Notation and structure of the number system
Notation and Structure of the Number System
Types and Properties of Numbers
- Understand the different types of numbers in the number system, including natural numbers, integers, rational numbers, irrational numbers, and real numbers.
- Natural numbers are those used for counting and ordering (e.g., 1, 2, 3).
- Integers include natural numbers, zero, and their negatives (e.g., -2, -1, 0, 1, 2).
- Rational numbers can be written as a fraction where both the numerator and denominator are integers (e.g., 7/3, -4/5). This category also includes integers and natural numbers.
- Irrational numbers cannot be written as a fraction. They do not terminate or repeat, and include numbers like π and √2.
- The real numbers are a combination of rational and irrational numbers.
Number Notations
- Get comfortable with standard form (or scientific notation) e.g., 3.4 x 10<sup>5</sup>.
- Understand the use of surds to represent the root of a number that cannot be simplified to a rational number (e.g., √2 is a surd).
- Know how to convert between fractions, decimals, and percentages and when to use each.
Order of Operations
- Remember the order of operations, often remembered by the acronym BIDMAS, standing for Brackets, Indices, Division, Multiplication, Addition, and Subtraction.
- Note that division and multiplication, as well as addition and subtraction, are of equal precedence and should be performed from left to right.
Sequences and Series
- Recognize and work with arithmetic sequences, where each term is a certain number (the common difference) more than the previous term.
- Similarly, understand geometric sequences, where each term is a certain number (the common ratio) times the previous term.
- Know how to calculate the sum of a finite arithmetic series and the sum of a finite and infinite geometric series.
Each of these key points serves as a solid foundation on which to build a deeper understanding of the notation and structure of the number system. Details of the mathematical operations and workings of these number types can then be studied individually for a comprehensive understanding of the subject.