TAOCP 1.2.2: Numbers, Powers, and Logarithms
Section 1.2.2 exercises: 30/30 solved.
Section 1.2.2. Numbers, Powers, and Logarithms
Exercises from TAOCP Volume 1 Section 1.2.2: 30/30 solved.
| # | Rating | Category | Status | Time |
|---|---|---|---|---|
| 1 | [00] | immediate | verified | 32s |
| 2 | [00] | immediate | verified | 2m19s |
| 3 | [02] | simple | verified | 31s |
| 4 | ▶ [05] | simple | verified | 49s |
| 5 | [05] | simple | verified | 38s |
| 6 | [10] | simple | verified | 1m58s |
| 7 | [M23] | math-medium | verified | 11m16s |
| 8 | [25] | medium | verified | 8m26s |
| 9 | [M23] | math-medium | verified | 2m06s |
| 10 | [18] | medium | verified | 38s |
| 11 | ▶ [10] | simple | verified | 3m59s |
| 12 | [02] | simple | verified | 36s |
| 13 | ▶ [M23] | math-medium | verified | 36s |
| 14 | [15] | simple | verified | 44s |
| 15 | [10] | simple | verified | 29s |
| 16 | [00] | immediate | verified | 30s |
| 17 | ▶ [05] | simple | verified | 33s |
| 18 | [10] | simple | verified | 26s |
| 19 | ▶ [20] | medium | verified | 32s |
| 20 | [10] | simple | verified | 36s |
| 21 | [15] | simple | verified | 29s |
| 22 | ▶ [20] | medium | solved | 1m31s |
| 23 | [M25] | math-medium | verified | 1m51s |
| 24 | [15] | simple | verified | 37s |
| 25 | [22] | medium | solved | 27s |
| 26 | [M27] | math-hard | solved | 29s |
| 27 | ▶ [M25] | math-medium | solved | 3m32s |
| 28 | [M30] | math-hard | solved | 27s |
| 29 | [HM20] | hm-medium | verified | 35s |
| 30 | [12] | simple | solved | 19s |
TAOCP 1.2.2 Exercise 1
There is no smallest positive rational number.
TAOCP 1.2.2 Exercise 2
No.
TAOCP 1.2.2 Exercise 3
By Eq.
TAOCP 1.2.2 Exercise 4
We have $(0.125)^{-2/3}$.
TAOCP 1.2.2 Exercise 5
We could define real numbers by binary expansions instead of decimal expansions.
TAOCP 1.2.2 Exercise 6
Let x=m+0.
TAOCP 1.2.2 Exercise 7
Assume throughout that $b\neq 0$.
TAOCP 1.2.2 Exercise 8
Let $u = a + 0.c_1c_2c_3\ldots$ be a positive real number.
TAOCP 1.2.2 Exercise 9
Let x=\frac pq,\qquad y=\frac rs, where $p,r\in\mathbb Z$ and $q,s\in\mathbb Z_{>0}$.
TAOCP 1.2.2 Exercise 10
Suppose, for the sake of contradiction, that $\log_{10} 2$ is rational.
TAOCP 1.2.2 Exercise 11
Let x_0=\log_{10}2, so that
TAOCP 1.2.2 Exercise 12
Equation (8) shows that 10^{0.
TAOCP 1.2.2 Exercise 13
(a) Let u=\sqrt[n]{1+x}.
TAOCP 1.2.2 Exercise 14
We are asked to prove Eq.
TAOCP 1.2.2 Exercise 15
The statement is true.
TAOCP 1.2.2 Exercise 16
By Eq.
TAOCP 1.2.2 Exercise 17
By definition (13), $\lg x = \log_2 x$.
TAOCP 1.2.2 Exercise 18
The statement is true.
TAOCP 1.2.2 Exercise 19
A 14-digit decimal integer $n$ satisfies $10^{13} \le n < 10^{14}.$ A computer word with 47 bits and a sign bit has 47 bits available for the magnitude.
TAOCP 1.2.2 Exercise 20
Yes.
TAOCP 1.2.2 Exercise 21
By equation (14), \log_b y = \frac{\ln y}{\ln b}.
TAOCP 1.2.2 Exercise 22
Using the change-of-base formula, \lg x=\frac{\ln x}{\ln 2}, \qquad \log_{10}x=\frac{\ln x}{\ln 10}.
TAOCP 1.2.2 Exercise 23
Let L(z) denote the area under the hyperbola
TAOCP 1.2.2 Exercise 24
The method based on Eqs.
TAOCP 1.2.2 Exercise 25
Let the value of $x$ at the beginning of an execution of step L3 be denoted by $x^{(t)}$, where $t = 0, 1, 2, \ldots$ counts the number of times step L4 has been performed.
TAOCP 1.2.2 Exercise 26
Let the computer work with fixed precision $\delta > 0$, meaning that every right shift and every subtraction is performed with an error whose absolute value is at most $\delta$.
TAOCP 1.2.2 Exercise 27
Let L'_k = n+\frac{b'_1}{2}+\cdots+\frac{b'_k}{2^k},
TAOCP 1.2.2 Exercise 28
Let $0 \le x < 1,$ and let $b>0$, $b\ne1$.