Number Systems. 5 th Year Maths Ordinary Level David Lewis

Transcription

1 5 th Year Maths Ordinary Level David Lewis Number Systems Anticipating problems and figuring out how to solve them is actually the opposite of worrying: it s productive. Chris Hadfield Astronaut No part of this publication may be copied, reproduced or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without prior written permission from The Dublin School of Grinds. (Notes reference: 5-mat-o-NumberSystems)

2 EASTER REVISION COURSES Looking to maximise your CAO points? Easter is well known as a time for students to vastly improve on the points that they received in their mock exams. To help students take advantage of this valuable time, The Dublin School of Grinds is running intensive exam-focused Easter Revision Courses. Each course runs for five days (90 minutes per day). The focus of these courses is to maximise students CAO points. Special offer: Buy 1st course and get 2nd course free. To avail of this offer, early booking is required as courses were fully booked last year. What do students get at these courses? EASTER REVISION COURSE FEES: PRICE TOTAL SAVINGS 1st Course nd Course FREE rd Course th Course th Course th Course ,075 7th Course ,270 8th Course ,465 9th Course ,660 NOTE: Any bookings for Junior Cert courses will also receive a weekly grind in one subject for the rest of the academic year, free of charge. This offer applies to 3rd and 2nd year students ONLY. FREE DAILY BUS SERVICE For full information on our Easter bus service, see 3 pages ahead minutes of intensive tuition per day for five days, with Ireland s leading teachers. Oral Preparation Courses 99 Comprehensive study notes. Separate to the Easter Revision Courses, The Dublin School of Grinds is also running Oral Preparation Courses. With the Oral marking component of the Leaving Certificate worth up to 40%, it is of paramount importance that students are fully prepared for these examinations. These courses will show students how to lead the Examiner towards topics that the student is prepared in. This will provide students with the confidence they need to perform at their peak. 99 A focus on simple shortcuts to raise students grades and exploit the critically important marking scheme. 99 Access to a free supervised study room. 99 Access to food and beverage facilities. NOTE: These courses are built on the fact that there are certain predicable trends that appear and reoccur over and over again in the State Examinations. ORAL PREPARATION COURSE FEES: PRICE To book, call us on or book online at TOTAL SAVINGS 1st Oral Course nd Oral Course

3 Timetable An extensive range of course options are available over a two-week period to cater for students timetable needs. Courses are held over the following weeks:»» Monday 21st March Friday 25th March 2016»» Monday 28th March Friday 1st April 2016 All Easter Revision Courses take place in The Talbot Hotel, Stillorgan (formerly known as The Stillorgan Park Hotel). 6th Year Easter Revision Courses SUBJECT LEVEL DATES TIME Accounting H Monday 21 st March Friday 25 th March 8:00am - 9:30am Agricultural Science H Monday 28 th March Friday 1 st April 2:00pm - 3:30pm Applied Maths H Monday 28 th March Friday 1 st April 8:00am - 9:30am Art History H Monday 28 th March Friday 1 April 8:00am - 9:30am Biology Course A* H Monday 21 st March Friday 25 th March 8:00am - 9:30am Biology Course A* H Monday 21 st March Friday 25 th March 12:00pm - 1:30pm Biology Course A* H Monday 28 th March Friday 1 st April 10:00am - 11:30am Biology Course B* H Monday 21 st March Friday 25 th March 10:00am - 11:30am Biology Course B* H Monday 21 st March Friday 25 th March 2:00pm - 3:30pm Biology Course B* H Monday 28 th March Friday 1 st April 8:00am - 9:30am Business H Monday 21 st March Friday 25 th March 12:00pm - 1:30pm Business H Monday 28 th March Friday 1 st April 8:00am - 9:30am Chemistry Course A* H Monday 28 th March Friday 1 st April 12:00pm - 1:30pm Chemistry Course B* H Monday 28 th March Friday 1 st April 2:00pm - 3:30pm Classical Studies H Monday 21 st March Friday 25 th March 8:00am - 9:30am Economics H Monday 21 st March Friday 25 th March 8:00am - 9:30am Economics H Monday 28 th March Friday 1 st April 10:00am - 11:30am English Paper 1* H Monday 21 st March Friday 25 th March 12:00pm - 1:30pm English Paper 2* H Monday 21 st March Friday 25 th March 10:00am - 11:30am English Paper 2* H Monday 21 st March Friday 25 th March 2:00pm - 3:30pm English Paper 2* H Monday 28 th March Friday 1 st April 10:00am - 11:30am English Paper 2* H Monday 28 th March Friday 1 st April 12:00pm - 1:30pm French H Monday 21 st March Friday 25 th March 10:00am - 11:30am French H Monday 28 th March Friday 1 st April 8:00am - 9:30am Geography H Monday 28 th March Friday 1 st April 8:00am - 9:30am Geography H Monday 28 th March Friday 1 st April 10:00am - 11:30am German H Monday 21 st March Friday 25 th March 10:00am - 11:30am History (Europe)* H Monday 21 st March Friday 25 th March 2:00pm - 3:30pm History (Ireland)* H Monday 21 st March Friday 25 th March 12:00pm - 1:30pm Home Economics H Monday 21 st March Friday 25 th March 10:00am - 11:30am Irish H Monday 21 st March Friday 25 th March 10:00am - 11:30am Irish H Monday 28 th March Friday 1 st April 12:00pm - 1:30pm Maths Paper 1* H Monday 21 st March Friday 25 th March 8:00am - 9:30am Maths Paper 1* H Monday 21 st March Friday 25 th March 12:00pm - 1:30pm Maths Paper 1* H Monday 28 th March Friday 1 st April 10:00am - 11:30am Maths Paper 1* H Monday 28 th March Friday 1 st April 2:00pm - 3:30pm Maths Paper 2* H Monday 21 st March Friday 25 th March 10:00am - 11:30am Maths Paper 2* H Monday 21 st March Friday 25 th March 2:00pm - 3:30pm Maths Paper 2* H Monday 28 th March Friday 1 st April 12:00pm - 1:30pm Maths Paper 2* H Monday 28 th March Friday 1 st April 4:00pm - 5:30pm Maths O Monday 21 st March Friday 25 th March 8:00am - 9:30am Maths O Monday 28 th March Friday 1 st April 12:00pm - 1:30pm Physics H Monday 28 th March Friday 1 st April 10:00am - 11:30am Spanish H Monday 21 st March Friday 25 th March 2:00pm - 3:30pm Spanish H Monday 28 th March Friday 1 st April 10:00am - 11:30am * Due to large course content, these subjects have been divided into two courses. For a full list of topics covered in these courses, please see 3 pages ahead. 6th Year Oral Preparation Courses SUBJECT LEVEL DATES TIME French H Sunday 20 th March 10:00am - 2:00pm German H Saturday 26 th March 10:00am - 2:00pm Irish H Saturday 26 th March 10:00am - 2:00pm Spanish H Saturday 19 th March 1:00pm - 5:00pm 5th Year Easter Revision Courses SUBJECT LEVEL DATES TIME Maths H Monday 28 th March Friday 1 st April 8:00am - 9:30am English H Monday 28 th March Friday 1 st April 4:00pm - 5:30pm Note: 5th year students are welcome to attend any 6th year course as part of our buy 1 get 1 free offer. 3rd Year Easter Revision Courses SUBJECT LEVEL DATES TIME Business Studies H Monday 28 th March Friday 1 st April 8:00am - 9:30am English H Monday 21 st March Friday 25 th March 8:00am - 9:30am English H Monday 28 th March Friday 1 st April 2:00pm - 3:30pm French H Monday 28 th March Friday 1 st April 12:00pm - 1:30pm Geography H Monday 28 th March Friday 1 st April 12:00pm - 1:30pm German H Monday 21 st March Friday 25 th March 8:00am - 9:30am History H Monday 21 st March Friday 25 th March 4:00pm - 5:30pm Irish H Monday 28 th March Friday 1 st April 2:00pm - 3:30pm Maths H Monday 21 st March Friday 25 th March 10:00am - 11:30am Maths H Monday 21 st March Friday 25 th March 12:00pm - 1:30pm Maths H Monday 28 th March Friday 1 st April 10:00am - 11:30am Maths O Monday 28 th March Friday 1 st April 12:00pm - 1:30pm Science H Monday 28 th March Friday 1 st April 2:00pm - 3:30pm Science H Monday 21 st March Friday 25 th March 2:00pm - 3:30pm Spanish H Monday 21 st March Friday 25 th March 12:00pm - 1:30pm 2nd Year Easter Revision Courses SUBJECT LEVEL DATES TIME Maths H Monday 21 st March Friday 25 th March 2:00pm - 3:30pm BUY 1ST COURSE GET 2ND COURSE FREE! NOTE: Any bookings for Junior Cert courses will also receive a weekly grind in one subject for the rest of the academic year, free of charge. This offer applies to 3rd and 2nd year students ONLY. BOOK EARLY TO AVAIL OF THE SPECIAL OFFER

4 Contents: Number Systems Number types Prime numbers Significant figures Scientific notation Putting the numbers in order Past and probable exam questions Solutions to Number Systems The Dublin School of Grinds Page 1 David Lewis

5 Number systems is worth 0% to 4% of the Leaving Cert. It appears on Paper 1. Number Systems 1. Number types After certain questions we often see the following symbols: x N x Z x R x Q x R/Q N stands for Natural numbers or as I say Normal numbers. Normal numbers are the ones we use in everyday life such as 1, 2, 3, 4, 5, etc. They are positive whole numbers. Note: Despite what some text books say, zero is not considered a natural number by The Irish State Examinations Commission. Example 1 Draw x 3 on the numberline, x N Notice that we only fill in the whole numbers and not the bits in between. We also fill in the arrow at the end to show x 3 goes on forever. Example 2 Draw x > 3 on the numberline, x N This is the same as eg1. except that we did not fill in the 3 itself, because the question said > 3, not 3. Comments The Dublin School of Grinds Page 3 David Lewis

6 Example 3 Draw x 4 on the numberline, x N Z stands for Integers They are positive or negative whole numbers. Examples of integers are: - 7, , 4, 12 Example 4 Draw x 3 on the numberline, x Z Notice that we fill in the arrow at the end to show x 3 goes on forever. Example 5 Draw x > - 4 on the numberline, x Z Notice that we did not fill in the - 4 itself, because the question said > - 4, not - 4. R stands for Real numbers or as I say Random numbers. Random numbers are any number, (except imaginary numbers which we meet in complex numbers). They can be plus or minus, they can be whole numbers, fraction numbers, decimal numbers.anything!! Examples of random numbers are: - 9,!, - 0.6, 0, 2, 7.3, 9! Example 8 Draw x 2 on the numberline, x R Notice that we fill in everything from 2 right up to and including the arrow. Comments The Dublin School of Grinds Page 4 David Lewis

7 Example 9 Draw x > 2 on the numberline, x R This is the same as eg6. except that we have not filled in the 2 itself, because the question said > 2, not 2. We used a hollow circle around the 2 to show we are excluding the 2. Q stands for Rational numbers or as I say fraction numbers. These are numbers which can be written as a fraction, for example:!,!,!!,!, 7 =!!!!!!, 12 =!!"! You will not be asked to draw these on the numberline, however you are required to know this at the end of certain questions, as shown in the example below Example 10 Write =!! +!"!! =!! +!!!!!!"!! as a + b 7 where a, b Q R/Q stands for Irrational number or as I say non- fraction numbers. These are numbers which cannot be written as a fraction, for example: 2, 3, 99, π NOTE: We have seen questions where you re told π =!!, but this is only a rough value of π. The actual value of π can t be written as a fraction, i.e. it is irrational.! The reason we use R/Q should be understood from our knowledge of the sets chapter. The back slash means take away the elements in. So R/Q means: Real numbers, take away the elements in rational numbers. If we put this in more simple English it means: Random numbers but leave out the fraction numbers. This clearly leaves us with the non- fraction numbers! Comments The Dublin School of Grinds Page 5 David Lewis

8 To find if a number is irrational, simply type it into your calculator and try to change it to a fraction by using the relevant button: Casio: Sharp: button button If it won t change to a fraction, then it s irrational! The types of numbers are often demonstrated by the Examiner on the following Venn diagram: As you see, Natural numbers are inside Integers, because Natural numbers are Integers! Then Natural numbers and Integers are inside Rational numbers, because Natural numbers and Integers are Rational numbers! And so on Comments The Dublin School of Grinds Page 6 David Lewis

9 2. Prime numbers Prime numbers are natural numbers that have two factors only. For example: 5 is a prime number (factors: 1 and 5) 7 is a prime number (factors: 1 and 7) 6 in not a prime number (factors: 1, 2, 3, 6) In your Leaving Cert you may be asked to express a number as a product of prime numbers. To do this you keep dividing in the lowest possible prime number until you get to 1. Let s look at some examples. Example 1 Express 42 as a product of prime numbers. Solution: => Example 2 Express 60 as a product of prime numbers: Solution: => = 2! 3 5 Comments The Dublin School of Grinds Page 7 David Lewis

10 3. Significant figures Rule for Significant Figures: Read from the left and start counting significant figures when you encounter the first non- zero digit. 1) All non zero numbers are significant. 613 has three significant figures has six significant figures 2) Zeros located between non- zero digits are significant has four significant figures 602 has three significant figures has 16 significant figures! 3) Trailing zeros (those at the end) are significant only if the number contains a decimal point; otherwise they are insignificant has four significant figures has six significant figures has two significant figures 4) Zeros to left of the first non- zero digit are insignificant has three significant figures 0.52 s two significant figures also has two significant figures! Comments The Dublin School of Grinds Page 8 David Lewis

11 4. Scientific notation Example 1 The diameters of Venus and Saturn are km and km. What is the difference between the diameters of the two planets? Give your answer in the form of a 10 n where n Z and 1 a < 10. Solutions: When you subtract these numbers on your calculator, and depending on the mode your calculator is in, you will either get ! (which is the answer) or To change to scientific notation, just count how many places you must move a decimal in from the right hand side so that just one number is remaining to the left hand side. In this case we must move in 5 places from the right hand side. Therefore the answer is !. You can ignore zeros at the end, so we have !. Example 2 Light travels at a speed of approximately km/sec. How many kilometres will light travel in 8 minutes? Express your answer in the form a 10!, where n N and 1 a < 10. Solution: Multiply by 60 to change to minutes, then multiply by 8 to change to 8 minutes: ! 60 8 = = ! = ! Question 4.1 Given that x = 2 10!! and y = 7 10!!, evaluate x + 8y. Express your answer in the form a 10!, where n N and 1 a < 10. Solution: Comments The Dublin School of Grinds Page 9 David Lewis

12 Question 4.2 In 1981 the population of Peru was approximately By 1988 the population had increased by 2.5 million. What would be the approximate population of Peru in 1988? Express your answer in the form a 10!, where n N and 1 a < 10. Solution Comments The Dublin School of Grinds Page 10 David Lewis

13 5. Putting the numbers in order A popular Leaving Cert exam question is to ask you to put a list of numbers in order. The numbers will be given in a mixture of decimals, fractions and square roots. Rule: Change all the numbers to decimals using your calculator. Example 1 By putting the smallest number first place the following number in order: Solution Changing to decimals: 10 7 = = = (1.19)! = => The order is 2, (1.19)!,!"!,!!! NOTE: Sometimes the Examiner can ask you to put them in order with the largest first, so watch out for the wording of the question. Question 5.1 By putting the largest number first, place the following numbers in order: NOTE: When you change the numbers to decimals they are usually close to each other (not always though!). If you get numbers that are totally different, this is a hint that you may be wrong, so make sure to check again on your calculator. Comments The Dublin School of Grinds Page 11 David Lewis

14 6. Past and probable exam questions Question 1 The Dublin School of Grinds Page 12 David Lewis

15 Question 2 Question 3 The Dublin School of Grinds Page 13 David Lewis

16 Question 4 The Dublin School of Grinds Page 14 David Lewis

17 7. Solutions to Number Systems Question 4.1 x + 8y = 2 10!! + 8(7 10!! ) = !! Note: If your calculator says!"!"## just press the change button to turn it into a decimal. Question ! + 2,500,000 = = ! = ! Question 5.1 Changing to decimals: 7 6 = = ! = = 1.16 => The order is 1.11!,!!,!!, Solutions to past and probable exam questions Question 1 i) ii) 5 cannot be written as a fraction Question 2 2!" = 16,77,216 = ! = ! (correct to three significant figures) The Dublin School of Grinds Page 15 David Lewis

18 Question 3 a) Note for students: Rational means it can be written as a fraction. Type each part into your calculator and see if it gives it as a fraction. This will tell you if it is rational or not. b) Question 4 a) Prime numbers are natural numbers that have 2 factors only e.g.: 5 or 7 b) c) 2!" 1 = !" = !" (correct to 2 significant figures) => 2652 = => 2652 = 2! d) There is the 2 in front of the decimal and then the 18 numbers to the right of the decimal. ð There are 19 digits. The Dublin School of Grinds Page 16 David Lewis