Elementary Algebra is based on unknown. Although number of value to be found or the number of solutions is known in advance by looking at the equations. These solutions are masked by the variables. A variable can take any value and the values which satisfy the equations are known as the solutions. A linear equation in one unknown has one solution. Similarly a linear equation in two variables has one solution if two equations are given,a linear equation in three unknowns also has one solution and is solvable if three equations are given in three unknowns. The set of equations which are used are called a system of equations. But a quadratic equation has two solutions and a cubic equation has three solutions. A variable raised to n power has n solutions.
Let us talk about what is taught in elementary algebra. We will talk as a whole that is what exist in elementary algebra from schools to colleges. The first thing to be taught is about sets. Set is the language of mathematics. This thing occurs in every branch of mathematics. The next thing is solution of linear equations, quadratic equations, cubic equations and quartic equations. The cubic equations are solved by Cardano's Method. The quartic equations are solved by Ferrari's or Descartes method. The next thing which comes under elementary algebra is complex numbers. The rise of complex numbers is attributed to the equation in which the square of a variable is equal to a negative number. Euler gave a definition by which he joined the complex number with trigonometry. It is worth to note it here.
eiθ = cos θ + i sin θ
One consequence of this formula is eiπ = −1
There is one more formula given by De Moivre and it is
(cos θ + i sin θ)n = cos nθ + i sin nθ
This formula is used in the expansion of cos θ and sin θ and it also ha many other uses.
The next thing we are taught is the system of equations. As described in the first paragraph it has solutions if a set of equations are given. The solution is possible under certain conditions and has a unique solution under a particular condition. The set of equations can have a unique solution or a dependent solution or no solution. The solution of linear systems can be found by Cramer's rule or by use of matrices. There is also matter related to inequalities. The inequalities known to the ancients (Inequality of the means, Triangle inequality), Less ancient inequalities (Cauchy-Schwarz inequality, Weierstrass’ inequalities, Tchebyshev’s inequalities).
Elementary algebra is taught from the beginning because it has vast applications in other fields. The expressions which we come across in elementary algebra takes shape of functions in calculus. The solution of quadratic, cubic and quartic equations are used to solve the maximum and minimum problems. The factor theorem taught also comes handy in calculus as well as statistics. Binomial theorem, multinomial theorem are used in statistics.
Every student will find it interesting if he finds instances where it is used in our life. I will try my best to bring those things come to life on this blog.
Let us talk about what is taught in elementary algebra. We will talk as a whole that is what exist in elementary algebra from schools to colleges. The first thing to be taught is about sets. Set is the language of mathematics. This thing occurs in every branch of mathematics. The next thing is solution of linear equations, quadratic equations, cubic equations and quartic equations. The cubic equations are solved by Cardano's Method. The quartic equations are solved by Ferrari's or Descartes method. The next thing which comes under elementary algebra is complex numbers. The rise of complex numbers is attributed to the equation in which the square of a variable is equal to a negative number. Euler gave a definition by which he joined the complex number with trigonometry. It is worth to note it here.
One consequence of this formula is eiπ = −1
There is one more formula given by De Moivre and it is
This formula is used in the expansion of cos θ and sin θ and it also ha many other uses.
The next thing we are taught is the system of equations. As described in the first paragraph it has solutions if a set of equations are given. The solution is possible under certain conditions and has a unique solution under a particular condition. The set of equations can have a unique solution or a dependent solution or no solution. The solution of linear systems can be found by Cramer's rule or by use of matrices. There is also matter related to inequalities. The inequalities known to the ancients (Inequality of the means, Triangle inequality), Less ancient inequalities (Cauchy-Schwarz inequality, Weierstrass’ inequalities, Tchebyshev’s inequalities).
Elementary algebra is taught from the beginning because it has vast applications in other fields. The expressions which we come across in elementary algebra takes shape of functions in calculus. The solution of quadratic, cubic and quartic equations are used to solve the maximum and minimum problems. The factor theorem taught also comes handy in calculus as well as statistics. Binomial theorem, multinomial theorem are used in statistics.
Every student will find it interesting if he finds instances where it is used in our life. I will try my best to bring those things come to life on this blog.
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