Data Representation

1 Convert the denary number 187 into 8 bit binary. Show your working. 3

One mark for each of the following, to a maximum of 3.

• Uses place values from 128 downwards
• Shows the subtraction working
• Correct answer 10111011

Model answer. Working from the largest place value: 128 fits into 187 leaving 59, so write 1. 64 does not fit into 59, write 0. 32 fits leaving 27, write 1. 16 fits leaving 11, write 1. 8 fits leaving 3, write 0 for 8 being used, so write 1. 4 does not fit into 3, write 0. 2 fits leaving 1, write 1. 1 fits leaving 0, write 1. This gives 10111011, and checking: 128 + 32 + 16 + 8 + 2 + 1 = 187.

2 Explain why hexadecimal is often used instead of binary. 3

One mark for each of the following, to a maximum of 3.

• One hexadecimal digit represents four binary digits
• Values are much shorter to write and read
• Shorter values are easier for people to remember and less prone to error

Model answer. Hexadecimal is used because each hexadecimal digit represents exactly four binary digits, so an eight bit value that needs eight characters in binary needs only two in hexadecimal. Values written in hexadecimal are therefore around a quarter of the length, which makes them far quicker for a person to write down and read back. Because the values are short, people make fewer transcription mistakes and any error that is made is much easier to spot, which is why colour codes, MAC addresses and memory addresses are all normally written in hexadecimal. The computer itself still works entirely in binary.

3 Add the binary numbers 01011010 and 00110110, showing your working, and state the denary value of the answer. 4

One mark for each of the following, to a maximum of 4.

• Adds column by column from the right with carries
• Correct binary answer 10010000
• Converts the answer to denary
• States that no overflow occurs as the answer fits in 8 bits

Model answer. Adding column by column from the right and carrying wherever two or three ones meet gives 10010000. Checking in denary, 01011010 is 90 and 00110110 is 54, and 90 plus 54 is 144, which is 10010000 in binary since 128 + 16 = 144. No overflow error occurs, because 144 is within the range 0 to 255 that eight bits can represent, so nothing is carried out of the most significant column and no data is lost.

4 A photograph is 1200 pixels wide and 800 pixels high with a colour depth of 24 bits. Calculate the file size in megabytes. 4

One mark for each of the following, to a maximum of 4.

• Multiplies colour depth by width by height
• Obtains 23,040,000 bits
• Divides by 8 to obtain 2,880,000 bytes
• States 2.88 MB

Model answer. File size in bits is colour depth multiplied by width multiplied by height, so 24 x 1200 x 800 = 23,040,000 bits. Dividing by 8 gives 2,880,000 bytes. Dividing by 1000 gives 2880 kilobytes, and dividing by 1000 again gives a file size of 2.88 megabytes.

5 Explain how sound is stored digitally, and describe the effect of increasing the sample rate. 4

One mark for each of the following, to a maximum of 4.

• The amplitude of the sound wave is measured at regular intervals
• Each measurement is stored as a binary number
• A higher sample rate means measurements are taken closer together in time
• So the recording is a more accurate copy of the original wave, but the file is larger

Model answer. Sound in the real world is a continuously varying wave, which cannot be stored directly because a computer can only store numbers. Instead the amplitude of the wave is measured at regular fixed intervals, a process called sampling, and each measurement is rounded to the nearest available level and stored as a binary number. Increasing the sample rate means those measurements are taken more often and therefore closer together in time, so less of the wave's shape is missed between samples and the stored data follows the original much more closely, producing a more accurate and better sounding recording. The cost is file size, which is directly proportional to sample rate, so doubling the sample rate doubles the amount of storage the recording needs.