Introduction to Algorithms
Week 1: Exercises

These are optional exercises for practicing the concepts we learned this week.

1. Powers of Two

What is the approximate value of

a) 2¹⁴

b) 2³²

c) 8⁸

2. Logarithms

What is the approximate value of

a) log₂(8,000,000,000)

b) log₁₆(16,000,000)

3. A Slow Connection

A notebook computer's disk holds 500 Gb of data. We would like to send the data across an extremely slow network that transfers only 1 byte/second. Approximately how long will the transfer take?

4. Squares Mod 4

Is this statement true or false?

For all n ∈ ℤ, n² mod 4 < 2.

Prove your answer.

5. Bases

Calculate by hand:

• What is 110100₂ in base 10? What is AB₁₆ in base 10?

• What is 95₁₀ in base 2? In base 16?

• In hexadecimal, what is the smallest and largest value that can be stored in a byte?

• How many digits does 100₁₀ have when written in base 5? How many digits does 500₁₀ have when written in the same base?

6. Bases, Continued

Calculate by hand:

1. What is 67₈ in base 10?

2. What is 67₁₀ in base 8?

3. What is 2376₁₀ in base 100? In base 1000?