How can I show that $3\times2^{-53} = 2^{-52}+2^{-53}$?

Question

I am taking a numerical analysis course and I am doing some practice problems dealing with IEEE floating point arithmetic. A big part of this process seems to be the identity: $3\times2^{-53} = 2^{-52}+2^{-53}$.

The book claims this to be true due to the mantissa (truncating) after the 52nd bit. However, I cannot comprehend how this can be true. The book also states that $3\times2^{-52} = 2^{-51}+2^{-52}$ is also true.

How can I show such an equality to be true?

Answer 1

$3\cdot \frac{1}{2^{53}} = 2\cdot \frac{1}{2^{53}}+\frac{1}{2^{53}} = \frac{1}{2^{52}}+\frac{1}{2^{53}}$

Answer 2

$2^{-52}+2^{-53} = 2\cdot2^{-53} + 2^{-53} = 3\cdot 2^{-53}.$

Answer 3

Multiply each side by $2^{53}$.

Answer 4

Hint

Write $3$ like this:

$$3=2+1$$

Answer 5

Notice this one..... ((

Hope it helps