2 Different Notations of Numbers

G11 – Mathematics – Study Guide

Exercise 2: Different Notations of Numbers

Not all number systems can be presented in the following ways: y Rational numbers cannot be tabulated. y Irrational numbers cannot be notated in any way. Thus we are limited in representing a group of irrational numbers. y In the following example, integers are used. (Study the given summary.

These notations are known since Gr. 8).

Summary: 1. English: All whole numbers between –5 and 2 2. Common notation –5 < �� < 2 3. Table notation = {– 4; –3; –2; –1; 0; 1} 4. Set-builder notation = {�� / – 4 ≤��≤ 1, ��∊ Z} 5. Interval notation ��∊ (– 4 ; 2) Note: This notation is only used for ��∊ R 6. Graphic notation (number line): – 4 –3 –2 –1 0 1 R Note: The rounded bracket means exclusive and the square bracket means inclusive. Note that only x may be written in this way.

Remember only for whole numbers … from there, the dots with the broken line between them. Dotted lines are no answer; it only shows the form of the line (graph or curve) For Grade 11

2.1* Sketch a Venn diagram of the following: The universal set is {1; 2; 3; 4; 5; 6; 7; 8; 9}. A sub-set A is {all the prime numbers}. A sub-set B is {all the odd numbers}. 2.1.1 Determine the intersection of A and B, i.e. A⋂B

A

Unit 1: Number Systems

y Number Sets ȇ Venn-diagram ȇ Notation y Set-builder notation y Interval notation

y Union y Intersection

B

The "and" and "or" , ⋃ and ⋂

C

Empty sets

{.....} or ∅

2.1.2 Determine the union of A and B, i.e. A⋃B

8

Examples:

{1; 2; 3; 4; 5}∩ {2; 3; 4; 7; 8; 9} = {2; 3; 4} This means the same as: intersection or “and” {1; 2; 3; 4; 5}∪ {2; 3; 4; 7; 8; 9} = {1; 2; 3; 4; 5; 7; 8; 9} This means the union of, and is represented by the English word “or” Remember the interval notation: ��∈ (2 ; 7] means 2 < ��≤ 7

2.2** Determine (4 ; 8] ⋂ [5 ; 9) if all the numbers are real.

2.3** Determine (4 ; 8] ⋃ [ 5 ; 9) if all the numbers are real.

2.4*** If (–∞ ; 10) ⋃ [–5 ; 19) is given and all the numbers are real. Write the answer in set-builder notation.

2.5** Write the answer of the following in interval notation −1 <m ≤ 6 and 0 ≤ m < 10

2.6** Write the answer of the following in interval notation −1 < m ≤ 6 or 0 ≤ m < 10

2.7* Write the answer of the following in interval notation ��<5 and��> 3 with�� ∊ N

2.8* Write the answer of the following in interval notation ��<5 and��> 3 with �� ∊ R

2.9* Write the answer of the following in interval notation ��<5 and��≥ 3 with�� ∊ R 2.10* Sketch the graph of f: {��/ −2≤ �� ≤ 4; �� ∈R and g: {��/ ��> 0, �� ∈R

Determine the intersection of the two and present the answer graphically.

2.11* Remember, if there are no elements in a set, it is called an empty set. The notation is as follows: {......} = ∅ .........symbol for an empty set.

2.11.1* Determine the intersection between the even natural numbers and odd natural numbers

2.11.2* If A= {prime numbers less than 100} and B = {compound factors less than 100} determine which numbers appear in A ⋂ B

2.11.3* Determine R ⋂ R'

Remember:

{.....} ≠ {0}

{1; 2; 3} or {2; 4}={1; 2; 3; 4} But {1; 2; 3} and {2; 4} = {2}