Exercises 5.4 Exercises
1.
The sequence \(2^n\)—that is, the powers of \(2\)—can be represented recursively in two different ways. What is one of those ways?
Hint.Hint: One of them involves addition, the other involves multiplication.
2.
Write \(2391\) in binary.
3.
Write \((11110101010)_2\) in decimal.
4.
Write \(954\) in binary.
5.
Write \(213\) in octal.
6.
Write \(756\) in hexadecimal.
7.
Write \((AF3)_{16}\) in decimal.
8.
Use binary numbers to prove that
Hint: What do you get when you add \(1\) to \((111)_2\text{?}\) What about \((1111)_2\text{?}\)
Solution: The number
can be represented as the \(n+1\) digit number \((111\ldots 11)_2\text{.}\) If we add \(1\) to this number, we will have to “carry the one” several times and the result will be the \(n+2\) digit number
which is \(2^{n+1}\text{.}\) So, we have shown
which can be rearranged to
9.
A number written in ternary is written in base-\(3\) with digits \(0\text{,}\) \(1\text{,}\) and \(2\text{.}\) Write the number \(143\) in ternary.
10.
Write the number \(143\) in base-\(9\text{.}\)