Information inside the computer

True or false? Justify each answer: (a) a bit can only be $0$ or $1$; (b) a byte consists of $10$ bits; (c) computers work in binary because their circuits only distinguish two states; (d) using the school convention, $1\ \text{KB}$ are $1000$ bytes.

Explain in your own words why computers use the binary system instead of our ten digits. Use the idea of a light switch, which can only be off or on, and the millions of tiny switches inside a processor to help you.

A bit is like a light bulb that can only be off ($0$) or on ($1$). How many "light bulbs" are needed to form a byte? Draw eight light bulbs in your notebook representing the byte $01100001$ and explain what each one means.

You can do it on paper and send a photo with your answer.

Explain why a computer does not directly store letters, colors or sounds, but zeros and ones: what the transistors in its circuits have in common with a light switch and what advantage working with only two possible states offers.

Match each unit to its definition: (a) bit, (b) byte, (c) kilobyte, (d) gigabyte. Definitions: 1. group of $8$ bits; 2. the smallest unit of information, a $0$ or a $1$; 3. using the school convention, $1024$ megabytes; 4. using the school convention, $1024$ bytes.

The figure shows eight boxes that together form a byte: seven already have a value and the last one is covered with a question mark, as shown in the diagram.

Which two values could the covered box hide? How many bits make up a byte? Write a byte created by you with its eight digits and explain what each box represents.

In the code computers use for text, each letter takes up $1$ byte. Calculate how many bytes the word "HOLA" takes up and how many bits that is in total, and explain the calculation you made.

Match each file to the most reasonable size: (a) a short text message, (b) a photo taken with a cell phone, (c) a song, (d) a movie. Sizes: 1. around $4\ \text{MB}$; 2. more than $1\ \text{GB}$; 3. a few bytes; 4. around $8\ \text{MB}$. Explain which of the four you based on your own experience.

Order these units from smallest to largest: megabyte, bit, gigabyte, byte, terabyte, kilobyte. Then explain, using the school convention $1\ \text{KB} = 1024\ \text{bytes}$, the relationship between each unit and the next one in the list.

The figure shows the ladder of information units: each step is climbed by multiplying, but the last step has lost its name, as shown in the diagram.

Which unit is missing from the last step? Write the full chain of equivalences from the bit up to that unit using the school convention ($1\ \text{KB} = 1024\ \text{bytes}$ and the same when going up to each larger unit) and explain in your own words why the step from bit to byte is different from the others.

Using the school convention $1\ \text{KB} = 1024\ \text{bytes}$, calculate and write down the operations: (a) how many bytes are in $3\ \text{KB}$; (b) how many kilobytes are in $2\ \text{MB}$; (c) how many bits are in $4$ bytes.

A cell phone photo takes up $4\ \text{MB}$. Using the school convention ($1\ \text{GB} = 1024\ \text{MB}$), calculate how many photos fit into $1\ \text{GB}$ and how many into $64\ \text{GB}$, and explain why in practice slightly fewer fit than calculated.

The bar in the figure shows how the $64\ \text{GB}$ of a cell phone are divided between the system, apps, photos and videos, and free space, but the section for photos is missing its value, as shown in the diagram.

How many gigabytes do the photos and videos take up? How many photos of $6\ \text{MB}$ would still fit into the free space? (Use the school convention $1\ \text{GB} = 1024\ \text{MB}$ and show your work.)

A song takes up $8\ \text{MB}$ and your player has $16\ \text{GB}$ of free space remaining. Using the school convention, calculate how many songs fit and explain why in practice slightly fewer fit than the calculation gives.

A page of text without images takes up about $2\ \text{KB}$. Using the school convention, calculate how many pages fit into $1\ \text{MB}$ and answer with reasoning: which takes up more space, a $300$-page novel or a single photo of $4\ \text{MB}$?

Your brother wants the $128\ \text{GB}$ cell phone and you say that $64\ \text{GB}$ is enough. Calculate exactly how many $5\ \text{MB}$ photos fit into the difference between both models (using the school convention) and explain for what type of person paying for the larger model is worth it.

Mara's cell phone has $64\ \text{GB}$ and is almost full: the system takes up $12\ \text{GB}$, apps take up $20\ \text{GB}$, photos take up $28\ \text{GB}$ and $4\ \text{GB}$ remain free. Check that the math adds up, calculate how many $7\ \text{MB}$ photos she would need to delete to free up $10\ \text{GB}$ (using the school convention) and suggest two other ways to gain space without deleting memories.

The school's audiovisual team is going to broadcast a charity festival that lasts $3$ hours and $45$ minutes without breaks. The camera records continuous video at a rate of $600\ \text{MB}$ per minute, and the school has a single $128\ \text{GB}$ memory card available.

1. a) Using the school convention ($1\ \text{GB} = 1024\ \text{MB}$), determine through exact calculations whether the card has enough capacity for the entire event or at what minute of the recording it will stop due to a lack of memory.
2. b) Design two viable technical alternatives to ensure that no footage of the festival is lost: one by modifying the camera's capture settings and another by planning the management of storage media.

You must submit a digital portfolio containing the technology projects for the course on a $16\ \text{GB}$ USB flash drive. Your team's folder contains: $1800$ photographs of $4.5\ \text{MB}$ each, $14$ demonstration videos of $650\ \text{MB}$ each, and $200$ technical diagrams in PDF format of $1024\ \text{KB}$ each.

1. a) Prove with detailed calculations (using the school convention) whether all the content fits on the USB flash drive.
2. b) If the space is not enough, calculate the excess in megabytes and in gigabytes. Design a technical strategy to reduce the file size without deleting any file and calculate what overall compression percentage you need to achieve so that the project fits on the flash drive.

The school's digital library server allocates a maximum quota of $25\ \text{GB}$ to store the annual work of your class of $30$ students. Each student must upload $3$ full projects during the course; each submission consists of a $12\ \text{MB}$ PDF report, a $68\ \text{MB}$ photo folder, and an explanatory video.

1. a) Applying the school convention ($1\ \text{GB} = 1024\ \text{MB}$), calculate the maximum size in megabytes that each project's video can occupy so that the submissions from the $30$ students fit on the server without exceeding the allocated limit.
2. b) If $10$ students upload their videos with a size of $400\ \text{MB}$ each while the others respect the calculated limit, predict how many gigabytes of excess will be generated and propose a technical protocol to solve the problem before the server space runs out.

When connecting a new external hard drive labelled by the manufacturer as "$500\ \text{GB}$", the operating system reports that its total capacity is approximately $465.66\ \text{GB}$.

1. a) Numerically demonstrate the origin of this difference of nearly $35\ \text{GB}$, knowing that manufacturers use the decimal system ($1\ \text{GB} = 10^9\ \text{bytes}$) while operating systems calculate in binary base ($1\ \text{GiB} = 1024^3\ \text{bytes}$).
2. b) Predict how many usable gigabytes the operating system will show if you connect a $2\ \text{TB}$ drive, and write a technical answer arguing whether this commercial discrepancy constitutes misleading advertising or is due to a legitimate difference in conventions.

A classmate makes the following claim during a practical session: "Since a gigabyte is one thousand megabytes and a megabyte is one thousand kilobytes, a $1\ \text{GB}$ file takes twice as long to download as two $500\ \text{MB}$ files, and a terabyte is exactly one million bytes".

1. a) Identify and refute each of the errors present in their claim, showing with the school convention ($1\ \text{KB} = 1024\ \text{B}$, $1\ \text{MB} = 1024\ \text{KB}$, etc.) the real numerical scale between the bit, the byte, the KB, the MB, the GB, and the TB.
2. b) If a connection transfers data at a constant real rate of $32\ \text{MB/s}$, predict with exact calculations how long it will take to download a $1\ \text{GB}$ file compared to a $1\ \text{MB}$ file, and explain why the difference is not a simple factor of one thousand.

Two tablets cost the same: one has $64\ \text{GB}$ and comes with free cloud storage; the other has $128\ \text{GB}$ but no cloud storage. Analyse the advantages and disadvantages of each option (what happens if the device breaks, what happens when there is no internet connection) and recommend one for a student of your age, justifying your choice.

A video game console is advertised with an internal drive of $500\ \text{GB}$. When you turn it on, the operating system and recovery software take up $85\ \text{GB}$, and the manufacturer always recommends leaving $10\,\%$ of the total capacity free for virtual memory and system updates.

1. a) Calculate how many full games of $48\ \text{GB}$ and how many gameplay videos of $1500\ \text{MB}$ can coexist at the same time on the console if you want to reserve at least $15\ \text{GB}$ exclusively for videos (use the school convention of $1\ \text{GB} = 1024\ \text{MB}$).
2. b) If you connect a $1\ \text{TB}$ external hard drive (with the same convention, $1\ \text{TB} = 1024\ \text{GB}$) dedicated exclusively to installing games, calculate how many more games of the same size you could install and justify why on this external drive you can actually use almost all the space, unlike on the internal drive.