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I am studying for an exam in my course, and I will certainly have a question of the kind:

In what base is the equation written, for example:

42-3=36

Another example:

(8*5+11)/4=12

I am wondering how to approach this kind of exercises, and how to solve them. I know how to convert from one base to another, but is there a method to solve these equations?

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    \$\begingroup\$ This question appears to be off-topic because it is about math, not EE. \$\endgroup\$ – Matt Young Jun 23 '14 at 19:44
  • \$\begingroup\$ @MattYoung, Actually I have to solve this question in a digital systems course, so I figuered it would be helpful to put it here. \$\endgroup\$ – Alan Jun 23 '14 at 19:48
  • \$\begingroup\$ @Tut not everything that is used in electronics is on topic, and this question is not on topic if you look at the help center. \$\endgroup\$ – user17592 Dec 16 '14 at 13:09
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Call the base "N". Then write the equation in base conversion form: The first one being:

4*N + 2 - 3 = 3*N + 6

Then solve for N

In general, digits in the second column (from the right) are multiplied by N, digits in the third column would be multiplied by \$N^2\$, digits in the fourth column would be multiplied by \$N^3\$, etc.

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  • \$\begingroup\$ thank you. Can you, please, explain why did you choose to put the N after the 4 and the 3? and what does the digit 7 represent? @Tut \$\endgroup\$ – Alan Jun 23 '14 at 19:52
  • \$\begingroup\$ I think you mean 4*N + 2 - 3 = 3*N + 6 which is easier to simplify... \$\endgroup\$ – Brian Drummond Jun 23 '14 at 19:55
  • \$\begingroup\$ Your question asked to find what base the equations are written. The value of N is the base. The 4 and the 3 are in the second column so are multiplied by N. If you had a three column number, you would multiply that digit by N squared, etc. \$\endgroup\$ – Tut Jun 23 '14 at 19:57
  • \$\begingroup\$ The 7 was a mistake. Alan, do you know how the decimal system works insofar as the columns and their values? The ones column has a value of 10^0, the tens a value of 10^1, the hundreds a value of 10^2 .. and it continues. The N here represents the unknown base (10 in my example). \$\endgroup\$ – sherrellbc Jun 23 '14 at 19:57
  • \$\begingroup\$ @BrianDrummond Ooops! I got a little careless in the rush to get my answer out :-) \$\endgroup\$ – Tut Jun 23 '14 at 20:01
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Do note that the the base is going to at least be 1 integral number larger than the largest value seen in the expression. For example,

42 - 3 = 36

It at least of base >= 7.


This slipped my mind. I was thinking about logs for some reason. Tut had it first.

(4*x) + (2) - (3) = (3*x) + (6)

4x + 2 - 3 = 3x + 6

x = 7

Therefore, base 7

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  • \$\begingroup\$ thank you. I thought that the only way to do it is by trial and error. If I understand your example, than if I have an equation like 12=8, than the base is at least 6? \$\endgroup\$ – Alan Jun 23 '14 at 19:46
  • \$\begingroup\$ At least of base 9 in that case. Actually, check out Tuts answer above. You can actually analytically solve in that way. \$\endgroup\$ – sherrellbc Jun 23 '14 at 19:49
  • \$\begingroup\$ No - in 12 = 8, the base must be at least 9 - the base must be at least one count larger than the largest digit used. \$\endgroup\$ – Peter Bennett Jun 23 '14 at 19:49

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