Question

Convert each of the following 5-digit 6’s complement base 7 numbers to 9-digit 4’s complement base 4. Be careful, show all your reasoning/arithmetic, and make no unwarranted assumptions!

(a) 50135

(b) 11402

Answer 1

Solution

Step 1: By definition, the formula of the base or radix complement of an n digit number y in radix b is bⁿ-y. There is a diminished radix complement given as (bⁿ-1)-y.

For example Take the number (1010)₂. The base or radix=b=2. Number of digits=n=4. Then 1's complement is diminished radix complement which is (2⁴-1)-(1010)

=(16-1)-(1010)=(15)-(1010)=(1111-1010)=0101

Step 2: Similarly we have

(i)(50135)₇ in 6's complement=((bⁿ-1)-y)₇. We need (b1ⁿ¹-y1)₄.

(7^5-1)-y=50135

y=((16807)₁₀-1)-(50135)₇)

=16807₁₀-1-12080₁₀=4726₁₀ = 16531₇

(bⁿ-1)-y=(b1ⁿ¹-y1)

-> (50135)₇=(4⁹-y1)₄

Convert 50135 from base 7 to base 4.

Converting 50135 to base 10.
5*7⁴=12005
0*7³=0
1*7²=49
3*7¹=21
5*7⁰=5
Adding all to get Ans=12080₁₀

converting 12080₁₀ to 4

The equation calculation formula for 12080₁₀ number to 4 is like this below.
4|12080
4|3020|0
4|755|0
4|188|3
4|47|0
4|11|3
4|2|3
4|2|2
Ans:2330300₄

->2330300₄=(4⁹-y1)₄

->y1=4⁹-2330300₄

->y1=(262144)₁₀ - 2330300₄

Converting 262144 to base 10.
2*10⁵=200000
6*10⁴=60000
2*10³=2000
1*10²=100
4*10¹=40
4*10⁰=4
Adding all to get Ans=262144₁₀

converting 262144₁₀ to 4

The equation calculation formula for 262144₁₀ number to 4 is like this below.
4|262144
4|65536|0
4|16384|0
4|4096|0
4|1024|0
4|256|0
4|64|0
4|16|0
4|4|0
4|1|0
4|1|1
Ans:1000000000₄

y1=1000000000₄ - 2330300₄ = (262144)₁₀ - 12080₁₀ =250064₁₀=331003100₄.

But we need (4⁹-331003100)₄ = 262144₁₀-250064₁₀ = 12080₁₀ = 2330300₄

Hence the answer is 2330300₄

Step 3 :

Number of digits n1=9, base=b1=4 and we need y1 value. Or, let us find the value of z ie (b1ⁿ¹-y1)₄

(11402)₇=(b1ⁿ¹-y1)₄

So we need to convert 11402 to base 4 which is as follows

Converting 11402 to base 10.
1*7⁴=2401
1*7³=343
4*7²=196
0*7¹=0
2*7⁰=2
Adding all to get Ans=2942₁₀

Step2 converting 2942₁₀ to 4

The equation calculation formula for 2942₁₀ number to 4 is like this below.
4|2942
4|735|2
4|183|3
4|45|3
4|11|1
4|2|3
4|2|2
Ans:231332₄

Hence the answer is 231332₄