? 9 8 ?» e 8 ? ? f g »> i g ?» 9 (■ .. . e ' g . . f g ? ? 8 ; T S JA II-lj 0 [ S'J AH Ol e X S.) AL '.) | t ' I S OA UJI o UI OJ J e II IO JJ f UI OJ J X; lU OJ J g XS.1 AIMI g UIO JJ X • a 'I a V J, 01 X 0 V a X fi Û s o 5U IAV O[[O J O T{X U.IP O[ p up OXJ J^ •N oi io va xa ns5 0 S U B T R A C T I O i î . S U B T R A C T I O N . 5 1
convenient to write tlie subtraJiend under the niinuend,
f)lacing the remainder beneath : iîius,
1 2 M h i u e t i d . 7 S u b t r a h e n d . 5 I t e m n i n d e r.
In this manner work the following
E X A M P L E S . I G !) ' 7 1 8 4 1 2 3 1 0 7 (^■) (0.) (r-) (S.) (9.) (10. 9 8 1 2 2 0 1 6 21 4 5 3 9 7
lo yjrove the add the remainder to the
snbtra-hendj and if the sum is equal to the minuend the work is
presumed to be right.
Perform and prove the following
E X A M P L E S .
{IQ,) (17,) (IS.) (19.)
From 0 7 11 15 8 13 17 10 U
Subtract 2 J J j j 5 9 g J
{^0.) (21.) (22.) (28) (24.) (2J.) (^6'.)
I T ' $ 1 3 1 6 w . l O l b s . 1 5
FrolV InV
10?is. 9yas. n/t. liin. 17 «5 33 clays.
" ~ ifL ijn. _8 $9 21
clayf-^^Caa 9 pou,Ids be subtracted fro.u li y«r&TVhy
E x e r c i s e s f o r " W r i t t e x W o r k .
45. When the ptfuves of the .^ubtrohend
are equal to or less than the
eorrespond-iuq fiqnres of the m innend.
E x P U A X . V T i o x . — w r i t e t h e s u b - i l l u p t o a t i c k .
tnihencl under the ininueud so tliat From 705
u n i t s s t a n d u n d e r u n i t s , t e n s u n d e r s u b t r a c t 5 3 2
tens, and linndrcds under iiundreds. Wo begin at the riglit and subtract 2 u n i t A f r o m 5 a n d w r i t e t h e
remainder, 3 unitK, beneath. We then subtract 3 tens from 0 tens and write tlie result, (i trnn, in the column of tens. Then wo
aiib-tnict .1 htnidrc'd-'< from 7 hundreds, and write the difference in the
c o l u m n o f h u n d r e d s .
P r o o f .
Remaiudcr, 263
Add the subtrahend and remainder, and if the sum equals the minuend
the work is right.
532 Siiblrahcncl. 263 Rcniainflcr. 795 = the Miuncnd. S e c o n d M e t u o d o f P r o o f .
Subtract the reraaiiuler from the 705 Minnend.
263 Remainder.
532 SiibtraUciid.
lu the same manner work and prove the following
minuend, and if the result equals the subtrahend the work is right.
E X A M P L E S . I ' r o m 8 7 S u b t r a c t 3 4
