Arithmetic

Arithmetic or arithmetics (from the Greek word αριθμός, meaning "number") is the oldest and most fundamental branch of mathematics. It is used by almost everyone, for tasks ranging from simple daily counting to advanced science and business calculations. Some have called it the "science of numbers." Our knowledge of and skill in using arithmetic operations is part of our definition of literacy.

In common usage, arithmetic refers to a branch of mathematics that records elementary properties of certain operations on numbers. Professional mathematicians sometimes use the term higher arithmetic[1] as a synonym for number theory, but this should not be confused with elementary arithmetic.

The traditional arithmetic operations are addition, subtraction, multiplication, and division, although more advanced operations (such as manipulations of percentages, square root, exponentiation, and logarithmic functions) are also sometimes included in this subject. 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 is the simplest form and combines two numbers, such as 1+1=2. This can be used for simple tasks such as adding grocery amounts or the money in one's pocket.

Subtraction is the process of finding the difference between two numbered quantities, such as 5-3=2. This process can be used in tasks such as calculating the balance in a bank account after withdrawing some cash.

Multiplication consists of adding a number (the multiplicand) to itself a certain number of times. For example, adding 3 to itself 5 times gives 15, which is written as 3x5=15.

Division is the inverse of multiplication. It consists of dividing a number into groups of equal amounts. For example, to divide the number 20 into several groups, each containing 4 units, one would write 20/4 (or 20÷4), which would yield 5. In other words, 20 can be divided into 5 equal groups, with 4 units in each group.

History

Our knowledge of the prehistory of arithmetic is limited by a small number of artifacts indicating a clear conception of addition and subtraction, the best-known being the Ishango Bone[2] from Africa, dating from 18,000 B.C.E.

It is clear that the Babylonians had solid knowledge of almost all aspects of elementary arithmetic circa 1850 B.C.E., historians can only infer the methods utilized to generate the arithmetical results. Likewise, a definitive algorithm for multiplication and the use of unit fractions can be found in the Rhind Mathematical Papyrus dating from Ancient Egypt circa 1650 B.C.E.

In the Pythagorean school, in the second half of the sixth century B.C.E., arithmetic was considered one of the four quantitative or mathematical sciences (Mathemata). These were carried over in medieval universities as the Quadrivium, which consisted of arithmetic, geometry, music, and astronomy. Together with the Trivium of grammar, rhetoric, and dialectic, they constituted the septem liberales artes (seven liberal arts). All these were thought to be fundamentally interconnected.

The book Introduction to Arithmetic was written by Nicomachus of Gerasa (ca. 60? - 120 C.E.) almost 2,000 years ago and contains both philosophical prose and very basic mathematical ideas. Nichomachus, one of the first mathematicians, was schooled in Alexandria. His book covers pythagorean number theory and contains the multiplication table of Greek origin. Compared to Euclid's book, which represents numbers by lines, Nichomachus used arithmetical notation expressed in ordinary language. Nicomachus referred to Plato (429 - 347 B.C.E.) quite often, and wrote about how philosophy can be possible only if one knows enough math. This is his only complete book that has survived to our day. Nicomachus describes how natural numbers and basic mathematical ideas are eternal and unchanging, and in an incorporeal realm.

Modern algorithms for arithmetic (for both hand and electronic computations) were made possible by the introduction of Arabic numerals and decimal place notation for numbers. By contrast, the ancient mathematician Archimedes (c. 287 - c. 212 B.C.E.) devoted an entire work, The Sand Reckoner, to devising a notation for a certain large integer. 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.

Decimal arithmetic

Decimal notation is based on ten parts and constructs all real numbers from the basic digits, and 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 (102), plus 0 tens (101), plus 7 units (100), plus 3 tenths (10-1) plus 6 hundredths (10-2). Decimals can also be noted in base ten, example: 0.34 = 34/100 (10-2)or 0.344 = 344/1000 (103). Algorithm 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.

An example of addition: 3 + 2 = 5, using apples, a popular choice in textbooks.[3]

Addition is the basic operation of arithmetic. In its simplest form, addition combines two numbers.

The result of adding two quantities a and b is a + b. It is sometimes phrased as "a more than b," or "b more than a." For example, 3 + 2 = 5, since 5 is 2 more than 3.

Addition is used to model many related processes, such as:

1. joining two collections of objects,
2. repeated incrementation,
3. moving a point across the number line,
4. representing two successive translations as one.

Terminology

The numbers or the objects to be added are generally called the "terms," the "addends," or the "summands"; this terminology carries over to the summation of multiple terms. The resultant number is called the sum. Therefore, from the above example, the terms are 3,2, and 5. The addends are 3 and 2. The sum is 5.

The word terms is to be distinguished from factors, which are multiplied. Some authors call the first addend the augend. In fact, during the Renaissance, many authors did not consider the first addend an "addend" at all. Today, due to the symmetry of addition, "augend" is rarely used, and both terms are generally called addends.[4]

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.

Notation

Addition is written using the plus sign "+" between the terms; that is, in infix notation. The result is expressed with an equals sign. For example,

1 + 1 = 2
2 + 2 = 4
5 + 4 + 2 = 11 (see "associativity" below)
3 + 3 + 3 + 3 = 12 (see "multiplication" below)

There are also situations where addition is "understood" even though no symbol appears:

• A column of numbers, with the last number in the column underlined, usually indicates that the numbers in the column are to be added, with the sum written below the underlined number.
• A whole number followed immediately by a fraction indicates the sum of the two, called a mixed number.[5] For example,
312 = 3 + 12 = 3.5.
This notation can cause confusion, since in most other contexts, juxtaposition denotes multiplication instead.

Properties

Addition is said to have "commutative" and "associative" properties. The term commutative come from "commute" or "move around," and in addition it means that terms can be interchanged. For instance, "a+b = b+a." The order in which the terms are added does not matter. The associative property means to "associate" or "group," and in addition it means that terms can be added in different groups. For example, "(a+b) + c = a + (b+c)."

The "identity element" of addition (or 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.

Subtraction (−)

An example of subtraction: "5 - 2 = 3".
A subtraction problem.

Subtraction is essentially the opposite of addition. It is denoted by a minus sign "−" in infix notation.

Subtraction is removing objects from a group. For example, 5 - 3 = 2, which means that three objects taken away from a total of five leaves two.

Subtraction is used to model several closely related processes:

1. From a given collection, take away (subtract) a given number of objects.
2. Combine a given measurement with an opposite measurement, such as a movement right followed by a movement left, or a deposit and a withdrawal.
3. Compare two objects to find their difference. For example, the difference between $800 and$600 is $800 −$600 = \$200.

Terminology

The traditional names for the parts of the formula

cb = a

are minuend (c)subtrahend (b) = difference (a). The words "minuend" and "subtrahend" are virtually absent from modern usage; Linderholm charges "This terminology is of no use whatsoever."[6] However, "difference" is very common.

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. For example:

5 - 3 = 2;
3 - 5 = -2;
3 - 3 = 0.

Notation

Imagine a line segment of length b with the left end labeled a and the right end labeled c. Starting from a, it takes b steps to the right to reach c. This movement to the right is modeled mathematically by addition:

a + b = c.

From c, it takes b steps to the left to get back to a. This movement to the left is modeled by subtraction:

cb = a.

Now, imagine a line segment labeled with the numbers 1, 2, and 3. From position 3, it takes no steps to the left to stay at 3, so 3 − 0 = 3. It takes 2 steps to the left to get to position 1, so 3 − 2 = 1. This picture is inadequate to describe what would happen after going 3 steps to the left of position 3. To represent such an operation, the line must be extended.

To subtract arbitrary natural numbers, one begins with a line containing every natural number (0, 1, 2, 3, 4, ...). From 3, it takes 3 steps to the left to get to 0, so 3 − 3 = 0. But 3 − 4 is still invalid since it again leaves the line. The natural numbers are not a useful context for subtraction.

The solution is to consider the integer number line (…, −3, −2, −1, 0, 1, 2, 3, …). From 3, it takes 4 steps to the left to get to −1, so

3 − 4 = −1.

Properties

Subtraction is neither commutative nor associative. For this 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.

In mathematics, it is often useful to view or even define subtraction as a kind of addition, the addition of the opposite. We can view 7 − 3 = 4 as the sum of two terms: seven and negative three. This perspective allows us to apply to subtraction all of the familiar rules and nomenclature of addition. Although subtraction is not associative or commutative, the addition of signed numbers is both.

Multiplication (× or ·)

Multiplication is in essence repeated addition, or the sum of a list of identical numbers.

${{a \times n = } \atop {\ }} {{\underbrace{a + \cdots + a}} \atop n}$

For example, 7 × 4 is the same as 7 + 7 + 7 + 7.

Fractions are multiplied by separately multiplying their denominators and numerators: a/b × c/d = (ac)/(bd). For example, 2/3 × 3/4 = (2×3)/(3×4) = 6/12 = 1/2.

Multiplication is used to determine the total of the amounts in many groups of the same size. For example, if there are 6 apples in 1 bag, and you buy 4 bags, then 6+6+6+6 = 24 (repeated addition), or 6 x 4 = 24 (multiplication), giving a total of 24 apples. Multiplication is used to increase a number by a fixed amount stepwise, or to compute a product. Simple numbers are incorporated in a multiplication table ("times table") as shown below.

Terminology

The two numbers being multiplied are formally called the multiplicand and the multiplier, where the multiplicand is usually written first. (Some write the multiplier first, and say that 7 × 4 stands for 4 + 4 + 4 + 4 + 4 + 4 + 4, but this usage is less common.) The difference was important in Roman numerals and similar systems, where multiplication is transformation of symbols and their addition.[7]

Because of the commutative property of multiplication, there is generally no need to distinguish between the two numbers so they are more commonly referred to as factors. The result of the multiplication is called the product.

Notation

Multiplication can be denoted in several equivalent ways. For example, the expression "5 multiplied by 2" can be written in any of the following ways:

5×2
5·2
(5)2, 5(2), (5)(2), 5[2], [5]2, [5][2]
5*2
5.2

The asterisk (*) is often used on computers because it is a symbol on every keyboard, but it is rarely used when writing math by hand. This usage originated in the FORTRAN programming language. Frequently, multiplication is implied by juxtaposition rather than shown in a notation. This is standard in algebra, taking forms such as

5x or xy.

This notation is not used with numbers alone: 52 never means 5 × 2. Also, this notation is potentially confusing if variables are permitted to have names longer than one letter, as in computer programming languages.

If the terms are not written out individually, then the product may be written with an ellipsis to mark out the missing terms, as with other series operations (like sums). Thus, the product of all the natural numbers from 1 to 100 can be written as:

$1 \cdot 2 \cdot \ldots \cdot 99 \cdot 100$ or $1 \cdot 2 \cdot \cdots \cdot 99 \cdot 100$.

Properties

Multiplication is really repeated addition, 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, that is, multiplying the reciprocal of any number by the number itself will yield the multiplicative identity, 1.

In summary the four properties are :

1. a*0 = 0*a = 0 the zero property
2. a*1 = 1*a = a the identity property
3. a*b = b*a the commutative property
4. a(b+c) = ab+ac the distributive property.

For integers, fractions, real and complex numbers, multiplication has certain properties:

• the order in which two numbers are multiplied does not matter. This is called the commutative property,
x · y = y · x.
• The associative property means that for any three numbers x, y, and z,
(x · y)z = x(y · z).
Note: the parentheses mean that the operations inside the parentheses must be done before anything outside the parentheses is done.
• Multiplication also has what is called a distributive property with respect to the addition,
x(y + z) = xy + xz.
• Also of interest is that any number times 1 is equal to itself, thus,
1 · x = x.
and this is called the identity property. In this regard the number 1 is known as the multiplicative identity.
• The sum of zero numbers is zero.
This fact is directly received by means of the distributive property:
m · 0 = (m · 0) + mm = (m · 0) + (m · 1) − m = m · (0 + 1) − m = (m · 1) − m = mm = 0.
So,
m · 0 = 0
no matter what m is (as long as it is finite).
• Multiplication with negative numbers also requires a little thought. First consider negative one (-1). For any positive integer m:
(−1)m = (−1) + (−1) +...+ (−1) = −m
This is an interesting fact that shows that any negative number is just negative one multiplied by a positive number. So multiplication with any integers can be represented by multiplication of whole numbers and (−1)'s.
All that remains is to explicitly define (−1)(−1):
(−1)(−1) = −(−1) = 1
• Every number x, except zero, has a multiplicative inverse, 1/x, such that x × 1/x = 1.
• Multiplication by a positive number preserves order: if a > 0, then if b > c then ab > ac. Multiplication by a negative number reverses order: if a < 0, then if b > c then ab < ac.

Division (÷ or /)

Division is essentially the inverse of multiplication. Specifically, if c times b equals a, written:

$c \times b = a$

where b is not zero, then a divided by b equals c, written:

$\frac ab = c$

For instance,

$\frac 63 = 2$

since

$2 \times 3 = 6\,$.

Division is the act or process of dividing. The arithmetic process is opposite of multiplication. Division is used to find out how many times a number will go into another number. For example, two goes into nine, four and a half times. This can also be written down as 9 ÷ 2 = 4.5, or 9 / 2 = 4.5 or spoken verbally as "nine over two is four and a half." The numbers in the operation have special names: Dividend ÷ divisor = quotient.

Terminology

In the above expression, a is called the dividend, b the divisor and c the quotient.

Division by zero (i.e., where the divisor is zero) is usually not defined.

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 and negative one). The quotient multiplied by the divisor always yields the dividend.

Notation

Division is most often shown by placing the dividend over the divisor with a horizontal line, also called a vinculum, between them. For example, a divided by b is written

$\frac ab.$

This can be read out loud as "a divided by b" or "a over b." A way to express division all on one line is to write the dividend, then a slash, then the divisor, like this:

$a/b.\,$

This is the usual way to specify division in most computer programming languages since it can easily be typed as a simple sequence of characters.

A typographical variation which is halfway between these two forms uses a slash but elevates the dividend, and lowers the divisor:

ab.

Any of these forms can be used to display a fraction. A fraction is a division expression where both dividend and divisor are integers (although typically called the numerator and denominator), and there is no implication that the division needs to be evaluated further.

A less common way to show division is to use the obelus (division sign) in this manner:

$a \div b.$

This form is infrequent except in elementary arithmetic. The obelus is also used alone to represent the division operation itself, as for instance as a label on a key of a calculator.

In some non-English-speaking cultures, "a divided by b" is written a : b. However, in English usage the colon is restricted to expressing the related concept of ratios (then "a is to b").

Properties

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 × 1b. When written as a product, it will obey all the properties of multiplication.

Division also has its own simple rules:

1. (2) All even numbers are divisible by 2.
2. (3) Add up all the digits of a number. If the sum is divisible by 3, then so is the number. For example, consider the number 1275. In this case, 1+2+7+5=15, and 15/3=5; therefore, 1275 is divisible by 3.
3. (4) In a number, if the group of last two digits is divisible by 4, then so is the number. For example, consider the number 1316. In this case, 16/4 = 4; therefore, 1316 is divisible by 4.
4. (5) All numbers ending in 5 or 0 are divisible by 5.
5. (6) If the number is divisible by 2 and 3, then it is divisible by 6.
6. (8) In a number, if the group of last 3 digits is divisible by 8, then so is the number. For example, consider the number 57144. In this case, 144/8 = 18; therefore, 57144 is divisible by 8.
7. (9) Add up all the digits of a number. If the sum is divisible by 9, then so is the number.
8. (10) If the last digit of a number is 0, then the number is divisble by 10.

Examples

+ 1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10 11
2 3 4 5 6 7 8 9 10 11 12
3 4 5 6 7 8 9 10 11 12 13
4 5 6 7 8 9 10 11 12 13 14
5 6 7 8 9 10 11 12 13 14 15
6 7 8 9 10 11 12 13 14 15 16
7 8 9 10 11 12 13 14 15 16 17
8 9 10 11 12 13 14 15 16 17 18
9 10 11 12 13 14 15 16 17 18 19
10 11 12 13 14 15 16 17 18 19 20

Multiplication table

× 1 2 3 4 5 6 7 8 9 10
1 1 2 3 4 5 6 7 8 9 10
2 2 4 6 8 10 12 14 16 18 20
3 3 6 9 12 15 18 21 24 27 30
4 4 8 12 16 20 24 28 32 36 40
5 5 10 15 20 25 30 35 40 45 50
6 6 12 18 24 30 36 42 48 54 60
7 7 14 21 28 35 42 49 56 63 70
8 8 16 24 32 40 48 56 64 72 80
9 9 18 27 36 45 54 63 72 81 90
10 10 20 30 40 50 60 70 80 90 100

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 [8].

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 central to the "Math Wars" over California's primary school curriculum in the 1990s, and continues today [9].

• Elementary arithmetic
• Finite field arithmetic
• Mathematics
• Number
• Number theory
• Arithmetic mean
• Arithmetic progression

Notes

1. Harold Davenport, (1999). The Higher Arithmetic: An Introduction to the Theory of Numbers, 7th ed. (Cambridge, England: Cambridge University Press. ISBN 0521634466).
2. "Ishango Bone" Mathematics of the African Diaspora.[1]. Mathematics dept. University of Buffalo. Retrieved June 22, 2008.
3. From Herbert Enderton. Elements of set theory. (Academic Press, 1977), 138: "…select two sets K and L with card K = 2 and card L = 3. Sets of fingers are handy; sets of apples are preferred by textbooks."
4. Steven Schwartzman. The words of mathematics: An etymological dictionary of mathematical terms used in English. (MAA, 1994), 19
5. D. Devine, J. Olson, and M. Olson. Elementary mathematics for teachers. (Wiley, 1991), 263
6. Carl Linderholm. Mathematics made difficult. (Wolfe, 1971), 42
7. For example, to multiply VII by XV one changes the VII to LXX (multiplying VII by X) plus XXV (V times V) plus X (II times V), but to multiply XV by VII one changes XV into LXXV (XV times V) plus XV plus XV (each XV times I).
8. Glossary of Terms. [2].mathematicallycorrect.com. Retrieved June 22, 2008.
9. "Math Wars" [3].education-world. Retrieved June 22, 2008.

References

History
• Bunt, Jones, and Bedient. The historical roots of elementary mathematics. Prentice-Hall, 1976. ISBN 0133890155
• Cunnington, Susan. The story of arithmetic, a short history of its origin and development. London: Swan Sonnenschein, 1904.
• Dickson, Leonard Eugene. History of the theory of numbers. Three volumes. Reprints: Carnegie Institute of Washington, Washington, 1932. Chelsea, New York, 1952, 1966.
• Ferreirós, José. Labyrinth of thought: A history of set theory and its role in modern mathematics. Birkhäuser, 1999. ISBN 0817657495
• Fine, Henry Burchard (1858-1928). The number system of algebra treated theoretically and historically. Boston: Leach, Shewell & Sanborn, 1891.
• Kaplan, Robert. The nothing that is: A natural history of zero. Oxford UP, 2000. ISBN 0195128427
• Karpinski, Louis Charles. The history of arithmetic. (original 1925) Reprint: New York: Russell & Russell, 1965.
• Ore, Øystein. Number theory and its history. New York: McGraw-Hill, 1948.
• Schwartzman, Steven. The words of mathematics: An etymological dictionary of mathematical terms used in English. MAA, 1994. ISBN 0883855119
• Williams, Michael. A history of computing technology. Prentice-Hall, 1985. ISBN 0133899179
• Weil, Andre. Number theory: an approach through history. Boston: Birkhauser, 1984. Reviewed: Math. Rev. 85c:01004.
Elementary mathematics
• Davison, Landau, McCracken, and Thompson. Mathematics: Explorations & Applications. Prentice Hall, 1999. ISBN 0134358171
• Sparks, F. and C. Rees. A survey of basic mathematics. McGraw-Hill, 1979. ISBN 0070599025
Education
Cognitive science
• Baroody and Tiilikainen. "Two perspectives on addition development." The development of arithmetic concepts and skills. 2003 ISBN 080583155X
• Fosnot and Dolk. Young mathematicians at work: Constructing number sense, addition, and subtraction. Heinemann, 2001 ISBN 032500353X
• Weaver, J. Fred. "Interpretations of number operations and symbolic representations of addition and subtraction." Addition and subtraction: A cognitive perspective. 1982 ISBN 0898591716
• Wynn, Karen. "Numerical competence in infants." The development of mathematical skills. | 1998 ISBN 086377816X
Mathematical exposition
• Bogomolny, Alexander. 1996. "Addition" Interactive Mathematics Miscellany and Puzzles [5] cut-the-knot.org. accessdate February 3, 2006
• Dunham, William. The mathematical universe. Wiley, 1994. ISBN 0471536563
• Johnson, Paul. From sticks and stones: Personal adventures in mathematics. Science Research Associates, 1975 ISBN 0574191151
• Linderholm , Carl. Mathematics made difficult. Wolfe, 1971 ISBN 0723404151
• Smith, Frank. The glass wall: Why mathematics can seem difficult. Teachers College Press, 2002 ISBN 0807742422
• Smith, Karl. The nature of modern mathematics. Wadsworth, 1980 ISBN 0818503521
• Bergman, George. An invitation to general algebra and universal constructions. General Printing, 2005 ISBN 0965521141 [6]
• Burrill, Claude. Foundations of real numbers. McGraw-Hill, 1967
• Davenport, Harold. The Higher Arithmetic: An Introduction to the Theory of Numbers, 7th ed. Cambridge, England: Cambridge University Press, 1999. ISBN 0521634466
• Dummit, D. and R. Foote. Abstract algebra. =Wiley, 1999 ISBN 0471368571
• Enderton, Herbert. Elements of set theory. Academic Press, 1977 ISBN 0122384407
• Lee, John. Introduction to smooth manifolds. Springer, 2003 ISBN 0387954481
• Martin, John. Introduction to languages and the theory of computation. McGraw-Hill 2003 ISBN 0072322004
• Rudin, Walter. Principles of mathematical analysis. McGraw-Hill, 1976 ISBN 007054235X
• Stewart, James. Calculus: Early transcendentals. Brooks/Cole, 1999. ISBN 0534362982
Mathematical research
Computing
• Flynn, M. and S. Oberman. Advanced computer arithmetic design. Wiley. 2001 ISBN 0471412090
• Horowitz, P. and W. Hill. The art of electronics. Cambridge UP, 2001 ISBN 0521370957
• Jackson, Albert. Analog computation. McGraw-Hill, 1960
• Truitt, T. and A. Rogers. Basics of analog computers. John F. Rider, 1960