8.6 Exercises

8.6.1 Converting decimal numbers into base N

1. Create a conversion function dec2base(x, base) that converts a decimal number x into a positional number of different base (with \(2 \leq\) base \(\leq 10\)). Thus, the dec2base() function provides a complement to base2dec() from the i2ds package:

library(i2ds)
base2dec(11, base = 2)
#> [1] 3
dec2base(3,  base = 2)
#> [1] "11"

2. Use your dec2base() function to compute the following conversions:

dec2base(100, base =  2)
dec2base(100, base =  3)
dec2base(100, base =  5)
dec2base(100, base =  9)
dec2base(100, base = 10)

3. Create a brief simulation that samples \(N = 20\) random decimal numbers \(x_i\) and base values \(b_i\) \((2 \leq b_i <= 10)\) and shows that

base2dec(dec2base(\(x_i,\ b_i\)),\(b_i\)) ==\(x_i\)

(i.e., converting a numeric value from decimal notation into a number in base \(b_i\) notation, and back into decimal notation yields the original numeric value).

Solution

Table 8.1: Convert integer values from decimal to base notation, and back to decimal notation.

n_org	base	n_base	n_dec	same
7179	10	7179	7179	TRUE
7548	8	16574	7548	TRUE
6252	10	6252	6252	TRUE
3759	6	25223	3759	TRUE
6676	8	15024	6676	TRUE
7605	2	1110110110101	7605	TRUE
3774	2	111010111110	3774	TRUE
4546	5	121141	4546	TRUE
542	6	2302	542	TRUE
9556	9	14087	9556	TRUE
8973	8	21415	8973	TRUE
3390	10	3390	3390	TRUE
5426	6	41042	5426	TRUE
4473	4	1011321	4473	TRUE
995	4	33203	995	TRUE
3116	10	3116	3116	TRUE
1681	8	3221	1681	TRUE
7150	4	1233232	7150	TRUE
890	7	2411	890	TRUE
2248	3	10002021	2248	TRUE

8.6.2 Exercise