Data Rep (Number Systems + Bases)

Question 1

Fraction Definition

Answer 1

• The ratio between two integers (numerator : denominator)

Question 2

Whole Numbers Definition

Answer 2

• A number with no fractional part, meaning there are no fractions or decimal place values after the number.

Question 3

Natural Numbers (Definition, uses and Symbol)

Answer 3

• Positive whole numbers of the base 10 number system, ranging from 0 to infinity.
• Uses for everyday counting and ordering.
• Mathematical Symbol: ℕ = {0, 1, 2, 3, 4, 5, 6…}

Question 4

Integer Numbers (Definition, Use in Programming, Symbol)

Answer 4

• Whole numbers that can hold a positive or negative value, including zero.
• Integers are one of the standard data types used when defining variables.
• Mathematical Symbol: ℤ = {…, -3, -2, -1, 0, 1, 2, 3…}

Question 5

Rational Numbers (Definition, Symbol)

Answer 5

• A number which can be expressed as a fraction/quotient.
• Mathematical Symbol: ℚ

Question 6

Irrational Numbers (Definition and Implication in Programming)

Answer 6

• A number that cannot be represented as a ratio of integers because their decimal expansions are non-terminating (infinite) and non-repeating (values after the decimal places never repeat in a pattern).
• As they have infinite decimal places, the programmer has to decide on the level of precision required and allocate a set amount of memory to storing the value.

Question 7

Real Numbers (Definition, sub-catergories, and Implication in Programming)

Answer 7

• Any positive or negative value, that can include a fractional part.
• Integers, rational numbers, and irrational numbers are all real numbers.
• As they have infinite decimal places, the programmer has to decide on the level of precision required and allocate a set amount of memory to storing the value.

Question 8

Ordinal & Cardinal Numbers (Definitions)

Answer 8

• Ordinal numbers identify the position of something within a list (first, second, third)
• Cardinal numbers indicate an amount, how much of something we have (0, 1, 2, 3)

Question 9

Well-ordered list

Answer 9

• A group of related numbers with a set and defined order.

Question 10

Note: To calculate the max number of permutations for bits, 2^n.

Answer 10

• If 8 bits are being used, we have 2^8 = 256 permutations (remember, this often includes zero, so 0-255).

Question 11

Binary Prefixes

Answer 11

Kibi-byte: Ki, 2^10 bits
Mebi-byte: Mi, 2^20 bits
Gibi-byte: Gi, 2^30 bits
Tebi-byte: Ti, 2^40 bits

Question 12

Decimal Prefixes

Answer 12

Kilo-byte: KB, 10^3 bits
Mega-byte: MB, 10^6 bits
Giga-byte: GB, 10^9 bits
Tera-byte: TBM 10^12 bits

Question 13

Number Bases

Answer 13

• A number base indicates how many different digits are available when using a particular number system.

Question 14

Is an unsigned integer positive or negative?

Answer 14

• A positive integer (it does not have a sign, which in this case is the ‘-‘ sign).

Question 15

Two’s Complement

Answer 15

• A way of representing negative integers in binary.
• The MSB (most significant bit) is -128, not 128.
• A total of 256 numbers, from -128 to 127 can be represented.

-128 | 64 | 32 | 16 | 8 | 4 | 2 | 1 |

Question 16

Converting from denary to binary using two’s complement

Answer 16

• Make the number positive, and write the number out normally in the binary table.
• Then, flip every digit (1s become 0s, vice versa)
• Add 1 to the number (00000001)
• The denary number is now in two’s complement

-128 | 64 | 32 | 16 | 8 | 4 | 2 | 1|

Question 17

Fixed Point Numbers

Answer 17

• Fixed point numbers are used to represent numbers with decimal places or fractional parts.
• A decimal point is added to a binary number (although it is not stored in the 8-bit code).
• A fixed point may look like: 1000.1100
• After the fixed point, the table looks like this:
  8 4 2 1 . 1/2 1/4 1/8 1/16
• The smallest numbers you can represent is 0000.0001 (1/16) and the largest is 1111.1111 (15.9375)
• Moving the fixed point around means we can either have more accurate decimals with a smaller range of whole numbers, or larger whole numbers with less accurate decimals.

1 | 0 | 0 | 1 | 1 | 1 | 0 | 0 |

Question 18

How many combinations does a 16 & 24-bit code allow?

Answer 18

• 65,536
• 16,777,216

Question 19

What range of whole numbers does a 16 -bit code allow with two’s complement?

Answer 19

-32,768 to 32767