I have a math class focusing on numerical analysis so I'm working with very small numbers. My professor has set $0.4 − 2^{−55} − 0.4×2^{−56} + 2^{−54} = 0.4 + 0.1×2^{-52}$ but shown no steps in between.
Clearly the initial $0.4$ can be taken out and we can set the remainder of both sides equal to one another, but that doesn't really help me. This is supposed to be a small step to look over so she has not explained it, but I can't seem to figure on my own why this is true. Specifically we're working on floating point representations of decimal numbers for double precision. Is $-2^{−55} − 0.4×2^{−56} + 2^{-54} = 0.1×2^{-52}$?
$$2^{-54}-2^{-55}-0.4\times2^{-56}$$
$$=4\times2^{-56}-2\times2^{-56}-0.4\times2^{-56}$$
$$=1.6\times2^{-56}$$
$$=0.1\times2^{-52}$$