Whole number are closed under dash and dash operation .
Answers
Operations under which a particular set is not closed require new sets of numbers:
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Counting Numbers: Subtraction requires 0 and negative integers; division requires rational numbers.
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Whole Numbers: Subtraction requires negative integers; division requires rational numbers.
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Integers: Division requires rational numbers.
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Rational Numbers: All four operations are okay here (with the exception of division by 0). However, solving problems with exponents would require us to expand from the rational numbers. For example, a problem like x2 = 3 can be solved using the real numbers, but not the rational numbers.
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Irrational Numbers: All operations require rational numbers.
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Real Numbers: All four operations are okay here (with the exception of division by 0).
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To go from one set to the next requires new types of numbers:
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To go from counting numbers to whole numbers, we need the additive identity 0.
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To go from whole numbers to integers, we need the additive inverses -- the opposites of the counting numbers.
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To go from integers to rational numbers, we need the multiplicative inverses of all non-zero counting numbers and their multiples. These are fractions with integer numerators and denominators, like 2/3 and -7/4.
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To go from rational numbers to real numbers, we need irrational numbers, such as and . Similarly, to go from irrational to real numbers, we need rational numbers.
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To go from real numbers to complex numbers, we need i (a number such that when squared it gives -1) and all its real multiples -- the imaginary numbers. Adding any real number and any imaginary number then forms a complex number, for example, 2 + 3i and -2/3 + 2.718i.
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