P2 T1 L1 - Number Systems

Question 1

What are natural numbers? (3 points)

Mathematical symbol?

Answer 1

1. A positive whole number
2. Usually consist of 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
3. This is known as base 10 as there are 10 different digits

The mathematical symbol for natural numbers is ℕ or N so you might see this represented as:

ℕ = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, …}

Question 2

What are integer numbers? (2 points)

Mathematical symbol?

Answer 2

1. A whole number which can be positive or negative
2. 0 included in the subset

The mathematical symbol for integer numbers is ℤ
or Z so you might see this represented as:

ℤ = {…, -3, -2, -1, 0, 1, 2, 3, …}

In theory you can have an infinite range, in practice some languages limit the range you can store (e.g. in VB you can store -247483648 to 2147483647)

Question 3

What are rational numbers? (3 points)

Mathematical symbol?

Answer 3

1. A number which can be expressed as a fraction (also known as a quotient) or a ratio of integers
2. Can be positive or negative
3. Fractions are made up of two integers with the value being the ratio between the two. Ratio can be expressed as a fraction or a decimal equivalent

E.g.
0.5 = 1/2

• 0.75 = (−6)/8
  1. 0 = 5/5

The mathematical symbol for Rational numbers is ℚ or Q so you might see this represented as:

ℚ = {…, - 0.3, -0.2, -0.1, 0.1, 0.2 …}

Question 4

What are irrational numbers? (1 point)

Answer 4

1. Any number which cannot be represented as a ratio of integers because the decimal equivalent would go on forever without a repeating pattern

E.g. π
In decimal form this number is infinite so has to be truncated to 3.14 (or slightly longer for accuracy)

Some square roots and cube roots are irrational numbers

If a number is terminating or repeating, it must be rational;
If it is both nonterminating and nonrepeating, the number is irrational.

Question 5

What are real numbers? (4 points)

Mathematical symbol?

Answer 5

1. Any positive or negative value which does or does not have a fractional part
2. Natural numbers/Integers, rational and irrational numbers are all real numbers
3. The fractional part can be any length, this means that the number can be as accurate as needed
4. Can be used to measure continuously or infinitely changing values.

The mathematical symbol for real numbers is ℝ or R so you might see this represented as:

ℝ = {…, 0.122, 0.123, 0.124 …}

Question 6

What are ordinal numbers? (1 point).

Answer 6

1. Used to identify the position of something in a list E.g. 1st, 2nd, 3rd

Often used with Cardinal numbers which are used to identify the size of something e.g. size of a list = 20

Can be used with data structures like lists, queues, stacks and arrays.

In Java you might use an array and use different methods to access the values in the array