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Number Session 1: Solutions
 
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Solutions for Session 1, Part C

See solutions for Problems: C1 | C2 | C3 | C4 | C5 | C6| C7


Problem C1

a. 

Yes, the set of counting numbers is closed for addition.

b. 

Yes, the set of counting numbers is closed for multiplication.

<< back to Problem C1


 

Problem C2

We must include 0 (to subtract things like 4 - 4) and negative integers (to subtract things like 23 - 831).

<< back to Problem C2


 

Problem C3

a. 

We must include all fractional numbers of the form p/q, where p and q are integers (positive, negative, or zero counting numbers), with the restriction that q cannot be 0 (dividing by 0 is not defined). For example, we will need numbers like 5/2 and 82/7 and -1/2.

b. 

Take a division problem like 53 = r. This is the equivalent to saying, "What number multiplied by 3 gives us 5?" The equation for this is 3 • r = 5. To solve this equation, we must isolate r on one side of it. Doing this requires dividing by 3 or multiplying 3 by its multiplicative inverse. The multiplicative inverse of 3 is usually written as 1/3. Multiplying both sides by 1/3 produces the following:

1/3 • (3 • r) = 1/3 • 5
(1/3 • 3) • r = 1/3 • 5
1 • r = 1/3 • 5
r = 5/3

c. 

No. If y is the multiplicative inverse of 0, then y • 0 = 1. But every real number multiplied by 0 equals 0, so y cannot be a real number -- and there is no multiplicative inverse for 0. That's why we can't divide by 0.

<< back to Problem C3


 

Problem C4

Counting numbers are not dense. There is no counting number between 2 and 3. The integers are not dense either. However, we can always find a rational number between any two given rational numbers; for example, the average of any two fractions must always be a fraction between the two given fractions. Therefore, rational numbers are dense. One rational number between 2.5 and 2.6 is 2.55. One rational number between 2.55 and 2.6 is 2.555.

<< back to Problem C4


 

Problem C5

a. 

Some major examples include raising a number to a power (exponentiation) and its inverse function (taking roots, such as square or cube roots), working with circles (and the number , approximately 3.141593), and solving equations with exponents (such as 2x = 3).

b. 

Such operations produce irrational numbers, like , , or e (the base of natural logarithms; e is a mathematical constant approximately equal to 2.7183). Roots such as and are algebraic irrationals since they can be solutions to polynomial equations: numbers such as and e are called transcendental irrationals since they cannot be solutions to polynomial equations. Other equations, like x2 = -1, do not have a solution on the number line at all; this solution would be an imaginary number.

<< back to Problem C5


 

Problem C6

a. 

No. Proving this is actually pretty difficult, but for the to be rational, we would have to be able to write it as p/q in reduced form, where p and q are integers that are relatively prime. This would mean that p and q are solutions to the equation p2 = 2q2, which cannot be solved if p and q can only be counting numbers.

b. 

The length of the is the hypotenuse of a right triangle whose legs are 1 and 1 (i.e., x2 = 12 + 12 or x2 = 2 or x = ). So we know it has a "physical" distance and therefore can be located on the number line. This is about 1.414, but no decimal could ever express the exactly.

<< back to Problem C6


 

Problem C7

Each is on the number line some specific distance from 0 (since each number is a constant). As with the , the distance cannot be expressed as a terminating or repeating decimal. is approximately 3.141593, while e is approximately 2.7183.

<< back to Problem C7


 

Problem C8

a. 

The real numbers could be represented as the horizontal axis (similar to the number line). All real numbers, like 2, 1/2, -3, and e, would be on this line.

b. 

The pure imaginary numbers, like 2i, (1/2)i, -3i, and ei could be represented as the vertical axis. The coordinates of a real number are (x,0), where x is the real part. The coordinates of a pure imaginary number are (0,yi), where yi is the imaginary part.

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