231(n) - 143(n) = 44(n). Find the number base n.?
Even though oobleck's method is the most direct way of doing this, you might also see it easily this way:
231(n) - 143(n) = 44(n), where n is the base of those numbers
so ...
2n^2 + 3n + 1 - (n^2 +4n +3) = 4n + 4
n^2 - 5n -6 = 0
(n-6)(n+1) = 0
so n = 6 or n = -1, and we probably don't want a negative base.
well, since 1 < 3, you have to borrow n from 3.
you have 31-3 ends in 4, so n+1-3=4, so n=6
Well, well, well, it looks like we're in the land of algebra! Let's solve this equation together, shall we?
231(n) - 143(n) = 44(n)
Now, to find the number base n, we need to find a value for n where this equation holds true.
Let's break it down step by step:
231(n) - 143(n) = 44(n)
231 - 143 = 44
88 = 44(n)
So, now we have a whole new equation:
88 = 44(n)
Divide both sides by 44:
2 = n
Voila! The number base n is 2! I hope you had fun on this math adventure with me.
To find the number base, we need to determine what value of n will make the equation true.
Let's simplify the equation by performing the arithmetic operations:
231(n) - 143(n) = 44(n)
To subtract numbers in different bases, we need to convert them to the same base. Since we're looking for the number base, let's convert all the numbers to base 10.
To convert a number from base n to base 10, we use the formula:
(number in base 10) = (digit1 * n^0) + (digit2 * n^1) + (digit3 * n^2) + ...
Let's apply this formula to convert the numbers in the equation to base 10:
231(n) = (1 * n^0) + (3 * n^1) + (2 * n^2)
143(n) = (3 * n^0) + (4 * n^1) + (1 * n^2)
44(n) = (4 * n^0) + (4 * n^1)
Now, let's rewrite the equation using base 10 numbers:
(1 * n^0) + (3 * n^1) + (2 * n^2) - (3 * n^0) - (4 * n^1) - (1 * n^2) = (4 * n^0) + (4 * n^1)
Simplifying the equation:
(1 - 3 + 4) * n^0 + (3 - 4 + 4) * n^1 + (2 - 1) * n^2 = 0 * n^0
2 * n^0 + 3 * n^1 + 1 * n^2 = 0 * n^0
Next, we collect like terms and set the coefficients equal to zero:
2 * n^0 = 0
3 * n^1 = 0
1 * n^2 = 0
Since, by definition, any number raised to the power of 0 is equal to 1, we have:
2 = 0
3 * n = 0
1 * n^2 = 0
The equation 2 = 0 has no solution. However, for 3 * n = 0 to be true, n must be 0. Similarly, for 1 * n^2 = 0 to be true, n must also be 0.
Therefore, the number base (n) that satisfies the equation is 0.
We move all terms to the left:
231n-143n-(44n)=0
We add all the numbers together, and all the variables
44n=0
n=0/44