Free «Real and Complex Numbers» Essay
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Mathematical and rather more specifically algebraic calculations have never been simple. Yet initially they do appear simple back they have been backed with years of research and experimentation of various mathematicians throughout history. Two such important inventions of mathematics are real numbers and complex numbers. Part of the similar algebraic world, both have different purposes for calculations and are based on different grounds. Both of these are discussed in the paper, though, not in much detail, as the paper’s limit confines that, but emphasis has been laid to their definition, basic characteristics and their developmental history.
When studying mathematics we often come across real numbers. Real numbers are basically used to measure quantities that are continuous. So what are they really? The real number system consists of rational numbers and irrational numbers. So to define real numbers we need to understand what are rational and irrational numbers.
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Rational Numbers are numbers which can be expressed as the ratio, or quotient, of two integers, with divisor being a non zero integer thus, a rational number is a number which can be expressed in the form a/b, where a and b are integers and b does not equal 0 (stated b ≠ 0). The numbers 1/6, -2/7, 23/455 and 137/ (-750) are all examples of rational numbers.
Because any integer can be written in the form of the quotient a/1, all integers are also rational numbers. Examples include -5 = -5/1 and 54 = 54/1. Zero is also considered to be an integer (neither negative nor positive), and it can be written in the quotient form 0/b = 0, b ≠ 0.
Irrational Numbers are real numbers which cannot be expressed as the ratio of two numbers. Numbers such as π = 3.14159265 … (which is the ratio of the circumference of a circle to its diameter), √2 = 1.4142 . . ., √3 = 1.7321 . . ., and √5 = 2.2361 . . . are all examples of irrational numbers.
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The set of real numbers can be represented using a number line. The number line has a zero point, often called the origin, which is used to represent the real number 0. To each and every point on the number line there corresponds a real number. The correspondence is that the real number represented by a point equals the directed distance travelled in moving from the origin to that point. Movements from left to right along the number line are considered to be in positive direction. Thus, points to the right of the origin correspond to positive real numbers whereas points to the left correspond to negative real numbers. Note that for each and every real number there corresponds a unique point on the number line.
The inequality symbols > or < are used to indicate that two numbers are not equal but they can be compared. When an inequality symbol is placed between two numbers, it “opens” in the direction of the larger number. Given two real numbers a and b, the notation a>b is read “a is greater than b.” the statement implies that on the real number line a is located to the right of b.
The absolute value of a real number is the magnitude or size of the number without the sign. The notation |a| denotes the absolute value of a. For example: the absolute value of the number +5 is |+5|= 5. The absolute value of -20 is |-20|= 20 and the absolute value of 0 is |0|= 0.
How did real numbers come into existence? The concept of real numbers and its foundation emerged during the 18th century when calculus was developed. The definition of real numbers, that we use today, was formulated by Dekekind in 1870’s. Although, his contribution as a mathematician was small but it laid the foundation of research work and development of real numbers. His introduction of the set of real numbers was based on Cauchy’s research about geometric lines (Miller, 2009).
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Real numbers are highly applicable in mathematical measures of continuous quantities, such as speed, mass, electric currents, etc. the calculations done on the computer in many mathematical software are also on the basis of real numbers.
An extension or rather advancement to real numbers is complex numbers. As the name suggest the numbers are truly complex as they take in to essence of real numbers together with imaginary numbers. These imaginary numbers are ‘fictitious’ as they have been called since their inception. The standard form of complex numbers is expressed in following form:
a + bi
Here, a and b represent real numbers and i represents an imaginary number. The following property is used for i:
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i2 = -1
Complex numbers are based on real numbers and contain real numbers but along with additional numbers to represent a closed field and to allow the effect of roots of a polynomial to be noted in the numerical equations. They expand the functions of multiplication and addition of integers.
The complex numbers, a +bi, are denoted by C. each complex number contains a real part and an imaginary part and two complex numbers can only equal if their real parts are similar and their imaginary parts equal.
The development and use of complex numbers is highly in depth and rigorous, but owing to the basic understanding of complex numbers, we can say that complex numbers fundamentally are based on group like structures, where real numbers are the subsets of complex numbers.
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Complex numbers can be added, divided, multiplied and subtracted like all numbers but for them specific methods are set up that take the form of equations that are too detailed to describe here.
Unlike real numbers, for complex numbers a number line is not defined, but they exist on a two-dimensional Cartesian plane where they are positioned as position or point vectors. Complex numbers are (Budnick, 2006):
Complex numbers became prominent in history by the 16th century with the help of an Italian mathematician named Niccolo Fontana Tartaglia. The concept of positive and negative numbers was already established during the early Greek civilization when their mathematics system was developed, but the core development as to the manipulation of the polynomials with roots took place when Italian mathematicians took on the research on them. By the 18th century, complex numbers had become very common and popular among many mathematicians. In 1748, complex analysis, based on complex numbers was used to establish the following formula (Fraenkel, 1965):
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Cauchy and Gauss, other famous historic mathematicians have had their research work and contributions to the development of complex numbers too.
Where real numbers represent rational and irrational numbers that are found on the number line and represent all the numbers that exist let it be in the form “9” or in the form “1/9”. Complex numbers are one step ahead. They include the imaginary numbers into account to represent the impact of the roots of polynomial equations. Where real numbers are represented on a simple number line, showing clear distinction of positive and negative numbers, complex numbers are represented on a Cartesian plane using position or point vectors.