Algebra Know-It-ALL

The length of the diagonal of a square that measures 1 unit on each edge. The length of the diagonal of a cube that measures 1 ...

126 Irrational and Real Numbers We can envision the reals as corresponding to points on a continuous, straight, infinitely long ...

The Number Hierarchy 127 S Q R= Everything inside the large rectangle ZN S N Q Z Figure 9-2 The relationship among the sets of r ...

128 Irrational and Real Numbers wires. The rational-number line is gray. When you pick it up, you find that it’s weightless. The ...

More About Irrationals and Reals 129 More About Irrationals and Reals Let’s explore “infinity” for a few minutes. Then we’ll pro ...

130 Irrational and Real Numbers where the digit b 1 is different from a 11 in the first list entry, the digit b 2 is different f ...

mathematics, and in which problems can be neatly worked out, simply defy any attempt at “seeing them in the mind’s eye.” It’s go ...

132 Irrational and Real Numbers Table 9-2. An S/R proof that 21/2 is not a rational number. Statements Reasons Assume 21/2 is ra ...

How Real Variables Behave Most of the properties of the natural numbers, integers, and rationals also apply to the reals. To get ...

134 Irrational and Real Numbers Multiplicative inverse (reciprocal) For every nonzero real number x, x(1/x)= 1 and (1/x)x= 1 Com ...

These rules also work for subtraction: x(y−z)=xy−xz and (x−y)z=xz−yz Variants of these rules work for division as long as the di ...

136 Irrational and Real Numbers Product of signs When two real numbers with plus signs (meaning positive, or larger than 0) or m ...

Product of sums For all real numbers w, x, y, andz, (w+x)(y+z)=wy+wz+xy+xz Cross multiplication For all real numbers w, x, y, an ...

138 Irrational and Real Numbers Quotient of quotients For all real numbers w, x, y, and z where x≠ 0, y≠ 0, and z≠ 0, (w /x)/(y ...

Quotient of powers For all nonzero real numbers x, y, andz, xy /z= (xy)^1 /z= (x1/z)y Power of product For all nonzero real numb ...

140 Irrational and Real Numbers is irrational. But you can “zoom in” on it. Divide out the fractions 59/19 and 60/19 with a calc ...

Suppose we have an irrational number and we display the first few digits of its endless nonrepeating decimal expansion. If we m ...

142 CHAPTER 10 Review Questions and Answers Part One This is not a test! It’s a review of important general concepts you learned ...

Part One 143 Question 1-3 The Hindu-Arabic system uses the numeral 0 to represent a quantity of nothing. Why is this significant ...

144 Review Questions and Answers Question 1-7 In the octal numeration system, what follows 7? What follows 47? What follows 77? ...