Answers to problems:
Page 72:
2^{10} ≈ 1000, so
2^{20} ≈ 1000000, a million, 10^{6};
2^{30} ≈ 1000000000, a billion, 10^{9};
2^{40} ≈ 1000000000000, a trillion, 10^{12}
and then
2^{80} ≈ a trillion trillion, 10^{24}.
Page 75:
81 is 3^{4}, so 80 ≈ 81 implies that 3^{12} ≈ 80^{3} Now use the approximation 3^{12} ≈ 2^{19} to give
2^{19} ≈ 80^{3} = 5^{3} × 2^{12}.
Dividing by 2^{9},
2^{10} ≈ 5^{3} × 2^{3} = 10^{3}, i.e. 1024 ≈ 1000.
A perfect harmonic seventh is given by the ratio 7/4. A perfect harmonic fourth is given by the ratio 4/3. Two fourths give 16/9, and the approximation 16/9 ≈ 7/4 is equivalent to 64 ≈ 63.
