Converting from Base b to Base 10
The next natural question is: how do we convert a number from another base into base 10? For example, what does 42015 mean? Just like base 10, the first digit to the left of the decimal place tells us how many 50’s we have, the second tells us how many 51’s we have, and so forth. Therefore:
42015 = (4.53 + 2.52 + 0.51 + 1.50)10
= 4.125 + 2.25 + 1
=55110
From here, we can generalize. Let x =(anan-1 … a1a0)b be an n + 1 -digit number in base b. In our example (274610) a3 = 2, a2 = 7, a1 = 4 and a0 = 6. We convert this to base 10 as follows:
x = (anan-1 … a1a0)b
= (bn.an + bn-1 . an-1 + … + b.a1 + a0)10
Converting from Base 10 to base b
It turns out that converting from base 10 to other bases is far harder for us than converting from other bases to base 10. This shouldn’t be a suprise, though. We work in base 10 all the time so we are naturally less comfortable with other bases. Nonetheless, it is important to understand how to convert from base 10 into other bases.
We’ll look at two methods for converting from base 10 to other bases.
Method 1
Let’s try converting 1000 base 10 into base 7. Basically, we are trying to find the solution to the equation 1000 = a0 + 7a1 + 49a2 + 343a3 + 2401a4 + …
Read more below-
SS 2 Mathematics Third Term: Problem Solving on Number Bases Expansion, Conversion and Relationship
