Elementary algebra

Short description: Basic concepts of algebra

x=−b±b2−4ac2a

The quadratic formula, which is the solution to the quadratic equation ax2+bx+c=0 where a≠0. Here the symbols a, b, and c represent arbitrary numbers, and x is a variable which represents the solution of the equation.

Two-dimensional plot (red curve) of the algebraic equation y=x2−x−2.

Elementary algebra, also known as high school algebra or college algebra,[1] encompasses the basic concepts of algebra. It is often contrasted with arithmetic: arithmetic deals with specified numbers,[2] whilst algebra introduces numerical variables (quantities without fixed values).[3]

This use of variables entails use of algebraic notation and an understanding of the general rules of the operations introduced in arithmetic: addition, subtraction, multiplication, division, etc. Unlike abstract algebra, elementary algebra is not concerned with algebraic structures outside the realm of real and complex numbers.

It is typically taught to secondary school students and at introductory college level in the United States,[4] and builds on their understanding of arithmetic. The use of variables to denote quantities allows general relationships between quantities to be formally and concisely expressed, and thus enables solving a broader scope of problems. Many quantitative relationships in science and mathematics are expressed as algebraic equations.

## Contents

* 1 Algebraic operations
* 2 Algebraic notation
* 2.1 Alternative notation
* 3 Concepts
* 3.1 Variables
* 3.2 Simplifying expressions
* 3.3 Equations
* 3.3.1 Properties of equality
* 3.3.2 Properties of inequality
* 3.4 Substitution
* 4 Solving algebraic equations
* 4.1 Linear equations with one variable
* 4.2 Linear equations with two variables
* 4.3 Quadratic equations
* 4.3.1 Complex numbers
* 4.4 Exponential and logarithmic equations
* 4.5 Radical equations
* 4.6 System of linear equations
* 4.6.1 Elimination method
* 4.6.2 Substitution method
* 4.7 Other types of systems of linear equations
* 4.7.1 Inconsistent systems
* 4.7.2 Undetermined systems
* 4.7.3 Over- and underdetermined systems
* 5 See also
* 6 References
* 7 External links

## Algebraic operations

This section is an excerpt from Algebraic operation[ edit ]

In mathematics, a basic algebraic operation is a mathematical operation similar to any one of the common operations of elementary algebra, which include addition, subtraction, multiplication, division, raising to a whole number power, and taking roots (fractional power).[5] The operations of elementary algebra may be performed on numbers, in which case they are often called arithmetic operations. They may also be performed, in a similar way, on variables, algebraic expressions,[6] and more generally, on elements of algebraic structures, such as groups and fields.[7]

An algebraic operation on a set S may be defined more formally as a function that maps to S the tuples of a given length of elements of S. The length of the tuples is called the arity of the operation, and each member of the tuple is called an operand. The most common case is the case of arity two, where the operation is called a binary operation and the operands form an ordered pair. A unary operation is an operation of arity one that has only one operand; for example, the square root. An example of a ternary operation (arity three) is the triple product.[8]

The term algebraic operation may also be used for operations that may be defined by compounding basic algebraic operations, such as the dot product. In calculus and mathematical analysis, algebraic operation is also used for the operations that may be defined by purely algebraic methods. For example, exponentiation with an integer or rational exponent is an algebraic operation, but not the general exponentiation with a real or complex exponent. Also, the derivative is an operation that is not algebraic.

## Algebraic notation

Main article: Mathematical notation

Algebraic notation describes the rules and conventions for writing mathematical expressions, as well as the terminology used for talking about parts of expressions. For example, the expression 3x2−2xy+c has the following components:

Algebraic expression notation:
1 – power (exponent)
2 – coefficient
3 – term
4 – operator
5 – constant term
c – constant
x y – variables

A coefficient is a numerical value, or letter representing a numerical constant, that multiplies a variable (the operator is omitted). A term is an addend or a summand, a group of coefficients, variables, constants and exponents that may be separated from the other terms by the plus and minus operators.[9] Letters represent variables and constants. By convention, letters at the beginning of the alphabet (e.g. a,b,c) are typically used to represent constants, and those toward the end of the alphabet (e.g. x,y and z) are used to represent variables.[10] They are usually printed in italics.[11]

Algebraic operations work in the same way as arithmetic operations,[12] such as addition, subtraction, multiplication, division and exponentiation,[13] and are applied to algebraic variables and terms. Multiplication symbols are usually omitted, and implied when there is no space between two variables or terms, or when a coefficient is used. For example, 3×x2 is written as 3x2, and 2×x×y may be written 2xy.[14]

Usually terms with the highest power (exponent), are written on the left, for example, x2 is written to the left of x. When a coefficient is one, it is usually omitted (e.g. 1x2 is written x2).[15] Likewise when the exponent (power) is one, (e.g. 3x1 is written 3x).[16] When the exponent is zero, the result is always 1 (e.g. x0 is always rewritten to 1).[17] However 00, being undefined, should not appear in an expression, and care should be taken in simplifying expressions in which variables may appear in exponents.

### Alternative notation

Other types of notation are used in algebraic expressions when the required formatting is not available, or can not be implied, such as where only letters and symbols are available. As an illustration of this, while exponents are usually formatted using superscripts, e.g., x2, in plain text, and in the TeX mark-up language, the caret symbol ^ represents exponentiation, so x2 is written as "x^2".[18][19] This also applies to some programming languages such as Lua. In programming languages such as Ada,[20] Fortran,[21] Perl,[22] Python[23] and Ruby,[24] a double asterisk is used, so x2 is written as "x**2". Many programming languages and calculators use a single asterisk to represent the multiplication symbol,[25] and it must be explicitly used, for example, 3x is written "3*x".

## Concepts

### Variables

Example of variables showing the relationship between a circle's diameter and its circumference. For any circle, its circumference c, divided by its diameter d, is equal to the constant pi, π (approximately 3.14).

Main article: Variable (mathematics)

Elementary algebra builds on and extends arithmetic[26] by introducing letters called variables to represent general (non-specified) numbers. This is useful for several reasons.

1. Variables may represent numbers whose values are not yet known. For example, if the temperature of the current day, C, is 20 degrees higher than the temperature of the previous day, P, then the problem can be described algebraically as C=P+20.[27]
2. Variables allow one to describe general problems,[4] without specifying the values of the quantities that are involved. For example, it can be stated specifically that 5 minutes is equivalent to 60×5=300 seconds. A more general (algebraic) description may state that the number of seconds, s=60×m, where m is the number of minutes.
3. Variables allow one to describe mathematical relationships between quantities that may vary.[28] For example, the relationship between the circumference, c, and diameter, d, of a circle is described by π=c/d.
4. Variables allow one to describe some mathematical properties. For example, a basic property of addition is commutativity which states that the order of numbers being added together does not matter. Commutativity is stated algebraically as (a+b)=(b+a).[29]

### Simplifying expressions

Main articles: Expression (mathematics) and Computer algebra § Simplification

Algebraic expressions may be evaluated and simplified, based on the basic properties of arithmetic operations (addition, subtraction, multiplication, division and exponentiation). For example,

* Added terms are simplified using coefficients. For example, x+x+x can be simplified as 3x (where 3 is a numerical coefficient).
* Multiplied terms are simplified using exponents. For example, x×x×x is represented as x3
* Like terms are added together,[30] for example, 2x2+3ab−x2+ab is written as x2+4ab, because the terms containing x2 are added together, and the terms containing ab are added together.
* Brackets can be "multiplied out", using the distributive property. For example, x(2x+3) can be written as (x×2x)+(x×3) which can be written as 2x2+3x
* Expressions can be factored. For example, 6x5+3x2, by dividing both terms by the common factor, 3x2 can be written as 3x2(2x3+1)

### Equations

Animation illustrating Pythagoras' rule for a right-angle triangle, which shows the algebraic relationship between the triangle's hypotenuse, and the other two sides.

Main article: Equation

An equation states that two expressions are equal using the symbol for equality, = (the equals sign).[31] One of the best-known equations describes Pythagoras' law relating the length of the sides of a right angle triangle:[32]

c2=a2+b2

This equation states that c2, representing the square of the length of the side that is the hypotenuse, the side opposite the right angle, is equal to the sum (addition) of the squares of the other two sides whose lengths are represented by a and b.

An equation is the claim that two expressions have the same value and are equal. Some equations are true for all values of the involved variables (such as a+b=b+a); such equations are called identities. Conditional equations are true for only some values of the involved variables, e.g. x2−1=8 is true only for x=3 and x=−3. The values of the variables which make the equation true are the solutions of the equation and can be found through equation solving.

Another type of equation is inequality. Inequalities are used to show that one side of the equation is greater, or less, than the other. The symbols used for this are: a>b where > represents 'greater than', and a<b where < represents 'less than'. Just like standard equality equations, numbers can be added, subtracted, multiplied or divided. The only exception is that when multiplying or dividing by a negative number, the inequality symbol must be flipped.

#### Properties of equality

By definition, equality is an equivalence relation, meaning it is reflexive (i.e. b=b), symmetric (i.e. if a=b then b=a), and transitive (i.e. if a=b and b=c then a=c).[33] It also satisfies the important property that if two symbols are used for equal things, then one symbol can be substituted for the other in any true statement about the first and the statement will remain true. This implies the following properties:

* if a=b and c=d then a+c=b+d and ac=bd;
* if a=b then a+c=b+c and ac=bc;
* more generally, for any function f, if a=b then f(a)=f(b).

#### Properties of inequality

The relations less than < and greater than > have the property of transitivity:[34]

* If  a<b and  b<c then  a<c;
* If  a<b and  c<d then  a+c<b+d;[35]
* If  a<b and  c>0 then  ac<bc;
* If  a<b and  c<0 then  bc<ac.

By reversing the inequation, < and > can be swapped,[36] for example:

* a<b is equivalent to b>a

### Substitution

Main article: Substitution (algebra)

See also: Philosophy:Substitution (logic)

Substitution is replacing the terms in an expression to create a new expression. Substituting 3 for a in the expression a*5 makes a new expression 3*5 with meaning 15. Substituting the terms of a statement makes a new statement. When the original statement is true independently of the values of the terms, the statement created by substitutions is also true. Hence, definitions can be made in symbolic terms and interpreted through substitution: if a2:=a×a is meant as the definition of a2, as the product of a with itself, substituting 3 for a informs the reader of this statement that 32 means 3 × 3 = 9. Often it's not known whether the statement is true independently of the values of the terms. And, substitution allows one to derive restrictions on the possible values, or show what conditions the statement holds under. For example, taking the statement x \+ 1 = 0, if x is substituted with 1, this implies 1 + 1 = 2 = 0, which is false, which implies that if x \+ 1 = 0 then x cannot be 1.

If x and y are integers, rationals, or real numbers, then xy = 0 implies x = 0 or y = 0. Consider abc = 0. Then, substituting a for x and bc for y, we learn a = 0 or bc = 0. Then we can substitute again, letting x = b and y = c, to show that if bc = 0 then b = 0 or c = 0. Therefore, if abc = 0, then a = 0 or (b = 0 or c = 0), so abc = 0 implies a = 0 or b = 0 or c = 0.

If the original fact were stated as "ab = 0 implies a = 0 or b = 0", then when saying "consider abc = 0," we would have a conflict of terms when substituting. Yet the above logic is still valid to show that if abc = 0 then a = 0 or b = 0 or c = 0 if, instead of letting a = a and b = bc, one substitutes a for a and b for bc (and with bc = 0, substituting b for a and c for b). This shows that substituting for the terms in a statement isn't always the same as letting the terms from the statement equal the substituted terms. In this situation it's clear that if we substitute an expression a into the a term of the original equation, the a substituted does not refer to the a in the statement "ab = 0 implies a = 0 or b = 0."

## Solving algebraic equations

See also: Equation solving

A typical algebra problem.

The following sections lay out examples of some of the types of algebraic equations that may be encountered.

### Linear equations with one variable

Main article: Linear equation

Linear equations are so-called, because when they are plotted, they describe a straight line. The simplest equations to solve are linear equations that have only one variable. They contain only constant numbers and a single variable without an exponent. As an example, consider:

Problem in words: If you double the age of a child and add 4, the resulting answer is 12. How old is the child?

Equivalent equation: 2x+4=12 where x represent the child's age

To solve this kind of equation, the technique is add, subtract, multiply, or divide both sides of the equation by the same number in order to isolate the variable on one side of the equation. Once the variable is isolated, the other side of the equation is the value of the variable.[37] This problem and its solution are as follows:

Solving for x 1\. Equation to solve:  | 2x+4=12

|

2\. Subtract 4 from both sides:  | 2x+4−4=12−4
3\. This simplifies to:  | 2x=8
4\. Divide both sides by 2:  | 2x2=82
5\. This simplifies to the solution:  | x=4

In words: the child is 4 years old.

The general form of a linear equation with one variable, can be written as: ax+b=c

Following the same procedure (i.e. subtract b from both sides, and then divide by a), the general solution is given by x=c−ba

### Linear equations with two variables

Solving two linear equations with a unique solution at the point that they intersect.

A linear equation with two variables has many (i.e. an infinite number of) solutions.[38] For example:

Problem in words: A father is 22 years older than his son. How old are they?
Equivalent equation: y=x+22 where y is the father's age, x is the son's age.

That cannot be worked out by itself. If the son's age was made known, then there would no longer be two unknowns (variables). The problem then becomes a linear equation with just one variable, that can be solved as described above.

To solve a linear equation with two variables (unknowns), requires two related equations. For example, if it was also revealed that:

Problem in words
In 10 years, the father will be twice as old as his son.
Equivalent equation
y+10=2×(x+10)y=2×(x+10)−10Subtract 10 from both sidesy=2x+20−10Multiple out bracketsy=2x+10Simplify

Now there are two related linear equations, each with two unknowns, which enables the production of a linear equation with just one variable, by subtracting one from the other (called the elimination method):[39]

{y=x+22First equationy=2x+10Second equation
Subtract the first equation from(y−y)=(2x−x)+10−22the second in order to remove y0=x−12Simplify12=xAdd 12 to both sidesx=12Rearrange

In other words, the son is aged 12, and since the father 22 years older, he must be 34. In 10 years, the son will be 22, and the father will be twice his age, 44. This problem is illustrated on the associated plot of the equations.

For other ways to solve this kind of equations, see below, System of linear equations.

### Quadratic equations

Main article: Quadratic equation

Quadratic equation plot of y=x2+3x−10 showing its roots at x=−5 and x=2, and that the quadratic can be rewritten as y=(x+5)(x−2)

A quadratic equation is one which includes a term with an exponent of 2, for example, x2,[40] and no term with higher exponent. The name derives from the Latin quadrus, meaning square.[41] In general, a quadratic equation can be expressed in the form ax2+bx+c=0,[42] where a is not zero (if it were zero, then the equation would not be quadratic but linear). Because of this a quadratic equation must contain the term ax2, which is known as the quadratic term. Hence a≠0, and so we may divide by a and rearrange the equation into the standard form

x2+px+q=0

where p=ba and q=ca. Solving this, by a process known as completing the square, leads to the quadratic formula

x=−b±b2−4ac2a,

where the symbol "±" indicates that both

x=−b+b2−4ac2aandx=−b−b2−4ac2a

are solutions of the quadratic equation.

Quadratic equations can also be solved using factorization (the reverse process of which is expansion, but for two linear terms is sometimes denoted foiling). As an example of factoring:

x2+3x−10=0,

which is the same thing as

(x+5)(x−2)=0.

It follows from the zero-product property that either x=2 or x=−5 are the solutions, since precisely one of the factors must be equal to zero. All quadratic equations will have two solutions in the complex number system, but need not have any in the real number system. For example,

x2+1=0

has no real number solution since no real number squared equals −1. Sometimes a quadratic equation has a root of multiplicity 2, such as:

(x+1)2=0.

For this equation, −1 is a root of multiplicity 2. This means −1 appears twice, since the equation can be rewritten in factored form as

[x−(−1)][x−(−1)]=0.

#### Complex numbers

All quadratic equations have exactly two solutions in complex numbers (but they may be equal to each other), a category that includes real numbers, imaginary numbers, and sums of real and imaginary numbers. Complex numbers first arise in the teaching of quadratic equations and the quadratic formula. For example, the quadratic equation

x2+x+1=0

has solutions

x=−1+−32andx=−1−−32.

Since −3 is not any real number, both of these solutions for x are complex numbers.

### Exponential and logarithmic equations

Main article: Logarithm

The graph of the logarithm to base 2 crosses the x axis (horizontal axis) at 1 and passes through the points with coordinates (2, 1), (4, 2), and (8, 3). For example, log2(8) = 3, because 23 = 8. The graph gets arbitrarily close to the y axis, but does not meet or intersect it.

An exponential equation is one which has the form ax=b for a>0,[43] which has solution

x=logab=ln⁡bln⁡a

when b>0. Elementary algebraic techniques are used to rewrite a given equation in the above way before arriving at the solution. For example, if

3⋅2x−1+1=10

then, by subtracting 1 from both sides of the equation, and then dividing both sides by 3 we obtain

2x−1=3

whence

x−1=log23

x=log23+1.

A logarithmic equation is an equation of the form loga(x)=b for a>0, which has solution

x=ab.

For example, if

4log5(x−3)−2=6

then, by adding 2 to both sides of the equation, followed by dividing both sides by 4, we get

log5(x−3)=2

whence

x−3=52=25

from which we obtain

x=28.

### Radical equations

x32≡x32

Radical equation showing two ways to represent the same expression. The triple bar means the equation is true for all values of x

A radical equation is one that includes a radical sign, which includes square roots, x, cube roots, x3, and nth roots, xn. Recall that an nth root can be rewritten in exponential format, so that xn is equivalent to x1n. Combined with regular exponents (powers), then x32 (the square root of x cubed), can be rewritten as x32.[44] So a common form of a radical equation is xmn=a (equivalent to xmn=a) where m and n are integers. It has real solution(s):

n is odd  | n is even
and a≥0 | n and m are even
and a<0 | n is even, m is odd, and a<0

|
|
|

x=amn

equivalently

x=(an)m
| x=±amn

equivalently

x=±(an)m
| x=±amn | no real solution

For example, if:

(x+5)2/3=4

then

x+5=±(4)3,x+5=±8,x=−5±8,

and thus

x=3orx=−13

### System of linear equations

Main article: System of linear equations

There are different methods to solve a system of linear equations with two variables.

#### Elimination method

The solution set for the equations x−y=−1 and 3x+y=9 is the single point (2, 3).

An example of solving a system of linear equations is by using the elimination method:

{4x+2y=142x−y=1.

Multiplying the terms in the second equation by 2:

4x+2y=14
4x−2y=2.

Adding the two equations together to get:

8x=16

which simplifies to

x=2.

Since the fact that x=2 is known, it is then possible to deduce that y=3 by either of the original two equations (by using 2 instead of x ) The full solution to this problem is then

{x=2y=3.

This is not the only way to solve this specific system; y could have been resolved before x.

#### Substitution method

Another way of solving the same system of linear equations is by substitution.

{4x+2y=142x−y=1.

An equivalent for y can be deduced by using one of the two equations. Using the second equation:

2x−y=1

Subtracting 2x from each side of the equation:

2x−2x−y=1−2x−y=1−2x

and multiplying by −1:

y=2x−1.

Using this y value in the first equation in the original system:

4x+2(2x−1)=144x+4x−2=148x−2=14

Adding 2 on each side of the equation:

8x−2+2=14+28x=16

which simplifies to

x=2

Using this value in one of the equations, the same solution as in the previous method is obtained.

{x=2y=3.

This is not the only way to solve this specific system; in this case as well, y could have been solved before x.

### Other types of systems of linear equations

#### Inconsistent systems

The equations 3x+2y=6 and 3x+2y=12 are parallel and cannot intersect, and is unsolvable. Plot of a quadratic equation (red) and a linear equation (blue) that do not intersect, and consequently for which there is no common solution.

In the above example, a solution exists. However, there are also systems of equations which do not have any solution. Such a system is called inconsistent. An obvious example is

{x+y=10x+0y=2.

As 0≠2, the second equation in the system has no solution. Therefore, the system has no solution. However, not all inconsistent systems are recognized at first sight. As an example, consider the system

{4x+2y=12−2x−y=−4.

Multiplying by 2 both sides of the second equation, and adding it to the first one results in

0x+0y=4,

which clearly has no solution.

#### Undetermined systems

There are also systems which have infinitely many solutions, in contrast to a system with a unique solution (meaning, a unique pair of values for x and y) For example:

{4x+2y=12−2x−y=−6

Isolating y in the second equation:

y=−2x+6

And using this value in the first equation in the system:

4x+2(−2x+6)=124x−4x+12=1212=12

The equality is true, but it does not provide a value for x. Indeed, one can easily verify (by just filling in some values of x) that for any x there is a solution as long as y=−2x+6. There is an infinite number of solutions for this system.

#### Over- and underdetermined systems

Systems with more variables than the number of linear equations are called underdetermined. Such a system, if it has any solutions, does not have a unique one but rather an infinitude of them. An example of such a system is

{x+2y=10y−z=2.

When trying to solve it, one is led to express some variables as functions of the other ones if any solutions exist, but cannot express all solutions numerically because there are an infinite number of them if there are any.

A system with a higher number of equations than variables is called overdetermined. If an overdetermined system has any solutions, necessarily some equations are linear combinations of the others.

## See also

* History of algebra
* Binary operation
* Gaussian elimination
* Mathematics education
* Number line
* Polynomial
* Cancelling out
* Tarski's high school algebra problem

## References

* Leonhard Euler,  Elements of Algebra, 1770. English translation Tarquin Press, 2007, ISBN 978-1-899618-79-8, also online digitized editions[45] 2006,[46] 1822.
* Charles Smith, A Treatise on Algebra, in Cornell University Library Historical Math Monographs.
* Redden, John. Elementary Algebra . Flat World Knowledge, 2011

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27. ↑ Lewis Hirsch, Arthur Goodman, Understanding Elementary Algebra With Geometry: A Course for College Students, Publisher: Cengage Learning, 2005, ISBN 0534999727, 9780534999728, 654 pages, page 48
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29. ↑ Charles P. McKeague, Elementary Algebra, Publisher: Cengage Learning, 2011, ISBN 0840064217, 9780840064219, 571 pages, page 49
30. ↑ Andrew Marx, Shortcut Algebra I: A Quick and Easy Way to Increase Your Algebra I Knowledge and Test Scores, Publisher Kaplan Publishing, 2007, ISBN 1419552880, 9781419552885, 288 pages, page 51
31. ↑ Mark Clark, Cynthia Anfinson, Beginning Algebra: Connecting Concepts Through Applications, Publisher Cengage Learning, 2011, ISBN 0534419380, 9780534419387, 793 pages, page 134
32. ↑ Alan S. Tussy, R. David Gustafson, Elementary and Intermediate Algebra, Publisher Cengage Learning, 2012, ISBN 1111567689, 9781111567682, 1163 pages, page 493
33. ↑ Douglas Downing, Algebra the Easy Way, Publisher Barron's Educational Series, 2003, ISBN 0764119729, 9780764119729, 392 pages, page 20
34. ↑ Ron Larson, Robert Hostetler, Intermediate Algebra, Publisher Cengage Learning, 2008, ISBN 0618753524, 9780618753529, 857 pages, page 96
35. ↑ "What is the following property of inequality called?". Stack Exchange. November 29, 2014. https://math.stackexchange.com/q/1043755.
36. ↑ Chris Carter, Physics: Facts and Practice for A Level, Publisher Oxford University Press, 2001, ISBN 019914768X, 9780199147687, 144 pages, page 50
37. ↑ Slavin, Steve (1989). All the Math You'll Ever Need. John Wiley & Sons. p. 72. ISBN 0-471-50636-2. https://archive.org/details/allmathyoullever00slav/page/72.
38. ↑ Sinha, The Pearson Guide to Quantitative Aptitude for CAT 2/ePublisher: Pearson Education India, 2010, ISBN 8131723666, 9788131723661, 599 pages, page 195
39. ↑ Cynthia Y. Young, Precalculus, Publisher John Wiley & Sons, 2010, ISBN 0471756849, 9780471756842, 1175 pages, page 699
40. ↑ Mary Jane Sterling, Algebra II For Dummies, Publisher: John Wiley & Sons, 2006, ISBN 0471775819, 9780471775812, 384 pages, page 37
41. ↑ John T. Irwin, The Mystery to a Solution: Poe, Borges, and the Analytic Detective Story, Publisher JHU Press, 1996, ISBN 0801854660, 9780801854668, 512 pages, page 372
42. ↑ Sharma/khattar, The Pearson Guide To Objective Mathematics For Engineering Entrance Examinations, 3/E, Publisher Pearson Education India, 2010, ISBN 8131723631, 9788131723630, 1248 pages, page 621
43. ↑ Aven Choo, LMAN OL Additional Maths Revision Guide 3, Publisher Pearson Education South Asia, 2007, ISBN 9810600011, 9789810600013, page 105
44. ↑ John C. Peterson, Technical Mathematics With Calculus, Publisher Cengage Learning, 2003, ISBN 0766861899, 9780766861893, 1613 pages, page 525
45. ↑ Euler's Elements of Algebra
46. ↑ Euler, Leonhard; Hewlett, John; Horner, Francis; Bernoulli, Jean; Lagrange, Joseph Louis (4 May 2018). "Elements of Algebra". Longman, Orme. https://books.google.com/books?id=X8yv0sj4_1YC&q=euler+elements.

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