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.

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 2^x = 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.

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.

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.