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Answers

A7.1 Number sets, bases and units

Data representation · A level · OCR H446 1.4.1, AQA 7517 4.5.1.1, Eduqas A500QS 2.3 · about 25 min

BugBotLab

What this lesson is about

Natural, integer, rational, irrational, real and ordinal numbers; any number base; bits, bytes, kilo and kibi.

Questions 6 marks in all

  1. [1 mark]Which set of numbers is the square root of 2 in, but not in the rational numbers?

    1. AIrrational numbers
    2. BNatural numbers
    3. CIntegers
    4. DOrdinal numbers
    Answer: A. The square root of 2 cannot be written as a fraction of two integers, so it is irrational (and real).

  2. [1 mark]Which of these are rational numbers?

    Tick every answer that is true.

    1. A0.75
    2. B-3
    3. C22/7
    4. DThe square root of 2
    5. EPi
    Answer: A, B, C. A rational number can be written as one integer divided by another: 3/4, -3/1 and 22/7. The square root of 2 and pi cannot.

  3. [1 mark]What are ordinal numbers used for?

    1. ATo describe a position in an ordered list, such as 1st, 2nd, 3rd
    2. BTo count how many things there are
    3. CTo measure a quantity such as a length
    4. DTo store negative values
    Answer: A. Ordinal numbers give position. Counting uses natural numbers; measuring uses real numbers.

  4. [1 mark]Write the denary number 255 in base 8.

    Answer: 377. 255 = 3 x 64 + 7 x 8 + 7, so the digits are 3, 7, 7.

  5. [1 mark]How many bytes are in 3 KiB?

    Answer: 3072. A kibibyte is 2 to the power 10 = 1,024 bytes, so 3 KiB is 3 x 1,024 bytes.

  6. [1 mark]What does this program print?

    n = 47
digits = ""
while n > 0:
    digits = str(n % 5) + digits
    n = n // 5
print(digits)
    Answer:
    142

    Repeated division by 5 gives remainders 2, 4, 1, read from last to first: 47 = 1 x 25 + 4 x 5 + 2.

The task: any base

Write to_base(n, base). The parameter n is a whole number, 0 or more; base is a whole number from 2 to 16. It returns the digits of n in that base as a string, using 0 to 9 then A to F, and returns "0" when n is 0. Use repeated division with // and %: you may not use bin, hex, oct or format. The loop at the bottom prints lines such as 173 in base 16 = AD.

# the two lines every program starts with: the commands, then the robot
from bugbot import *
connect()

DIGITS = "0123456789ABCDEF"

def to_base(n, base):
    # repeated division goes here
    return ""

for n, base in [(173, 2), (173, 8), (173, 16), (0, 2), (2024, 16)]:
    print(n, "in base", base, "=", to_base(n, base))

The hint students can ask for: Divide by the base again and again. Each remainder is one digit, and the first remainder you get is the rightmost digit, so build the string from the right. Decide what happens when n starts at 0.

A solution

# the two lines every program starts with: the commands, then the robot
from bugbot import *
connect()

DIGITS = "0123456789ABCDEF"

def to_base(n, base):
    if n == 0:
        return "0"
    digits = ""
    while n > 0:
        digits = DIGITS[n % base] + digits
        n = n // base
    return digits

for n, base in [(173, 2), (173, 8), (173, 16), (0, 2), (2024, 16)]:
    print(n, "in base", base, "=", to_base(n, base))

Any program that meets the task's checks is marked correct in the simulator; this is one way, not the only way.