Arithmetic tables for children,
Lausanne, 1835
Arithmetic or
arithmetics (from
the
Greek word
αριθμός =
number) is the oldest and most elementary branch of
mathematics, used by almost everyone, for tasks
ranging from simple day-to-day counting to advanced
science and
business
calculations. It involves the study of
quantity, especially as the result of combining
numbers. In common usage, it refers to the
simpler properties when using the traditional
operations of
addition,
subtraction,
multiplication and
division with smaller values of
numbers. Professional
mathematicians
sometimes use the term
(higher) arithmetic when referring
to more advanced results related to
number
theory, but this should not be confused with
elementary arithmetic.
History
The
prehistory of arithmetic is limited to a very small number of small
artifacts indicating a clear conception of addition and
subtraction, the best-known being the Ishango bone from central
Africa, dating from somewhere between 20,000 and 18,000
BC.
The earliest written records indicate the
Egyptians and
Babylonians used all the elementary
arithmetic operations as early as 2000 BC. These artifacts do not
always reveal the specific process used for solving problems, but
the characteristics of the particular
numeral system strongly influence the
complexity of the methods. The hieroglyphic system for
Egyptian numerals, like the later
Roman numerals, descended from
tally marks used for counting. In both cases,
this origin resulted in values which used a
decimal base but did not include
positional notation. Although addition
was generally straightforward, multiplication in
Roman arithmetic required the assistance of
a
counting board to obtain the
results.
The early number systems which included positional notation were
not decimal, including the
sexagesimal
system for
Babylonian numerals
and the
vigesimal system which defined
Maya numerals. Because of this
place-value concept, the ability to reuse the same digits for
different values contributed to simpler and more efficient methods
of calculation.
The continuous historical development of modern arithmetic starts
with the
Hellenistic
civilization of ancient Greece, although it originated much
later than the Babylonian and Egyptian examples. Prior to the works
of
Euclid around 300 BC,
Greek studies in mathematics overlapped
with philosophical and mystical beliefs. For example,
Nicomachus summarized the viewpoint of the
earlier
Pythagorean approach to
numbers, and their relationships to each other, in his
Introduction to
Arithmetic.
Greek numerals, derived from the
hieratic Egyptian system, also lacked positional notation, and
therefore imposed the same complexity on the basic operations of
arithmetic. For example, the ancient mathematician
Archimedes devoted his entire work
The Sand Reckoner merely to devising
a notation for a certain large integer.
The gradual development of
Arabic
numerals independently devised the place-value concept and
positional notation which corrected this omission, combining it
with a decimal base and the use of a digit representing
zero. This approach eventually replaced or
superseded all other systems. In the early 6th century AD, the
Indian mathematician
Aryabhata
incorporated an existing version of this system in his work, and
experimented with different notations. In the 7th century,
Brahmagupta established the use of zero as a
separate number and determined the results for multiplication,
division, addition and subtraction of zero and all other numbers,
except for the result of
division by
zero. His contemporary, the
Syriac bishop Severus Sebokht described
the excellence of this system as "valuable methods of calculation
which surpass description". The Arabs also learned this new method
and called it
hesab.
Although the
Codex Vigilanus
described an early form of Arabic numerals (omitting zero) by 976
AD,
Fibonacci was primarily responsible
for spreading their use throughout Europe after the publication of
his book
Liber Abaci in 1202.
He considered the significance of this "new" representation of
numbers, which he styled the "Method of the Indians" (Latin
Modus Indorum), so fundamental that all related
mathematical foundations, including the results of
Pythagoras and the
algorism describing the methods for performing
actual calculations, were "almost a mistake" in comparison.
In the
Middle Ages, arithmetic was one
of the seven
liberal arts taught in
universities.
The flourishing of
algebra in the
medieval Islamic world and
in
Renaissance Europe was an outgrowth of the enormous
simplification of
computation through
decimal notation.
Examples of calculating tools include
slide
rules (for multiplication, division, and trigonometry)and
nomographs.
Decimal arithmetic
Decimal notation constructs all
real numbers from the basic digits, the first ten non-negative
integers 0,1,2,...,9. A decimal numeral consists of a sequence of
these basic digits, with the "denomination" of each digit depending
on its
position with respect to the decimal point: for
example, 507.36 denotes 5 hundreds (10
^{2}), plus 0 tens
(10
^{1}), plus 7 units (10
^{0}), plus 3 tenths
(10
^{-1}) plus 6 hundredths (10
^{-2}). An essential
part of this notation (and a major stumbling block in achieving it)
was conceiving of zero as a number comparable to the other basic
digits.
Algorism comprises all of the rules of performing arithmetic
computations using a decimal system for representing numbers in
which numbers written using ten symbols having the values 0 through
9 are combined using a place-value system (positional notation),
where each symbol has ten times the weight of the one to its
right.This notation allows the addition of arbitrary numbers by
adding the digits in each place, which is accomplished with a 10 x
10 addition table. (A sum of digits which exceeds 9 must have its
10-digit carried to the next place leftward.) One can make a
similar algorithm for multiplying arbitrary numbers because the set
of denominations {...,10²,10,1,10
^{-1},...} is closed under
multiplication. Subtraction and division are achieved by similar,
though more complicated algorithms.
Arithmetic operations
The basic arithmetic operations are addition, subtraction,
multiplication and division, although this subject also includes
more advanced operations, such as manipulations of
percentages,
square
roots,
exponentiation, and
logarithmic functions. Arithmetic is
performed according to an
order of
operations. Any set of objects upon which all four operations
of arithmetic can be performed (except division by zero), and
wherein these four operations obey the usual laws, is called a
field.
Addition (+)
Addition is the basic operation of arithmetic. In its simplest
form, addition combines two numbers, the
addends or
terms, into a single
number, the
sum of the numbers.
Adding more than two numbers can be viewed as repeated addition;
this procedure is known as
summation and
includes ways to add infinitely many numbers in an
infinite series; repeated addition of
the number
one is the most basic form of
counting.
Addition is
commutative and
associative so the order in which the terms are
added does not matter. The
identity
element of addition (the
additive
identity) is 0, that is, adding zero to any number will yield
that same number. Also, the
inverse
element of addition (the
additive
inverse) is the opposite of any number, that is, adding the
opposite of any number to the number itself will yield the additive
identity, 0. For example, the opposite of 7 is -7, so 7 + (-7) =
0.
Addition can be given geometrically as follows:
- If a and b are the lengths of two sticks,
then if we place the sticks one after the other, the length of the
stick thus formed will be a + b.
Subtraction (−)
Subtraction is the opposite of addition. Subtraction finds the
difference between two numbers, the
minuend minus
the
subtrahend. If the minuend is larger than the
subtrahend, the difference will be positive; if the minuend is
smaller than the subtrahend, the difference will be negative; and
if they are equal, the difference will be zero.
Subtraction is neither commutative nor associative. For that
reason, it is often helpful to look at subtraction as addition of
the minuend and the opposite of the subtrahend, that is
a −
b =
a + (−
b).
When written as a sum, all the properties of addition hold.
There are several methods for calculating results, some of which
are particularly advantageous to machine calculation. For example,
digital computers employ the method of
two's complement. Of great importance is
the counting up method by which change is made. Suppose an amount P
is given to pay the required amount Q, with P greater than Q.
Rather than performing the subtraction P-Q and counting out that
amount in change, money is counted out starting at Q and continuing
until reaching P. Curiously, although the amount counted out must
equal the result of the subtraction P-Q, the subtraction was never
really done and the value of P-Q might still be unknown to the
change-maker.
See also:
Method of
complements
Multiplication (×, ·, or *)
Multiplication is the second basic operation of arithmetic.
Multiplication also combines two numbers into a single number, the
product. The two original numbers are called the
multiplier and the
multiplicand, sometimes both
simply called
factors.
Multiplication is best viewed as a scaling operation. If the real
numbers are imagined as lying in a line, multiplication by a
number, say x, greater than 1 is the same as stretching everything
away from zero uniformly, in such a way that the number 1 itself is
stretched to where x was. Similarly, multiplying by a number less
than 1 can be imagined as squeezing towards zero. (Again, in such a
way that 1 goes to the multiplicand.)
Multiplication is commutative and associative; further it is
distributive over addition and
subtraction. The
multiplicative
identity is 1, that is, multiplying any number by 1 will yield
that same number. Also, the
multiplicative inverse is the
reciprocal of any number
(except zero; zero is the only number without a multiplicative
inverse), that is, multiplying the reciprocal of any number by the
number itself will yield the multiplicative identity.
Division (÷ or /)
Division is essentially the opposite of multiplication. Division
finds the
quotient of two numbers, the
dividend
divided by the
divisor. Any dividend
divided by zero is undefined. For positive
numbers, if the dividend is larger than the divisor, the quotient
will be greater than one, otherwise it will be less than one (a
similar rule applies for negative numbers). The quotient multiplied
by the divisor always yields the dividend.
Division is neither commutative nor associative. As it is helpful
to look at subtraction as addition, it is helpful to look at
division as multiplication of the dividend times the
reciprocal of the divisor, that is
a ÷
b =
a ×
^{1}⁄
_{b}.
When written as a product, it will obey all the properties of
multiplication.
Examples
Multiplication table
× |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
11 |
12 |
13 |
14 |
15 |
16 |
17 |
18 |
19 |
20 |
21 |
22 |
23 |
24 |
25 |
1 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
11 |
12 |
13 |
14 |
15 |
16 |
17 |
18 |
19 |
20 |
21 |
22 |
23 |
24 |
25 |
2 |
2 |
4 |
6 |
8 |
10 |
12 |
14 |
16 |
18 |
20 |
22 |
24 |
26 |
28 |
30 |
32 |
34 |
36 |
38 |
40 |
42 |
44 |
46 |
48 |
50 |
3 |
3 |
6 |
9 |
12 |
15 |
18 |
21 |
24 |
27 |
30 |
33 |
36 |
39 |
42 |
45 |
48 |
51 |
54 |
57 |
60 |
63 |
66 |
69 |
72 |
75 |
4 |
4 |
8 |
12 |
16 |
20 |
24 |
28 |
32 |
36 |
40 |
44 |
48 |
52 |
56 |
60 |
64 |
68 |
72 |
76 |
80 |
84 |
88 |
92 |
96 |
100 |
5 |
5 |
10 |
15 |
20 |
25 |
30 |
35 |
40 |
45 |
50 |
55 |
60 |
65 |
70 |
75 |
80 |
85 |
90 |
95 |
100 |
105 |
110 |
115 |
120 |
125 |
6 |
6 |
12 |
18 |
24 |
30 |
36 |
42 |
48 |
54 |
60 |
66 |
72 |
78 |
84 |
90 |
96 |
102 |
108 |
114 |
120 |
126 |
132 |
138 |
144 |
150 |
7 |
7 |
14 |
21 |
28 |
35 |
42 |
49 |
56 |
63 |
70 |
77 |
84 |
91 |
98 |
105 |
112 |
119 |
126 |
133 |
140 |
147 |
154 |
161 |
168 |
175 |
8 |
8 |
16 |
24 |
32 |
40 |
48 |
56 |
64 |
72 |
80 |
88 |
96 |
104 |
112 |
120 |
128 |
136 |
144 |
152 |
160 |
168 |
176 |
184 |
192 |
200 |
9 |
9 |
18 |
27 |
36 |
45 |
54 |
63 |
72 |
81 |
90 |
99 |
108 |
117 |
126 |
135 |
144 |
153 |
162 |
171 |
180 |
189 |
198 |
207 |
216 |
225 |
10 |
10 |
20 |
30 |
40 |
50 |
60 |
70 |
80 |
90 |
100 |
110 |
120 |
130 |
140 |
150 |
160 |
170 |
180 |
190 |
200 |
210 |
220 |
230 |
240 |
250 |
11 |
11 |
22 |
33 |
44 |
55 |
66 |
77 |
88 |
99 |
110 |
121 |
132 |
143 |
154 |
165 |
176 |
187 |
198 |
209 |
220 |
231 |
242 |
253 |
264 |
275 |
12 |
12 |
24 |
36 |
48 |
60 |
72 |
84 |
96 |
108 |
120 |
132 |
144 |
156 |
168 |
180 |
192 |
204 |
216 |
228 |
240 |
252 |
264 |
276 |
288 |
300 |
13 |
13 |
26 |
39 |
52 |
65 |
78 |
91 |
104 |
117 |
130 |
143 |
156 |
169 |
182 |
195 |
208 |
221 |
234 |
247 |
260 |
273 |
286 |
299 |
312 |
325 |
14 |
14 |
28 |
42 |
56 |
70 |
84 |
98 |
112 |
126 |
140 |
154 |
168 |
182 |
196 |
210 |
224 |
238 |
252 |
266 |
280 |
294 |
308 |
322 |
336 |
350 |
15 |
15 |
30 |
45 |
60 |
75 |
90 |
105 |
120 |
135 |
150 |
165 |
180 |
195 |
210 |
225 |
240 |
255 |
270 |
285 |
300 |
315 |
330 |
345 |
360 |
375 |
16 |
16 |
32 |
48 |
64 |
80 |
96 |
112 |
128 |
144 |
160 |
176 |
192 |
208 |
224 |
240 |
256 |
272 |
288 |
304 |
320 |
336 |
352 |
368 |
384 |
400 |
17 |
17 |
34 |
51 |
68 |
85 |
102 |
119 |
136 |
153 |
170 |
187 |
204 |
221 |
238 |
255 |
272 |
289 |
306 |
323 |
340 |
357 |
374 |
391 |
408 |
425 |
18 |
18 |
36 |
54 |
72 |
90 |
108 |
126 |
144 |
162 |
180 |
198 |
216 |
234 |
252 |
270 |
288 |
306 |
324 |
342 |
360 |
378 |
396 |
414 |
432 |
450 |
19 |
19 |
38 |
57 |
76 |
95 |
114 |
133 |
152 |
171 |
190 |
209 |
228 |
247 |
266 |
285 |
304 |
323 |
342 |
361 |
380 |
399 |
418 |
437 |
456 |
475 |
20 |
20 |
40 |
60 |
80 |
100 |
120 |
140 |
160 |
180 |
200 |
220 |
240 |
260 |
280 |
300 |
320 |
340 |
360 |
380 |
400 |
420 |
440 |
460 |
480 |
500 |
21 |
21 |
42 |
63 |
84 |
105 |
126 |
147 |
168 |
189 |
210 |
231 |
252 |
273 |
294 |
315 |
336 |
357 |
378 |
399 |
420 |
441 |
462 |
483 |
504 |
525 |
22 |
22 |
44 |
66 |
88 |
110 |
132 |
154 |
176 |
198 |
220 |
242 |
264 |
286 |
308 |
330 |
352 |
374 |
396 |
418 |
440 |
462 |
484 |
506 |
528 |
550 |
23 |
23 |
46 |
69 |
92 |
115 |
138 |
161 |
184 |
207 |
230 |
253 |
276 |
299 |
322 |
345 |
368 |
391 |
414 |
437 |
460 |
483 |
506 |
529 |
552 |
575 |
24 |
24 |
48 |
72 |
96 |
120 |
144 |
168 |
192 |
216 |
240 |
264 |
288 |
312 |
336 |
360 |
384 |
408 |
432 |
456 |
480 |
504 |
528 |
552 |
576 |
600 |
25 |
25 |
50 |
75 |
100 |
125 |
150 |
175 |
200 |
225 |
250 |
275 |
300 |
325 |
350 |
375 |
400 |
425 |
450 |
475 |
500 |
525 |
550 |
575 |
600 |
625 |
Number theory
The term
arithmetic is also used to refer to number
theory. This includes the properties of integers related to
primality,
divisibility, and the
solution of equations in integers, as
well as modern research which is an outgrowth of this study. It is
in this context that one runs across the
fundamental theorem of
arithmetic and
arithmetic
functions.
A Course in Arithmetic by
Jean-Pierre Serre reflects this usage, as
do such phrases as
first order arithmetic or
arithmetical algebraic geometry. Number theory is also
referred to as
the higher arithmetic, as in the title of
Harold Davenport's book on the
subject.
Arithmetic in education
Primary education in mathematics
often places a strong focus on algorithms for the arithmetic of
natural numbers,
integers,
rational
numbers (
vulgar fractions), and
real numbers (using the decimal
place-value system). This study is sometimes known as
algorism.
The difficulty and unmotivated appearance of these algorithms has
long led educators to question this curriculum, advocating the
early teaching of more central and intuitive mathematical ideas.
One notable movement in this direction was the
New Math of the 1960s and 1970s, which attempted to
teach arithmetic in the spirit of axiomatic development from set
theory, an echo of the prevailing trend in higher
mathematics.
Since the introduction of the electronic
calculator, which can perform the algorithms far
more efficiently than humans, an influential school of educators
has argued that mechanical mastery of the standard arithmetic
algorithms is no longer necessary. In their view, the first years
of school mathematics could be more profitably spent on
understanding higher-level ideas about what numbers are used for
and relationships among number, quantity, measurement, and so on.
However, most research
mathematicians
still consider mastery of the manual algorithms to be a necessary
foundation for the study of algebra and computer science. This
controversy was one element of the "
math
wars" over California's primary school curriculum in the
1990s.
See also
Related topics
Footnotes
- Davenport, Harold, The Higher
Arithmetic: An Introduction to the Theory of Numbers (7th
ed.), Cambridge University Press, Cambridge, UK, 1999, ISBN
0-521-63446-6
- Mathematically Correct: Glossary of Terms
- Education World - Curriculum: MATH WARS!
References
- Cunnington, Susan, The Story of Arithmetic: A Short History
of Its Origin and Development, Swan Sonnenschein, London,
1904
- Dickson, Leonard Eugene,
History of the Theory of Numbers (3 volumes), reprints:
Carnegie Institute of Washington, Washington, 1932; Chelsea, New
York, 1952, 1966
- Euler, Leonhard, Elements of Algebra, Tarquin Press, 2007
- Fine, Henry Burchard
(1858–1928), The Number System of Algebra Treated Theoretically
and Historically, Leach, Shewell & Sanborn, Boston,
1891
- Karpinski, Louis Charles
(1878–1956), The History of Arithmetic, Rand McNally,
Chicago, 1925; reprint: Russell & Russell, New York, 1965
- Ore, Øystein, Number Theory
and Its History, McGraw-Hill, New York, 1948
- Weil, André, Number Theory:
An Approach through History, Birkhauser, Boston, 1984;
reviewed: Mathematical Reviews
85c:01004
External links