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# Number Systems : Base Conversions, Pairwise Distances in a 3-Cube, Weighted Decimal Codes and Gray Codes

Please show answers with all steps. See attached file for full problem description.

keywords: converting bases

#### Solution Preview

1.1 (a)

8 ) 523
8 ) 65 r 3 ^
8 ) 8 r 1 |
8 ) 1 r 0 |
0 r 1 | read backwards gives 1013 (base 8)

0.1 × 8 = 0.8 + 0
0.8 × 8 = .4 + 6
0.4 × 8 = .2 + 3
0.2 × 8 = .6 + 1
0.6 × 8 = .8 + 4
0.8 × 8 = .4 + 6

Ans: 523.1 dec = 1013.0[6314] oct
[] means recurring
write a vertical line over those recurring digits or
put a dot on the first and the last recurring digit
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1.1 (b)

from 1.1(a) octal, convert to binary according the
following scheme
0 -> 000 1 -> 001 2 -> 010 3 -> 011
4 -> 100 5 -> 101 6 -> 110 7 -> 111

Ans: 101010011.000[110011001100]
[] means recurring
write a vertical line over those recurring digits or
put a dot on the first and the last recurring digit
---------------------------------------------------------------
1.1 (c)

apply the reverse method of 1.1 (b)
101.11 bin = 101.110 bin = 5.6 oct (Ans)

---------------------------------------------------------------
1.1 (d)

continue from 1.1 (c)
101.11 bin = 5.6 oct = 5 + 6/8 = 5.75 (Ans)
---------------------------------------------------------------
1.1 (e)

convert to base 10 first
1100.11 bin
= 1 × 2^3 + 1 × 2^2 + 0 × 2^1 + 0 × 2^0 + 1/2 + 1/2^2
= 12 + 3/4
= 12.75

7 ) 12
7 ) 1 r 5 ^ read backwards gives 15 (base 7)
0 r 1 |

0.75 × 7 = 0.25 + 5
0.25 × 7 = 0.75 + 1
0.75 × 7 = 0.25 + 5 ... etc

Ans: 1100.11 bin = 15.[51] (base 7)

[] means recurring
write a vertical line over those recurring digits or
put a dot on the first and the last recurring digit
===============================================================
1.2 A number in base b can be represented as
... d2 d1 d0 (base b)
which has value
... + d2×b^2 + d1×b + d0 (*)

In base b, the maximum digit has value (b-1), and the
maximum product of two digits has value
(b-1)^2
= b^2 ...

#### Solution Summary

Base Conversions, Pairwise Distances in a 3-Cube, Weighted Decimal Codes and Gray Codes are investigated.

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