Question

A gigabyte is one billion bytes; a terabyte is one trillion bytes. A byte is 8 bits, each a 0 or a 1 . Because \(2^{10}=1024\), which is about 1000 , you can store about three digits (any number between 0 and 999) in 10 bits. About how many decimal digits could you store in five gigabytes of memory (a gigabyte is \(2^{30}\), or approximately one billion bytes)? About how many decimal digits could you store in five terabytes of memory (a terabyte is \(2^{40}\), or approximately one trillion bytes)? How does this compare with the number \(10^{120}\) ? (To do this problem, it is reasonable to continue to assume that 1024 is about 1000 .)

Short answer

5 GB stores approximately 4 billion digits. 5 TB stores about 4 trillion digits; both are far less than \(10^{120}\) digits.

Step-by-step solution

Understanding Memory Units

First, we need to confirm the numbers given: 1 gigabyte (GB) is approximately \(10^9\) bytes, and 1 terabyte (TB) is approximately \(10^{12}\) bytes. Since 1 byte has 8 bits and 10 bits can store approximately 3 digits, we calculate how many digits 1 byte can store. Since 8 bits are slightly less than 10 bits, we estimate that 1 byte, which is less than 10 bits, stores less than 3 digits.

Calculating Digits in Gigabytes

For 5 gigabytes, we have \(5 \times 10^9\) bytes. Since each byte contains 8 bits, the number of bits in 5 GB is \(5 \times 10^9 \times 8\). Dividing by 10, we determine the number of 10-bit groups: \(5 \times 10^9 \times \frac{8}{10} = 4 \times 10^9\). This number tells us how many decimal digits can be stored because each 10-bit group stores approximately 3 digits.

Calculating Digits in Terabytes

Similarly, for 5 terabytes, we have \(5 \times 10^{12}\) bytes. Calculate the total number of bits: \(5 \times 10^{12} \times 8\). Dividing by 10 yields the number of 10-bit groups: \(5 \times 10^{12} \times \frac{8}{10} = 4 \times 10^{12}\). Hence, this is how many decimal digits can be stored.

Comparison with \(10^{120}\)

The total number of decimal digits that can be stored in 5 terabytes is \(4 \times 10^{12}\), which is significantly less than \(10^{120}\). Therefore, the storage capacity is minimal compared to \(10^{120}\).

Conclusion

To compare, 5 gigabytes can store approximately \(4 \times 10^9\) digits and 5 terabytes can store \(4 \times 10^{12}\) digits, both of which are much less than a number with \(10^{120}\) digits.

Key concepts

Bits and Bytes

Every digital memory storage system is fundamentally built on the idea of bits and bytes. Let's break down these terms so you fully understand what they mean. A **bit** is the smallest unit of data in a computer, representing a binary value of 0 or 1. Bits are like tiny switches that computers use to represent and store information. Just like in the real world, we can control systems by turning switches on or off.
- **1 Byte** is equal to 8 bits.
- Since each bit is a binary number, a byte can represent numbers from 0 to 255. Understanding the bit-byte relationship is crucial because it helps us measure and manage data storage capacities. Whether saving a small text file or a large video, it all boils down to how many bits and bytes you need.

Decimal Representation in Memory

When we talk about storing information in computers, we often think in terms of bytes and bits. However, sometimes we need to understand how this translates into storing regular numbers that we use daily, called decimal numbers. To grasp this concept better, imagine needing a certain number of bits to represent numbers from 0 to 999 (which is approximately 3 digits). From our understanding that 10 bits can store about three digits, we can see how useful these simple units can be in representing larger numbers.
- **10 bits** can hold numbers up to 1023, but for simplicity and ease of calculation, they're estimated to store about 1000 (or 3 decimal digits).
When translating this to memory storage, as mentioned in step-by-step calculations, a byte (which has 8 bits) can hold slightly less than three digits, mostly used to store two full digits comfortably. This kind of approximation is handy when calculating how many digits five gigabytes or terabytes of memory can handle.

Gigabyte and Terabyte Comparisons

Storage on a computer or any digital device is typically measured in bytes, which can escalate quickly to larger units such as gigabytes (GB) and terabytes (TB). Understanding these units can help you manage your own data more efficiently and make informed decisions based on your needs.A **gigabyte** is approximately equal to one billion bytes. But when it comes to how computers use the storage, it's actually a multiple of base 2 (binary), precisely 2 to the power of 30 bytes. Similarly, a **terabyte** follows the same principles, being about one trillion bytes or 2 to the power of 40 bytes.

• **5 GB** can store approximately 4 billion digits, calculated from the number of bytes times 8 (bits per byte), divided by 10 because 10 bits store roughly three digits.
  
• **5 TB** can store close to 4 trillion digits, with similar calculations but on a larger scale.

By comparing these numbers with an extremely large digit number like \(10^{120}\), the storage capacities of gigabytes and terabytes may seem limited. This is due to the exponential growth of numbers, which digital storage units cannot practically reach with today's technology and methods.