Ronald Hui Tak Sun Secondary School Mathematics Ronald Hui Tak Sun Secondary School

Mathematics I am English teacher! Mathematics is an English lesson We are English (EMI) School!!! Ronald HUI

Mathematics What should be ready in the next lesson? Textbook Erasable Pens Notebook & folder Ronald HUI

DSE Mathematics Further information Talk to me? http://www.hkeaa.edu.hk Talk to me? Facebook: Ronald Hui IG: ronaldhuisir Whatsapp: 9268 8529 Email: roh@tsss.edu.hk Homepage for F4 students: http://personal.tsss.edu.hk/roh Ronald HUI

Good Luck! Shall we start? Ronald HUI 4 September 2017

Real Number System

Positive integers are also called natural numbers. Positive integers, negative integers and zero together form the system of integers. Integers Negative integers …, –4, –3, –2, –1, Zero 0, Positive integers 1, 2, 3, 4, … Positive integers are also called natural numbers.

p and q are integers and q ≠ 0 Rational numbers Rational numbers are numbers which can be expressed as . p q p and q are integers and q ≠ 0 Fractions and integers together form the system of rational numbers. For example: , , , 1 5 2 –9 17 4 3 1 1 –6 1 3 = , 0 = , –6 = Integers These are fractions. Rational numbers These are integers. Fractions

By long division, all fractions can be converted into terminating decimals or recurring decimals. Consider and . (i) (ii) ∴ ∴

Can all terminating decimals and recurring decimals be converted into fractions? For terminating decimal, (i) (ii)

All terminating decimals and recurring decimals are rational numbers. For recurring decimal, Let i.e. (2)  (1): All terminating decimals and recurring decimals are rational numbers. ∴

p and q are integers and q ≠ 0 Irrational numbers Irrational numbers are numbers which cannot be expressed as . p q p and q are integers and q ≠ 0 They can only be written as non-terminating and non-recurring decimals. For example:

Real numbers Rational and irrational numbers together form the real number system. Real numbers Rational numbers Irrational numbers ,  –4, When plotting each real number above on a real number line, we have

Now, we summarize the relationships among different kinds of real numbers as follows:

Follow-up question Consider the following numbers. , 8, , , 3.5, 4 , 8, , , 3.5, 4 Rational numbers Integer 8 Fraction Terminating decimal 3.5 Recurring decimal Irrational numbers , 4