1. Compare and contrast the negative integers with the complex numbers. Name at least two ways in which these numbers sets are alike and at least two ways these number sets are different. Use complete sentences in your response.
2. Write and solve a problem that involves operating on complex numbers where the result of the operation is a real number. You can use any of the four operations, addition, subtraction, multiplication, or division to write your problem. Explain why the answer is not an element of the complex number set.
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Answer:
The natural numbers N , integers Z , rational numbers Q , real numbers R , and complex numbers C are nested within each other as follows:
N⊂Z⊂Q⊂R⊂C.
To see how these are different, let’s start with N and extend step by step to reach C , seeing what distinguishing properties emerge with each step.
N
Intuitively, we can think of the natural numbers as the set of positive whole numbers {0,1,2,…} together with operations of addition and multiplication. It has the algebraic structure of a Semiring. They are closed under operations of addition and multiplication.
If we try to do subtraction of natural numbers, we find that the naturals are not closed under this operation, i.e., subtracting two natural numbers does not always give a result that is a natural number (e.g., 10–4=6 , but 4–10 does not exist in the natural numbers). To get closure under subtraction, we need numbers to have additive inverses.
If we extend natural numbers by including all additive inverses, we get the integers…
Z
The integers supplement all the natural numbers with the negative whole numbers: {…,−3,−2,−1,0,1,2,3,…}. The integers have the structure of a ring. They are closed under operations of addition, subtraction, and multiplication.
If we try to do division of integers, we find that the integers are not closed under this operation, i.e., dividing two integers does not always give a result that is an integer (e.g., 12/4=3 , but 4/12 is not an integer). To get closure under division, we need numbers to have multiplicative inverses.
If we extend the integers by including multiplicative inverses, we get the rational numbers…
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