Converting between base 7 to 3

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In summary: Base 10--> 10312. Thanks for catching that.In summary, you could convert 4325 to its base-4 equivalent by doing the following: 1. multiplying 4325 by 11710, which gives you 1xxx2. multiplying 53 by 16, which gives you 13xx3. multiplying 5 by 1, which gives you 131x4. multiplying 40 by 12, which gives you 4325. multiplying 152 by 40, which gives you 8006. multiplying 3 by 1, which gives you 37. subtracting 1 from each number, which gives you 08. adding 1 to each number, which gives you 19. dividing each number by 3, which gives you 010
  • #1
tnutty
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Homework Statement



convert (352416)_7 to (...)_3

the _7 means base 7 and similarly, _3 means base 3.

Idea : convert base 7 to base 10 then base 10 to base 3.

I know I can use expansion, 3 * 7^6 + 5 * 7^6 ... to convert it to base 10.

but in my exam, we are not allowed to use calculators, and I really not want to spend time
on figuring out the power of X. So do you know a faster way to convert this to base 10?
From then on I could convert it to base 3.
 
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  • #2
I can't think of any way other than brute force, which is what you're doing. BTW, it would be 3*75 + 5*74 + ... + 1*71 + 6

You could make a couple of shortcuts, though. 74 = 492 = (50 - 1)2 = 502 - 2*50 + 1 = 2500 - 100 + 1 = 2401.
 
  • #3
Thanks for the tip.
 
  • #4
Also when say the number turned out to be something like 443267. Then to change this
number of base 10 to base 3, I could divide by 3 and get the remainder. But is there another way that you can think of.
 
  • #5
And could I have just divided the base 7 number by 3 to turn it into base 3?
 
  • #6
Once you have converted to a base-10 number, subtract the highest power of 3 that is <= your number. It's probably easier to just show an example than to explain the process.

Suppose you want to convert 4325 to its base-4 equivalent. 4325 = 11710.

The powers of 4 are 1, 4, 16, 64, 256, and so on.

117 = 64 + 53, so my base-4 representation is going to be 1xxx.
53 = 3*16 + 5, so now I have 13xx
5 = 4 + 1, so I have 131x.
My final remainder is 1, so the number we want is 13114.

It might be possible to convert directly from base 7 to base 3, but that requires you to know arithmetic in both bases; e.g. to be able to know that 6*4 = 21 in base 7, that sort of thing.
 
  • #7
tnutty said:
And could I have just divided the base 7 number by 3 to turn it into base 3?

Yes, it's pretty simple in this case but be sure to express everything in base 7:

[tex](352416)_7=3 \times (115352)_7+0[/tex]

[tex](115352)_7=3 \times (26340)_7 +2[/tex]

[tex](26340)_7=3 \times (6560)_7 +0[/tex]

[tex](6560)_7=3 \times (2164)_7 +2[/tex]

[tex](2164)_7=3 \times (521)_7 +1[/tex]

[tex](521)_7=3 \times (152)_7 +2[/tex]

[tex](152)_7=3 \times (40)_7 +2[/tex]

[tex](40)_7=3 \times (12)_7 +1[/tex]

[tex](12)_7=3 \times (3)_7 +0[/tex]

[tex](3)_7=3 \times (1)_7 +0[/tex]

[tex]\Rightarrow (352416)_7=(10012212020)_3[/tex]
 
  • #8
Mark44 said:
e.g. to be able to know that 6*4 = 21 in base 7, that sort of thing.

I think that [tex]6\times 4=33[/tex] in that base.
 
  • #9
Mark44 said:
Once you have converted to a base-10 number, subtract the highest power of 3 that is <= your number. It's probably easier to just show an example than to explain the process.

Suppose you want to convert 4325 to its base-4 equivalent. 4325 = 11710.

The powers of 4 are 1, 4, 16, 64, 256, and so on.

117 = 64 + 53, so my base-4 representation is going to be 1xxx.
53 = 3*16 + 5, so now I have 13xx
5 = 4 + 1, so I have 131x.
My final remainder is 1, so the number we want is 13114.

It might be possible to convert directly from base 7 to base 3, but that requires you to know arithmetic in both bases; e.g. to be able to know that 6*4 = 21 in base 7, that sort of thing.

Yea, that what I do with power of 2's.

Thanks everyone.
 
  • #10
Donaldos said:
I think that [tex]6\times 4=33[/tex] in that base.
My error.
 

1. What is the process of converting a number from base 7 to base 3?

The process of converting a number from base 7 to base 3 involves dividing the original number by 7 and then converting the remainder into base 3. This process is repeated until the original number becomes zero.

2. How do I convert a number with decimals from base 7 to base 3?

To convert a number with decimals from base 7 to base 3, the number must first be converted into a fraction. The decimal part of the number is then multiplied by 3 and the resulting number is converted into base 7. This process is repeated until the decimal part of the number becomes zero.

3. Can a number in base 7 be converted directly to base 3 without going through base 10?

No, a number in base 7 cannot be directly converted to base 3 without going through base 10. This is because base 10 is the most commonly used base system and serves as a bridge between different base systems.

4. What is the significance of converting between base 7 and base 3?

Converting between base 7 and base 3 is significant in various fields of mathematics and computer science, such as data compression, error correction, and number representation. It also helps in understanding the concept of different number systems.

5. Are there any online tools or calculators available for converting between base 7 and base 3?

Yes, there are many online tools and calculators available for converting between base 7 and base 3. These tools allow users to easily convert numbers between different base systems without having to perform manual calculations.

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