Solutions for Session 2, Part A

See solutions for Problems: A1 | A2 | A3

 Problem A1

Problem A2

For each set, you can only do operations for which that set is closed:

 • Counting Numbers: This set is closed only under addition and multiplication. In other words, we can solve all addition and multiplication problems, but not all subtraction and division problems are solvable. • Whole Numbers: This set is closed only under addition and multiplication. • Integers: This set is closed only under addition, subtraction, and multiplication. • Rational Numbers: This set is closed under addition, subtraction, multiplication, and division (with the exception of division by 0). • Irrational Numbers: This set is closed for none of the operations (e.g., • = 2, a rational number). • Real Numbers: This set is closed only under addition, subtraction, multiplication, and division (with the exception of division by 0).

Problem A3

a.

Operations under which a particular set is not closed require new sets of numbers:

 • Counting Numbers: Subtraction requires 0 and negative integers; division requires rational numbers. • Whole Numbers: Subtraction requires negative integers; division requires rational numbers. • Integers: Division requires rational numbers. • 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. • Irrational Numbers: All operations require rational numbers. • Real Numbers: All four operations are okay here (with the exception of division by 0).

b.

To go from one set to the next requires new types of numbers:

 • To go from counting numbers to whole numbers, we need the additive identity 0. • To go from whole numbers to integers, we need the additive inverses -- the opposites of the counting numbers. • 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. • 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. • 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.