I read the page 2 of Mathematical Induction, I have difficulty of understanding of
The Induction Hypothesis is “If m is the integer represented by the binary array b[1,2 . . . k], then n = 2 ^ k * t + m”
1) What is m?
2) And for t is even the m is unchangeable while for t is odd the m=m+2^k, how?
Please explain above points. Thanks
m is an integer represented by the binary array in the step k, that is probably the information that is not stressed enough clearly.
Let's take an example. When you are converting n = 11 to binary, the final result will be 1011. After step k = 2 you converted two lowest significant bits, so the current binary array is 11, which gives m = 3. You divided 11 by 2 twice, so t = 11/2/2 = 2. Obviously, 2 ^ k * t + m = 2^2 * 2 + 3 = 11 = n. That explains your second question as well, because for odd t the next bit in the binary representation will be 1, which increases the current binary representation by 2^k.
