Abstract
Definitions. An algebraic expression is a collection of one or more algebraic quantities (numbers or letters) connected by the signs of operations (+, -, :, √ etc.) with brackets of different kinds indicating the succession of operations.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Notes
For equations see p. 158.
See p. 164.
I.e., the term containing x in the highest power.
I.e., denominator and numerator have no common factor containing x.
If we do not confine ourselves to the real numbers, then the case (3) will not differ from (1) and the case (4) from (2). From this point of view, each fraction R(x) can be transformed into a sum of partial fractions of the form \(\frac{A}{(x-\alpha )^{k}}\), where A and α are complex numbers. This is applied in solving linear differential equations (see p. 543).
For simple and multiple roots see p. 164.
The following formula is easy to remember \(\textup{log}_{a}\;N=\frac{\textup{log}\;N}{\textup{log}\;a}\), where the logarithms on the right hand side are taken to an arbitrary, but to the same base.
For the number e see p. 331.
For tables of common logarithms see pp. 50–52; tables of antilogarithms pp. 53–55; tables of natural logarithms pp. 68–71.
To find a logarithm of an expression involving addition and subtraction, this expression should be first transformed into a suitable form (i.e., into a form containing multiplication and division).
An equality that is true for arbitrary values of the variable x is called an identity.
Here and in the sequel, the coefficients a0, a1,…, an are assumed to be real, with exception of certain cases which are discussed separately.
For multiple roots see p. 164.
The coefficient a0 at the highest power of x has been made equal to 1 (by dividing the equation by this coefficient).
For conjugate complex numbers see p. 587.
A way of obtaining such an equation has been given above. Practically, one can leave the multiple roots and at once form Sturm sequence of functions; if the last remainder Pm is not a constant, then P(x) has multiple roots and they should be abandoned.
To simplify the computation, the obtained remainders can be multiplied by constant positive factors; this will not change the result.
If some of these number are zero, then they should be omitted in counting the changes of sign.
For continuity of a function see p. 335.
The first index, i, of the element aij indicates that the element is taken from the i-th row of the determinant; the second index, j, indicates that the element is taken from the j-th column (i—number of the row counted from the top; j—number of the column counted from the left).
For permutations see p. 192.
A system of solutions of the form x1 = α1, x2 = α2, …, xn = αn will be denoted by α1, α2, …, αn.
See footnote on p. 177.
There can be infinitely many basic systems of solutions (see below).
Degree of a polynomial of two variables x and y is the highest sum of exponents of these variables in members of the polynomial. For example, the polynomial x3 + x2y2 + y3 is of the fourth degree.
If the symbol (3) concerns the values for which the relations of “greater” or “less” are not defined, e.g., complex numbers (p. 585), vectors (p. 613), then it can not be replaced by the symbol (3a). In this section, we are concerned only with real numbers.
For a particular case (n = 2) of the inequality see p. 190.
If n = 3, then a1, a2, a3 and b1, b2, b3 may be regarded as rectangular Cartesian coordinates of a vector; then Buniakowsky-Cauchy inequality reads that the scalar product of vectors does not exceed the product of their moduli (see p. 617). For n > 3, this formulation is extended to vectors in the n-dimensional space. An analogy of Buniakowsky-Cauchy inequality for infinite convergent series: \(\left ( \sum_{i-1}^{\infty} a_{i}b_{i}\right )^2\leqslant \left ( \sum_{i-1}^{\infty} a_{i}^{2} \right )\left ( \sum_{i-1}^{\infty} b_{i}^{2} \right )\) An analogy of the same inequality for definite integrals: \(\left (\int\limits_{a}^{b}f(x)\varphi(x)dx \right)^2\leqslant \left (\int\limits_{a}^{b}[f(x)]^2dx \right)\left (\int\limits_{a}^{b}[\varphi (x)]^2dx \right)\).
For a table of infinite series see pp. 353–354.
And also to the complex numbers.
For complex x, if re (x) >0.
For the symbol “n!” (the factorial) see p. 190.
Rights and permissions
Copyright information
© 1973 Verlag Harri Deutsch, Zürich
About this chapter
Cite this chapter
Bronshtein, I.N., Semendyayev, K.A. (1973). Algebra. In: A Guide Book to Mathematics. Springer, New York, NY. https://doi.org/10.1007/978-1-4684-6288-3_4
Download citation
DOI: https://doi.org/10.1007/978-1-4684-6288-3_4
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4684-6290-6
Online ISBN: 978-1-4684-6288-3
eBook Packages: Springer Book Archive