Questions tagged [elementary-algebra]
Elementary algebra, also called algebra precalculus, is the mathematical field studying basic properties of linear, exponential, logarithmic, polynomial, rational, and trigonometric functions, graphs, and the solving of equations and systems of equations.
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The Root of a Geometric Progression
Good people!
I'm presently in the process of putting something together on Euler and Gauss and cyclotomy and modular arithmetic, and I noticed that when it comes to the terminology primitive root ...
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What is the origin of the method of undetermined coefficients?
This MSE post asked about a specific integration technique that appears to be attributed to Charles Hermite, per a comment. The OP's source calls the technique el método alemán, i.e. the German method....
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Where did Lagrange write his technique of resolvents for solving polynomials?
From Andrew Ellinor & Satyajit Mohanty's article on a technique for solving cubics:
Using Lagrange's resolvents, to solve the cubic, one has to first solve a quadratic. Given the general cubic, $$...
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Why was the cubic specifically so hard to solve?
I'm a huge fan of the history of Algebra and, recently, I've noticed a bit of an oddity. Degree one equations have been known (and solved) for as long as human history. For degree two equations, we ...
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Why was solving polynomial equations historically considered so interesting?
From reading a few accounts of the unsolvability of the quintic, I am told that, e.g., there were public contests in which people competed to solve polynomial equations, and that over the course of ...
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Why did al-Khwarizmi use al jebr, ‘the reuniting of broken parts’ to signify algebra?
By "algebra", I'm assuming the solving of polynomial equations like every school child does. I accept that many kids loathe algebra, and that solving equations can be dreary and weariful, ...
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Fourth powers and quartic equations before Descartes
How did mathematicians interpret quartic equations and fourth powers before Descartes propose to perform elementary arithmetic on line segments?
I ask this because it seems strange to me that ...
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Has the idea that the result of division of positive number by negative number should be negative ever been controversial?
If we divide a positive number by another positive number, the result becomes greater as the divisor becomes smaller. If we continue this logic, division by a negative number should be greater than ...
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Who invented multiplying by the conjugate to rationalize denominators and when?
1860 Manual of Algebra describes a method which is now taught in upper secondary schools worldwide:
To rationalize the denominators of fractions which consist of binomial quadratic surds, use the ...
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Who discovered the indeterminate forms like 0/0?
Who discovered the indeterminate forms and how did they discover them? How did someone come to know that a particular form (fraction, product, sum/difference, exponent) is indeterminate? For example, $...
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Were ancient Romans so bad at computations before Arab numerals?
It is often said that Romans (see below) had a terrible number system, which made computations a mess. I do believe this, but I'm suspicious of the claim that nobody had better ways to do computations ...
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How did Ruffini discover his method of polynomial division?
How did Ruffini discover his method of polynomial division? At that time was it known that polynomial division works similar to integer division?
user9281
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When were polynomial equations first factored?
The question pretty much says it all, though I have a specific example in mind. In the mid-1500s while working on his Ars Magna Cardano asked Tartaglia to solve the cubic $x^3=9x+10$. Using ...
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Etymology of Some Terms Used in Ratio and Proportion in Old Algebra Textbooks
In older algebra textbooks for high school (mainly 19th century) four properties of ratio and proportions were used. The properties were Invertendo, Alternendo, Componendo, and Dividendo. This ...
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Do these trigonometric identities belong to Antonio Cagnoli?
I'm new to this stack community, please bear with me as I try to explain my question properly. Recently I came across with these trigonometric identities (where $ \omega + \phi + \psi = 180^\circ $):
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How was invented quadratic equation? [duplicate]
How actually people invented the quadratic equation? What practical problem did they solve?
There are some info about this exist in internet, but it is very abstract - I cant find the concrete task ...
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How was the sum of squares formula discovered by Archimedes?
AFAIK, Archimedes is credited with discovering the following formula for computing the sum of squares:
$$1^2 + 2^2 + 3^2 + \cdots + n^2 = \frac{n(n+1)(2n+1)}{6}$$
This seems to have come up in his ...
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Where can I find the list of problems from the (Chinese) "Nine Chapters on Mathematical Art"?
For the sake of curiosity, I'm interested in the "list of problems" that were laid out in the ancient Chinese text on Math. However, I haven't found a "list" in English anywhere. Only a few excerpts ...
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When was the idea of exponents generalized from "repeated multiplication"?
Recently I became curious about when the following ideas came about, and I couldn't really find information about them with some google searches.
$a^0 = 1$
$a^{\frac pq} = \sqrt [q] {2^p}$
$a^{-x} =...
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Why $x_a$ (or $x_o$) and not $a_x$? (conventions for algebraic quantities)
It's my understanding that the convention of using letters from the end of the alphabet ($x$, $y$, $z$) to represent $variables$, and letters from the start of the alphabet ($a$, $b$, $c$) to ...
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Who first introduced the longhand square-rooting method into European mathematics?
A previous question credits François Viète with introducing the well-known longhand method for the computation of square roots in digit-by-digit manner. This method is related to the binomial theorem.
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Who was the first to show that this quintic equation has five radical roots?
Despite this is being a very trivial question, I would like to know who was the first (with reference evidence) to show that the following quintic equation has five radical roots: $$x^5 + x + 1 = 0$$
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The origin of quadratic equation in actual practice
I read that in ancient times the quadratic equation of this kind $$x^2+10x=39$$ had been solved long ago. I read that this kind of equation originated in the geometric question of "Given an area of 39,...
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Exploring problems about quadratic function (in one variable) across the ages
I'm looking for problems about quadratic function across the ages. For example, in the Babylonian civilization, there are problems which are related with quadratic equation. Besides that, the concept ...
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How were the phenomena relating to symmetric polynomials discovered?
The "fundamental theorem of symmetric polynomials" states that any symmetric polynomial can be expressed as a polynomial in the elementary symmetric polynomials. This, or at least variants on it or ...
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Was 18th century algebra more symbolic/formal than the modern conception?
I've found Lagrange's Sur la résolution des équations algébriques to be a very confusing and difficult read, and I think I'm starting to see why: it seems that Lagrange thinks of algebra in a much ...
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Does anyone know about Ramanujan's method of solving the quartic?
I have read (probably) in Kanigel's book The Man Who Knew Infinity that S. Ramanujan devised his own method of solving the Quartic Equation after he learnt to solve the Cubic Equation. Does anyone ...
user459
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Why does the "Principle Of Permanence" have two different definitions?
This question is a sub-question of previous question on MSE. I feel that on this website I have better chances of knowing more things.
For quite some time now, I have been searching about the "...
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Was the "polynomial remainder theorem" known before polynomial long division was discovered?
Nowadays we can easily prove the following fact using polynomial long division:
If $a$ is a root of the polynomial $f$, then there exists a polynomial $g$ such that $f(x) = (x - a)g(x)$.
I can't ...
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What does Lagrange mean in this passage from Reflexions sur la résolution?
I'm having difficulty with section 6 of Lagrange's Réfléxions sur la résolution algébrique des équations. That's page 11 of the paper, 215 of his Oeuvres. Annoyingly, this is one of the most critical ...
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When was the method of getting square roots (invented by Viète in 1610 and developed by Harriot in 1631) first taught to school children?
François Viète's On the Numerical Resolution of Powers by Exegetics published
in 1610 (Viete, 2006, pp. 311-370) introduced one way of numerically solving polynomial equations, a special case of which ...
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What mathematical developments/discoveries caused imaginary numbers to gain acceptance at the time (18th century) they did?
In a Wiki article on imaginary numbers it was asserted that "the use of imaginary numbers was not widely accepted until the work of Leonhard Euler (1707–1783) and Carl Friedrich Gauss (1777–1855)."
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