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7. Complex Numbers Instead of starting with a formal definition of complex number and then proceeding to study the properties of the new number system, we shall begin with a brief review of our old number systems N, ?L, Q and IR in order to discover a certain inadequacy in each concerning the provision of solutions of equations. This inadequacy will be partially overcome by successive extensions. Ultimately the new number system C of complex members will be seen as the final result of the effort in removing this inadequacy. 7.1. EQUATIONS AND NUMBER SYSTEMS An equation can be regarded as a mathematical expression of a certain condition or specification on an unknown number. Take, for instance, an example from primary school mathematics. John has saved a sum of money. If his father gives him 45 dollars, he will have altogether a total of 200 dollars. How much has John saved? The condition that 45 added to the unknown number x of dollars that John has saved will give 200 can be written as a simple equation x + 45 =200. We may say that this equation is formulated in the number system N in the sense that all its numbers belong to N and all its symbols have meaning in the system No The equation admits a solution in the system N because we can find a number 155 in N such that 155 + 45 = 200. Similarly x + 50 = 214, x + 7 = 83, 5y = 25 are equations which are formulated in N and admit solutions in N. However, it is not always true that an equation which is formulated in N would admit a solution in N. Take, for instance, the equation x + 45 = 5. This can well be the mathematical expression of a real life situation. For example, after I put 45 dollars into my account, the balance will show 5 dollars and I wish to know the balance before that. The equation is formulated in N but admits no solution in N, because there is no natural 194 Fundamental Concepts of Mathematics number that will give 5 when added to 45. The equations 2x + 3 =2, x 2 + 5x + 6 =0, x 2 =2, x 2 + 1 =0 are all formulated in N but admit no solution in No Undoubtedly this shows that N is too small to admit solutions of all equations that are formulated in N. This inadequancy of N can be overcome, at least partially, by extending N to the next larger system 7L of intergers. By saying that the system 1L is an extension of the system N, we mean not only that the set 7L contains the set N as a proper subset, but also that addition and multiplication are extended as well. Under the extension. equations which are formulated in N remain formulated in lL. Now some equations that are formulated in N but admit no solution in it will admit solutions in lL. For example, x + 45 = 5 admits the solution -40 in 1L and the quadratic equation x2 + 5x + 6 = 0 admits two solutions -2 and -3 in 1L. However, the system 7L is still not large enough to admit solutions of the equation 2x + 3 = 2 which is formulated in 7L. Similar to what is done before, this inadequancy is partially overcome when 7L is extended to the number system 0 of rational numbers. In 0, 2x + 3 = 2 admits the solution -1/2. However x 2 = 2 and x 2 + 1 =0 still have no solutions in O. This means that we have to extend 0 one step further to the number system [R of real numbers. Then x2 = 2 has solutions ±V2 in [R while x2 + 1 = 0 which is formulated in N, 7L, 0 as well as in ~ still admits no solutions in [R since the squares of all real numbers are non-negative. The successive extensions leading finally to the system [R fail to remove the inadequancy entirely. There are still many equations, which are expressed in terms of real numbers and their addition and multiplication, but admit no solution in R This means that [R has to be extended to some larger number system C. The set C should contain [R as a proper subset; moreover, it must at least contain a solution of the equation x 2 + 1 = O. Furthermore, addition and multipl ication of numbers of the system should also extend those of real numbers, in the sense that the sum...

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