Other Numeral Systems

In computer science, the smallest unit of information is the bit, which can only take the values $0$ or $1$ (binary system). An old processor in a retro console uses 8-bit registers. If a character has an energy level represented by the decimal number $43$, how would that value appear in the console's binary code?

Computers use the binary system (base 2) to process information, using only the digits $0$ and $1$. To understand how a machine "thinks", convert the decimal number $13$ to binary code using the method of successive divisions by $2$.

The diagram shows the divisions you need to do. In each row, the quotient from the previous row is divided by $2$, and you continue until the quotient is $0$.

1. a) Fill in the boxes of the diagram: the quotient and remainder of each division, until you reach a quotient of $0$.

1. b) Read the remainders from bottom to top (from the last to the first) and write the number $13$ in base 2.

In a computer system, multiplying a binary number by $2^n$ is equivalent to shifting its digits $n$ positions to the left (adding zeroes at the end).

Given the decimal number $21$:

1. a) Express it in the binary system.
2. b) Perform the operation of shifting its digits $3$ positions to the left.
3. c) Convert the resulting number back to the decimal system and check that the result is equivalent to multiplying $21 \times 2^3$.

A file server for an esports gaming community has a total capacity of $2^{35}$ bits. Currently, the server contains three folders with game screenshots: folder A occupies $1.2$ GB, folder B occupies $950$ MB and folder C contains $2^{30}$ bytes. Given that $1$ GB = $1024$ MB and $1$ MB = $1024$ KB, determine whether it is possible to save an additional video of $1.5$ GB without exceeding the server's total capacity. Express all calculations as powers of base 2 where possible.

On the façade of an old theater in your city, the year of its opening is carved in Roman numerals: MCMXCVIII. Convert this number to our decimal system to find out which year the theater opened.

Calculate the result of the following operation with Roman numerals, expressing the final value both in Roman numerals and in the decimal number system:

$(MMCDLXIV - MDCCCLXXXIX) \times XII$

The Mayan number system was vigesimal (base 20) and vertically positional. Look at the following number represented by its levels and convert it into our decimal system:

• Level 3 (top): One dot.

• Level 2 (middle): Two bars and three dots.

• Level 1 (bottom): One bar.

(Remember that in the Mayan system, the dot is worth $1$ and the bar is worth $5$).

In an archaeological dig, an inscription with an unknown number system was found. After analyzing it, experts determined that it is a base-4 system, where the symbols are: $\triangle = 0$, $\square = 1$, $\bigcirc = 2$ and $\star = 3$.

If the treasure is located at a distance of $\bigcirc \star \square \triangle$ meters from the main entrance, calculate this distance in our decimal system. Then, express the result obtained in the binary system.

An intergalactic explorer has arrived on a planet where the natives use a base 5 numeral system. You are given the following equivalences to understand their language:

$1_{5} = \text{I}$, $2_{5} = \text{V}$, $3_{5} = \text{X}$, $4_{5} = \text{L}$.

If they tell you that there are $\text{XXL}$ hours left until the mothership takes off:

1. a) Write the number of hours in the decimal system.
2. b) Calculate how many minutes that time is equivalent to.