Number Systems - Dublin Academy Of Education · 2016. 2. 5. · 9 90 minutes of intensive tuition...

5th 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

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Oral Preparation CoursesSeparate 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.

ORAL PREPARATION COURSE FEES:

PRICE TOTAL SAVINGS

1st Oral Course €140 €140 -

2nd Oral Course €100 €240 €40

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?

9 90 minutes of intensive tuition per day for five days, with Ireland’s leading teachers.

9 Comprehensive study notes.

9 A focus on simple shortcuts to raise students’ grades and exploit the critically important marking scheme.

9 Access to a free supervised study room.

9 Access to food and beverage facilities.

EASTERREVISION COURSES EASTER REVISION COURSE FEES:

PRICE TOTAL SAVINGS

1st Course €295 €295 -

2nd Course FREE €295 €295

3rd Course €100 €395 €490

4th Course €100 €495 €685

5th Course €100 €595 €880

6th Course €100 €695 €1,075

7th Course €100 €795 €1,270

8th Course €100 €895 €1,465

9th Course €100 €995 €1,660

To book, call us on 01-442 4442 or book online at www.dublinschoolofgrinds.ie

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.

FREE DAILY BUS SERVICE For full information on our Easter bus service, see 3 pages ahead.

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.

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).

BOOK EARLY TO AVAIL OF THE SPECIAL OFFER

BUY 1ST COURSE GET 2ND COURSE

F R E E ! 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 Easter Revision CoursesSUBJECT LEVEL DATES TIME

Accounting H Monday 21st March – Friday 25th March 8:00am - 9:30am

Agricultural Science H Monday 28th March – Friday 1st April 2:00pm - 3:30pm

Applied Maths H Monday 28th March – Friday 1st April 8:00am - 9:30am

Art History H Monday 28th March – Friday 1 April 8:00am - 9:30am

Biology Course A* H Monday 21st March – Friday 25th March 8:00am - 9:30am

Biology Course A* H Monday 21st March – Friday 25th March 12:00pm - 1:30pm

Biology Course A* H Monday 28th March – Friday 1st April 10:00am - 11:30am

Biology Course B* H Monday 21st March – Friday 25th March 10:00am - 11:30am

Biology Course B* H Monday 21st March – Friday 25th March 2:00pm - 3:30pm

Biology Course B* H Monday 28th March – Friday 1st April 8:00am - 9:30am

Business H Monday 21st March – Friday 25th March 12:00pm - 1:30pm

Business H Monday 28th March – Friday 1st April 8:00am - 9:30am

Chemistry Course A* H Monday 28th March – Friday 1st April 12:00pm - 1:30pm

Chemistry Course B* H Monday 28th March – Friday 1st April 2:00pm - 3:30pm

Classical Studies H Monday 21st March – Friday 25th March 8:00am - 9:30am

Economics H Monday 21st March – Friday 25th March 8:00am - 9:30am

Economics H Monday 28th March – Friday 1st April 10:00am - 11:30am

English Paper 1* H Monday 21st March – Friday 25th March 12:00pm - 1:30pm

English Paper 2* H Monday 21st March – Friday 25th March 10:00am - 11:30am

English Paper 2* H Monday 21st March – Friday 25th March 2:00pm - 3:30pm

English Paper 2* H Monday 28th March – Friday 1st April 10:00am - 11:30am

English Paper 2* H Monday 28th March – Friday 1st April 12:00pm - 1:30pm

French H Monday 21st March – Friday 25th March 10:00am - 11:30am

French H Monday 28th March – Friday 1st April 8:00am - 9:30am

Geography H Monday 28th March – Friday 1st April 8:00am - 9:30am

Geography H Monday 28th March – Friday 1st April 10:00am - 11:30am

German H Monday 21st March – Friday 25th March 10:00am - 11:30am

History (Europe)* H Monday 21st March – Friday 25th March 2:00pm - 3:30pm

History (Ireland)* H Monday 21st March – Friday 25th March 12:00pm - 1:30pm

Home Economics H Monday 21st March – Friday 25th March 10:00am - 11:30am

Irish H Monday 21st March – Friday 25th March 10:00am - 11:30am

Irish H Monday 28th March – Friday 1st April 12:00pm - 1:30pm

Maths Paper 1* H Monday 21st March – Friday 25th March 8:00am - 9:30am

Maths Paper 1* H Monday 21st March – Friday 25th March 12:00pm - 1:30pm

Maths Paper 1* H Monday 28th March – Friday 1st April 10:00am - 11:30am

Maths Paper 1* H Monday 28th March – Friday 1st April 2:00pm - 3:30pm

Maths Paper 2* H Monday 21st March – Friday 25th March 10:00am - 11:30am

Maths Paper 2* H Monday 21st March – Friday 25th March 2:00pm - 3:30pm



Maths O Monday 21st March – Friday 25th March 8:00am - 9:30am

Maths O Monday 28th March – Friday 1st April 12:00pm - 1:30pm

Physics H Monday 28th March – Friday 1st April 10:00am - 11:30am

Spanish H Monday 21st March – Friday 25th March 2:00pm - 3:30pm

Spanish H Monday 28th March – Friday 1st April 10:00am - 11:30am

6th Year Oral Preparation CoursesSUBJECT LEVEL DATES TIME

French H Sunday 20th March 10:00am - 2:00pm

German H Saturday 26th March 10:00am - 2:00pm

Irish H Saturday 26th March 10:00am - 2:00pm

Spanish H Saturday 19th March 1:00pm - 5:00pm

5th Year Easter Revision CoursesSUBJECT LEVEL DATES TIME

Maths H Monday 28th March – Friday 1st April 8:00am - 9:30am

English H Monday 28th March – Friday 1st 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 CoursesSUBJECT LEVEL DATES TIME

Business Studies H Monday 28th March – Friday 1st April 8:00am - 9:30am

English H Monday 21st March – Friday 25th March 8:00am - 9:30am

English H Monday 28th March – Friday 1st April 2:00pm - 3:30pm

French H Monday 28th March – Friday 1st April 12:00pm - 1:30pm

Geography H Monday 28th March – Friday 1st April 12:00pm - 1:30pm

German H Monday 21st March – Friday 25th March 8:00am - 9:30am

History H Monday 21st March – Friday 25th March 4:00pm - 5:30pm

Irish H Monday 28th March – Friday 1st April 2:00pm - 3:30pm

Maths H Monday 21st March – Friday 25th March 10:00am - 11:30am

Maths H Monday 21st March – Friday 25th March 12:00pm - 1:30pm

Maths H Monday 28th March – Friday 1st April 10:00am - 11:30am

Maths O Monday 28th March – Friday 1st April 12:00pm - 1:30pm

Science H Monday 28th March – Friday 1st April 2:00pm - 3:30pm

Science H Monday 21st March – Friday 25th March 2:00pm - 3:30pm

Spanish H Monday 21st March – Friday 25th March 12:00pm - 1:30pm

2nd Year Easter Revision CoursesSUBJECT LEVEL DATES TIME

Maths H Monday 21st March – Friday 25th March 2:00pm - 3:30pm

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.

©The Dublin School of Grinds Page 1 David Lewis

Contents: Number Systems ....................................................................... 3

1. Number types ............................................................................................................................................................................... 3 2. Prime numbers ............................................................................................................................................................................. 7 3. Significant figures ....................................................................................................................................................................... 8 4. Scientific notation ....................................................................................................................................................................... 9 5. Putting the numbers in order ............................................................................................................................................. 11 6. Past and probable exam questions ................................................................................................................................... 12 7. Solutions to Number Systems ............................................................................................................................................. 15


Comments

Number Systems Number systems is worth 0% to 4% of the Leaving Cert.

It appears on Paper 1.

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

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, -‐3. 0, 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

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

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: button

Sharp: 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

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:

=> 2×3×7

Example 2 Express 60 as a product of prime numbers: Solution:

=> 2×2×3×5 = 2!×3×5


Comments

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

123456 has six significant figures

2) Zeros located between non-‐zero digits are significant.

5004 has four significant figures

602 has three significant figures

6000000000000002 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.

5.640 has four significant figures

120000.0 has six significant figures

120000 has two significant figures

4) Zeros to left of the first non-‐zero digit are insignificant.

0.000456 has three significant figures

0.52 s two significant figures

0. 000000000000000000000000000000000052 also has two significant figures!


Comments

4. Scientific notation

Example 1

The diameters of Venus and Saturn are 1.21×104km and 1.21×105km. What is the difference between the diameters of the two planets? Give your answer in the form of 𝑎×10n 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 1.089×10! (which is the answer) or 108900. To change 108900 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 1.08900×10!. You can ignore zeros at the end, so we have 1.089×10!.

Example 2 Light travels at a speed of approximately 2.9×105km/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: 2.9 × 10! × 60 × 8 = 139200000 = 1.39200000 × 10! = 1.392 × 10!

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

Question 4.2 In 1981 the population of Peru was approximately 1.8×107. 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

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: 107= 1.428571429

2 = 1.414213562 72 6

= 1.428869017

(1.19)! = 1.4161 => 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.


6. Past and probable exam questions

Question 1


Question 2

Question 3


Question 4


7. Solutions to Number Systems

Question 4.1

𝑥 + 8𝑦 = 2 × 10!! + 8(7 ×10!!) = 7.6 × 10!! Note: If your calculator says !"

!"## just press the ‘change’ button to turn it into a decimal.

Question 4.2 1.8 × 10! + 2,500,000 = 20500000 = 2.050000 ×10! = 2.05 × 10! Question 5.1

Changing to decimals: 76= 1.166666667

62= 1.224744871

1.11 ! = 1.2321 1.3456 = 1.16

=> The order is 1.11 !, !!, !!, 1.3456

Solutions to past and probable exam questions

Question 1

i)

ii) 5 cannot be written as a fraction

Question 2

2!" = 16,77,216 = 1.6772116×10! = 1.67×10! (correct to three significant figures)


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)

=> 2652 = 2×2×3×13×17 => 2652 = 2!×3×13×17

c) 2!" − 1 = 2.305843009×10!" = 2.3×10!" (correct to 2 significant figures)

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.

Number Systems - Dublin Academy Of Education · 2016. 2. 5. · 9 90 minutes of intensive tuition...

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Transcript of Number Systems - Dublin Academy Of Education · 2016. 2. 5. · 9 90 minutes of intensive tuition...