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Learning Math Home
Number and Operations: Solutions
 
Session 2 Part A Part B Part C Homework
 
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A B C 
Homework

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Solutions for Session 2, Part A

See solutions for Problems: A1 | A2 | A3


Problem A1

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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).

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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.

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