This is one of those questions that is difficult to answer out of context, which is why folks are answering with more questions. The answer depends on where the base 10 and binary numbers are, and how they are being used, so a little more background would help. I'm not sure about the "algebraic equation" part. Algebra deals with variables and numbers, and the base is largely irrelevant. The algebraic equation a = b + c is valid regardless of the base of the variables.

So I'm not sure there is such a thing as an algebraic equation to do base conversion. But as others pointed out, there certainly are

*algorithms* to do base conversion, and some are even built into the Arduino environment.

On the other hand, the internal representation of a number in hardware is always binary. I like to say (usually when the boss is really steamed about something) "It's all just ones and zeroes." Or as Tom Lehrer said, "Base 8 is just like base 10 really, if you're missing two fingers."

Ultimately, for humans to be able to read and understand a number from a computer, it has to be converted to

*characters* and displayed on some device. As @retrolefty indicated, the Arduino Serial.print() function can take a number stored in the MCU and convert it to a

*character representation* in any of several bases: binary (base 2), octal (base eight), decimal (base 10), or hexadecimal (base 16).

If the problem here is to output a number in a given base, then Serial.print() is one way. On the other hand, if the problem is for a person to

*input* a number in a given base, and convert it to the proper internal representation, then that is a different algorithm.

Sorry for the long-winded reply, we need more information to help, the question as asked is too open to interpretation.

Tom Lehrer: