TAOCP 4.1: Positional Number Systems
Section 4.1 exercises: 34/34 solved.
Section 4.1. Positional Number Systems
Exercises from TAOCP Volume 2 Section 4.1: 34/34 solved.
| # | Rating | Category | Status | Time |
|---|---|---|---|---|
| 1 | [15] | simple | solved | 4m03s |
| 2 | ▶ [24] | medium | solved | 4m40s |
| 3 | [29] | hard | verified | 13m |
| 4 | [**] | solved | 3m49s | |
| 5 | [**] | verified | 1m17s | |
| 6 | [**] | verified | 1m07s | |
| 7 | [**] | verified | 2m28s | |
| 8 | [**] | verified | 1m09s | |
| 9 | ▶ [**] | verified | 2m21s | |
| 10 | [**] | verified | 3m48s | |
| 11 | [**] | verified | 1m27s | |
| 12 | [**] | verified | 2m35s | |
| 13 | ▶ [**] | solved | 3m58s | |
| 14 | [**] | solved | 4m02s | |
| 15 | [**] | verified | 2m41s | |
| 16 | [**] | solved | 5m20s | |
| 17 | [**] | solved | 3m26s | |
| 18 | [**] | solved | 6m35s | |
| 19 | ▶ [**] | verified | 3m28s | |
| 20 | [HM28] | hm-hard | solved | 5m37s |
| 21 | [M22] | math-medium | verified | 1m55s |
| 22 | [HM25] | hm-medium | solved | 3m13s |
| 23 | [HM30] | hm-hard | solved | 2m12s |
| 24 | [M35] | math-hard | verified | 2m14s |
| 25 | [M25] | math-medium | verified | 5m01s |
| 26 | ▶ [HM30] | hm-hard | verified | 5m21s |
| 27 | [**] | solved | 10m13s | |
| 28 | ▶ [**] | verified | 9m35s | |
| 29 | [**] | verified | 5m05s | |
| 30 | [**] | solved | 16m40s | |
| 31 | ▶ [M35] | math-hard | verified | 3m49s |
| 32 | [M40] | math-project | verified | 5m45s |
| 33 | [M40] | math-project | solved | 7m10s |
| 34 | ▶ [22] | medium | verified | 3m41s |
TAOCP 4.1 Exercise 1
We are asked to express the integers -10, -9, \ldots, 9, 10 in the number system whose radix is $-2$.
TAOCP 4.1 Exercise 2
We are asked to represent the numbers -49, \quad -3\frac12, \quad \pi in four number systems: (a) binary (signed magnitude),
TAOCP 4.1 Exercise 3
**Exercise 4.
TAOCP 4.1 Exercise 4
In MIX fixed-point arithmetic, the radix point is understood to be fixed relative to the register positions.
TAOCP 4.1 Exercise 5
A negative integer $-N$ has a nines' complement representation obtained by replacing each decimal digit $d$ of $N$ with $9 - d$.
TAOCP 4.1 Exercise 6
In signed magnitude notation, one bit is reserved for the sign, leaving $p-1$ bits for magnitude.
TAOCP 4.1 Exercise 7
Represent each real number by a decimal expansion extending infinitely in both directions, x=\cdots d_2d_1d_0.
TAOCP 4.1 Exercise 8
Equation (5) asserts that a distribution function $F(x)$ is monotonically nondecreasing, with $F(x_1) \le F(x_2) \quad \text{if } x_1 \le x_2; \qquad F(-\infty) = 0, \quad F(+\infty) = 1.$ By definiti...
TAOCP 4.1 Exercise 9
Each octal digit corresponds to three binary digits, and each hexadecimal digit corresponds to four binary digits.
TAOCP 4.1 Exercise 10
**Exercise 4.
TAOCP 4.1 Exercise 11
Let s_i=a_i+b_i+t_i, where $t_i$ is the carry entering position $i$.
TAOCP 4.1 Exercise 12
**Exercise 4.
TAOCP 4.1 Exercise 13
x=(0.
TAOCP 4.1 Exercise 14
**Exercise 4.
TAOCP 4.1 Exercise 15
**Exercise 4.
TAOCP 4.1 Exercise 16
Let \beta=i-1.
TAOCP 4.1 Exercise 17
**Exercise 4.
TAOCP 4.1 Exercise 18
**Exercise 4.
TAOCP 4.1 Exercise 19
Assume that every integer in the interval l\le m\le u, \qquad l=-\frac{\max D}{b-1}, \qquad
TAOCP 4.1 Exercise 20
Let D=\{-1,0,1,0.
TAOCP 4.1 Exercise 21
Let B=\left\{-\frac92,-\frac72,-\frac52,-\frac32,-\frac12, \frac12,\frac32,\frac52,\frac72,\frac92\right\}.
TAOCP 4.1 Exercise 22
Let $x$ be an arbitrary real number and $\epsilon > 0$ be given.
TAOCP 4.1 Exercise 23
Let $b > 1$ be an integer and $D$ a set of $b$ real numbers containing $0$, such that every positive real number $x$ has a representation $x = \sum_{k \le n} a_k b^k, \quad a_k \in D.$ Define the set...
TAOCP 4.1 Exercise 24
Let D=\{d_0,d_1,\ldots,d_9\} be a set of ten nonnegative integers satisfying 1.
TAOCP 4.1 Exercise 25
Let \frac{u}{v}=a_0+\sum_{j\ge1}a_jb^{-j}, \qquad 0\le a_j\le b-1, be the standard radix-$b$ representation of $u/v$.
TAOCP 4.1 Exercise 26
Let S_n:=\sum_{k\le n}\epsilon_k\beta_k .
TAOCP 4.1 Exercise 27
**Exercise 4.
TAOCP 4.1 Exercise 28
Let R(e_0,\ldots,e_r) =(1+i)^{e_0}+i(1+i)^{e_1}-(1+i)^{e_2}-i(1+i)^{e_3} +\cdots+i^r(1+i)^{e_r}, \qquad e_0<\cdots<e_r.
TAOCP 4.1 Exercise 29
**Exercise 4.
TAOCP 4.1 Exercise 30
**Exercise 4.
TAOCP 4.1 Exercise 31
Let u=(\ldots u_3u_2u_1u_0.
TAOCP 4.1 Exercise 32
Let A=\left\{\sum_{i=0}^{m} a_i3^i \;:\; a_i\in\{0,1\},\ m\ge0\right\} be the set of nonnegative integers whose ternary representation uses only the digits $0$ and $1$.
TAOCP 4.1 Exercise 33
Let $D$ be any set of integers, let $b$ be any positive integer, and let $k_n$ be the number of distinct integers representable as $n$-digit numbers $(a_{n-1}\ldots a_0)_b$ with digits $a_i\in D$.
TAOCP 4.1 Exercise 34
We are asked to represent an integer $n$ in a **balanced binary form**, that is, as $(\ldots a_2 a_1 a_0)_2 = \sum_{i=0}^{\infty} a_i 2^i, \quad a_i \in \{-1, 0, 1\},$ using the **fewest nonzero digit...