Question

Convert the following numbers to binary and then to hexadecimal. If the answer is irrational,stop at four digits after number point:(a) 0.6640625^10 (The power of 10)(b) 27.625^10 (The power of 10)

Answer 1

Let us start with

a. To convert

`0.6640625_(10)\text{ to base 2}`

To do this, we keep multipllying by 2 and keeping track of the whole number part. I will demonstrate this using the equation tab

So 0.6640625 ten to binary =

`0.1010_2`

So to hexadecimal, 10 means A. So this becomes

`\begin{gathered} 0.6640625\text{ }*\text{ 16 = 10.625} \\ So\text{ we pick the whole number which is 10 } \\ \text{And 10 is A.} \\ \text{Next we multiply the decimal 0.625 by 16 = 10} \\ \text{And 10 is A} \\ \text{Therefore the answer is }0.6640625_2=0.AA_(16) \end{gathered}`

(b) 27.625 means 27 + 0.625

So we will convert both the whole number part and the decimal part to base 2 and base 16, then we combine to get the answer

`\begin{gathered} 27\text{ to base 2} \\ \text{Divide by 2 and keep track of the remainder } \\ \text{read the remainder from bottom to the top} \\ 27_(10)=11011_2 \end{gathered}`

Now the decimal part

`\begin{gathered} \text{Just like I did above,} \\ m\text{ultiply the decimal 0.625 by 2 and note the whole number part} \\ 0.625\text{ }*\text{ 2 = 1.250} \\ \\ 1.250\text{ }*\text{ 2= 0.500} \\ \text{0.500 }*\text{ 2 = 1.00} \\ \text{The whole number part from above is 101} \\ \text{This becomes 0.101}_2 \end{gathered}`

Therefore

`27.625_(10)\text{ to binary }=11011.101_2`

To hexadecimal we will do the same, but 10 becomes A, 11 becomes B, 12 becomes C.

`27_{10\text{ }}to\text{ decimal }=1B_(10)`
`\begin{gathered} 0.625\text{ }*\text{ 16 = 10.00 and 10 is A} \\ So\text{ }0.625_(10)\text{ to hexadecimal }=A_(16) \end{gathered}`

Therefore

`27.625_{10\text{ }}to\text{ hexadecimal }=1B.A_(16)`

That is 1B.A to base 16