Real and Complex Numbers
Real Numbers in mathematics are any digit or number which has a real value (Howie, 2005). This includes all the digits or digits that are part of any number. 42 and 34/43 are examples of real numbers. Real numbers are infinite and there values keep on moving on the right side of the decimals, thus real numbers are specifically used to demonstrate an open identity of equation. Other real numbers also includes pi whose value is 3.1416… which means that value on the right side of decimal will go to infinity and for the sake of simplicity 4 to 5 digits are used by rounding of the last digit after the decimal point. Real numbers also include rational or irrational numbers and algebraic equations. The value of real numbers can be positive or negative and zero is also included in this list.
Real numbers moderate the different physical constants. The force of gravitation, mass, position, velocity, speed etc. all includes real numbers. There are however, some discrepancies in the use of real numbers in these physical constants which are against its (Howie, 2005). Major discrepancy is the rationalizing of these physical constant to an infinite value in order to pursue better precision of measurement. The open bounded values are therefore sometimes not applicable in the analysis of real numbers for the sake of simplicity and precision. Computers on the other hand are unable to compute through with most of the real numbers. They work greatly when the digit form is given but when the limit of the numbers increase computers don’t work. Some mathematicians of the contemporary world describe only those numbers as real numbers that can be computed by the computers and this number is not large as opposed to the infinity that is prevailed in the real numbers (Howie, 2005). The notation that is used for real numbers or for a set that contains real numbers is R, a bold capital letter R. This letter differentiates between a set that belongs to real to number to any other set. If the power of R is 2 then it is considered to be a two dimensional plane and if 3, then means a three dimensional plane.
If we analyze historical perspectives of real numbers, it is believed that Egyptians used fraction terms in 1000 BC. Indians also followed some mathematical observance and they determined that square root of 2 and 61 cannot provide a discrete or closed value making them realize that real numbers have no boundaries on either side of decimal and even if the value is negative. Middle Aged mathematicians accepted the rational numbers, integrations and fractions. Arabic mathematicians were the first ones who employed real numbers as algebraic equations.
Complex numbers on other hand is the combination of a real number and an imaginary number (Joshi, 1989). It is most of the time denoted as a + bi, where a and b are real numbers and i shows imaginary part of the equation. The complex numbers have significantly close boundaries and imaginary part is used to show closeness of the equation or a plane. There are certain analogues that are used to depict that each real number is a complex number if the imaginary part of the equation is zero.
Complex numbers are used far more than real numbers in calculations and analysis. In fact in electrical engineering disciplines everything is analyzed as a complex number. A complex number is also determined as a position vector where the real number is plotted horizontally and imaginary number vertically which then allows the solving of the complex plane by different methods by showcasing answers in Cartesian, Rectangular and algebraic forms (Joshi, 1989). The techniques of multiplication, division, addition and subtractions all are done in the two dimensional planes by the use of position vectors. These operations can also be performed by the usage of different laws such as distributive and associative laws. Complex numbers are also expressed in the forms of matrix where different elements of rows and columns determine which part is real and imaginary and then the solving is done by the Cramer’s rule more specifically. Complex analysis is determined to be the study of the complex numbers and plays an important part in the mathematics and engineering disciplines. Various physical quantities and other constants all can only be proved by the usage of complex numbers, therefore implying there importance in every aspect of real life. Its applications include the Control Theory in which the transformation between time domain and frequency domain is conducted by the help of Laplace (Joshi, 1989). Another application is the Signal Analysis which is done by using Fourier Transform. The integration of complex numbers most of the times tries to emphasize on the frequency analysis of any signal that has to be transmitted through transmitter, an important application in the GSM technology for mobile communication and in the radiating systems of electrical engineering.
The complex numbers were first formed by Heron of Alexandria, a Greek Mathematician who emphasized on the square roots of negative numbers. In 16th century complex numbers gain popularity as the Italian scientists were able to find out the roots of cubes and cubic polynomials.
On the whole, the analysis of both real and complex numbers given above provides some basic information about them. The limitations and the extension of their operations is so much that it is critically impossible to determine their all roles which they play in the success of mathematics.
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