Topic 2 Revision
2.1 The sets of numbers:
You are expected to know the symbols and the definitions of each type of number that come up in the course. They are:
N – natural numbers {0, 1, 2, 3…}
Z – integers {…-3, -2, -1, 0, 1, 2, 3…}
Q – rational numbers {} i.e. rational numbers are numbers that can be written as a fraction.
Notice here that it is all numbers that can be written as a fraction. They may be given to you in decimal form. The following decimals are all rational as they can be written in fraction form:
0.321 /-1.4 /
0.333333… /
-5 /
Copy and complete the following passage using the missing words below.
decimals fraction rational fractions terminate irrational recur fraction decimals recur rational recur integers subset terminateRational numbers are numbers that can be written as ______. They are often also given as ______. Decimals which ______or _____ can all be written as ______and so they are ______. All ______can be written as ______so are ______. We say that the set of ______is a ______of rational numbers. Some ______(e.g. …} neither ______or ______and are therefore ______.
R – real numbers – these are any numbers that can be written on a number line. This includes every type of number you have come across in your maths so far.
Notice here that
This means that if a number is a natural number then it must be an integer. If it is an integer… (copy and complete).
The other types of number you are expected to be aware of include:
Multiples: This means ‘in the times table of’. E.g. multiplies of 5 are {5, 10, 15…}
Lowest common multiple: The smallest number which is a multiple of two numbers
e.g. To find the LCM of 6 and 8:
Multiples of 6: {6, 12, 18, 24, 30, 36, 42, 48…}
Multiples of 8: {8, 16, 24, 32, 40, 48, 56, 64…}
The LCM is therefore 24 as it is the smallest number that appears in both lists.
Factors: Numbers which divide exactly into a natural number
e.g. to find factors of 24:
1 x 24 = 24
2 x 12 = 24
3 x 8 = 24
4 x 6 =24
Notice that you do not need to continue the list as 5 does not divide into 24 and the next pair would give you 6 x 4 (which we already have).
Factors of 24 are therefore {1, 2, 3, 4, 6, 8, 12, 24}
Highest common factor: The largest number which is a factor of 2 numbers
e.g. to find the HCF of 12 and 18
Factors of 12 {1, 2, 3, 4, 6, 12}
Factors of 18 {1, 2, 3, 6, 9, 18}
The HCF is therefore 6 as it is the largest number which appears in both lists.
Prime numbers: Prime numbers have 2 factors – 1 and themselves
e.g. to see if 8 is prime:
Factors of 8 {1,2 , 4, 8} – therefore 8 is not prime as it has 4 factors
e.g. to see if 11 is prime:
Factors of 11 {1, 11} – therefore 11 is prime as it has 2 factors
e.g. to see if 1 is prime:
Factors of 1 {1} – therefore 1 is not prime as it only has 1 factor
Prime factors: Every natural number can be written as a product of prime numbers
e.g. to write 24 as a product of prime factors.
Divide the number by the smallest prime i.e. {2, 3, 5, 7..}
You are left with:
2 x 2 x 2 x 3 = 24
Or
Questions on 2.1:
1.Tick the correct boxes:
N / Z / Q / R4.2
-3
2.571571571…
8
2.Draw a Venn diagram showing the sets N, Z, Q and R and place the numbers in the correct position.
3.How many times larger is 6300 x 120 than 6.3 x 1.2
4.My bank account contained €12400. I withdraw €260 a month for 18 months. What is the new balance? How many months do I need to carry on withdrawing for until I am left with an overdraft?
5.List the first 5 powers of 4.
6.Write as a natural number
7.Write 660 as the product of prime factors.
8.List all the factors of 120.
9.List the multiples of 7 between 80 and 120.
10.Find the largest number which divides into 85 and 119.
11.Evaluate
12.Find the Lowest common multiple of 15 and 18.
13.A bag of sweets has to be able to be shared equally between either 5, 6 or 9 kids. What is the smallest number of sweets there could be in the bag?
2.2 Approximation:
Numbers can be rounded to any degree of accuracy. A number line often helps.
e.g. 2047 people attended a concert. Round this to the nearest 20.
Notice that only multiples of 20 are on the line.
2047 is closest to 2040 so we can say that 2047 rounded to the nearest 20 is 2040.
2047 rounded to 2040 because it was less than 2050 (the half-way point). By convention, if a number is exactly half way between the two rounded possibilities we would round up.
So 2090 rounded to the nearest 20 would be 2100.
Rounding to decimal places:
The first number after the decimal point is the 1stdecimal place
The second number after the decimal point is the 2nddecimal place
e.g. the number given to the right is ∏ to 9 decimal places.
i)Round ∏ to 1 decimal place
Firstly we underline the first decimal place. We then circle the next digit. If the circled digit is 5 or larger then the underlined digit increases by 1 otherwise the underlined digit remains the same.
4 is less than 5 therefore ∏=3.1 (to 1 d.p.)
ii)Round ∏ to 3 d.p.
The circled 5 causes the underlined digit to increase by 1 therefore ∏=3.142 to 3 d.p.
If you are really lazy you can do these on a calculator:
- Press mode
- On the 2nd row change ‘float’ to 3
- Type in 3.141592654 and press enter
- Make sure you change your mode back to ‘float’
Note that when the last digit of a rounded number is 0 you must leave the 0 in.
e.g. 31.402 to 2 d.p. is 31.40 (not 31.4)
Rounding to significant figures:
When counting significant figures your count should begin at the first non-zero number.
We can now round in exactly the same way as before.
e.g. round 0.0035478 to 3 s.f.
We underline the third s.f. and circle the next digit
The circled digit is 5 or larger so the underlined increases by 1.
So 0.0035478=0.00355 (to 3 s.f.)
Accuracy of measurement:
Judy’s height ismeasured as 175 cm to the nearest cm. What is the possible range in Judy’s height?
The number line above shows that if Judy’s height were less than 174.5 cm it would have been rounded down to 174 cm.
If her height were more than 175.5 cm it would have been rounded up to 176 cm.
Remember our convention: If it had been exactly 174.5 cm then it would have been rounded up to 175. If exactly 175.5 cm it would have been rounded to 176 cm.
Judy’s height is greater than or equal to 174.5 cm but it must be less than 175.5 cm
e.g. Find the bounds of the area and perimeter of a square whose side length x is given as 2.4 cm to 1 d.p.
The boundaries for side length are
Percentage error:
Your formula book gives the following formula for percentage error.
e.g. John complains to his teacher saying that it will take him 1 hour 30 minutes to complete his homework. Actually it takes him 55 minutes. What is the percentage error in his calculation?
e.g. Find the percentage error when ∏ is given to 3 decimal places
Estimation:
Before performing a calculator operation you should have an idea of the order of magnitude of your answer (how large you expect it to be).
You can do this by rounding all numbers to 1 s.f.
e.g. Johnty claims that why must he be wrong?
Doing these sorts of calculations in your head will help you avoid silly answers.
Questions on 2.2:
1.The profit gained by a large company last year was €1,342,987,645. Give this number:
a)To the nearest thousand
b)To the nearest million
c)To the nearest billion
2.Evaluate giving your answer to 2 decimal places
3.Round the following numbers to 3 decimal places:
a)2.41655
b)8.1997
c)3.9299
4.Write the fraction as a decimal to 3 significant figures
5.Write the following to 3 s.f.
a)15648
b)0.0045798
c)1.0006
d)1.99999
6.A cuboid shaped house has dimensions 6m by 3m by 4m (measurements given to the nearest metre). Calculate the upper and lower bounds for the surface area and volume of the house.
7.Find the percentage error when 72,346 is given to the nearest 1000.
8.Round each of the numbers in question 2 to 1 s.f. and use this to estimate the answer. What is the percentage error in your estimation?
2.3 Standard Index Form / Scientific Notation
A number in standard form is given as
None of the following are in standard form:
The first one is not standard form as the base is not 10. The second one is not in standard form as a is too large and the third is not in standard form as the power is not an integer.
Converting from standard form to decimal form:
If k is positive then the decimal point moves k units to the right
If k is negative then the decimal point moves k units to the left
Zeroes may need to be inserted as place-holders.
e.g. convert the following to standard form:
i)
We move the decimal point five spaces to the right
We now need to insert zeroes in the blank spaces and we are left with:
ii)
We move the decimal point 2 spaces to the left
Again we insert zeroes and we are left with
On a calculator the button EE means (note that only one E appears on screen)
If you type 6.3 EE 5 Enter or 3.54 EE -2 Enter you should see the correct answer.
Notice that with small numbers, the value of k is negative while for large numbers it is positive.
Converting from Decimal Form to Standard Form
We do exactly the same process in reverse ensuring that the value of a is between 1 and 10.
e.g. Write 0.000356 in standard form
This is a small number so k will be negative, the decimal point must be moved so it is between the 3 and the 5. So we are moving it 4 spaces.
e.g. Write 287.2 in standard form
This is a large number so k will be positive, the decimal point must be moved so it is between the 2 and the 8. So we are moving it 2 spaces.
On the calculator:
1.Press MODE and change to SCI (scientific notation)
2.Type the number 0.000356 and press Enter
3.You get the answer 3.56 E -4. Remember this must be written as .
Calculations with standard form can all be done on your calculator
e.g.
Questions on 2.3:
1.Express the following in standard form:
a)0.00376
b)12500000
c)
2.Write in normal form:
a)
b)
3.
i)Arrange a, b and c in order of size
ii)Evaluate the following:
a)b)c)
2.4 Conversion of units:
Prefixes: You should be familiar with the following prefixes:
Giga / A billion of… /Mega / A million of … /
Kilo / A thousand of /
Centi / One hundredth of /
Milli / One thousandth of /
Micro / One millionth of /
Nano / One billionth of /
e.g. A running track is 450m long. I use a number of 15cm rulers placed end to end to measure it. How many rulers must I have?
1.Before calculating you must ensure that the units are both the same. You can convert both to metres or both to cm.
450m = 450 x 100 cm =45000 cm
2.Now calculate
Therefore I would need 3000 rulers
e.g. A rectangle has side lengths 1.2m and 3.6m
i)Find its area in
ii)Find its area in
A common error here is to multiply the above answer by 100. To be safe you must instead change the original units to cm:
Length=120cm Width=360cm
Other units: You may be asked to convert other units but you will be given the conversion factors.
e.g. 1760 yards is 1 mile, 1 mile is 1.62 km. How many yards are there in 5 km?
These are easiest to solve as ratio problems
So 5km is 3.086 x 1760 yards which is 5432 to the nearest yard
Questions on 2.4:
1.A bicycle wheel has radius 15cm. How many complete turns must it make on a journey of 3km?
2.How long would it take (to the nearest minute) to walk 10 km at 3km/h?
3.Which is faster 10km/h or 3m/s?
4.If convert 84°F to Centigrade. Convert 10°C to Fahrenheit.
5.A cube has side length 2m. What is its volume in ?
6.1 pound is 16 ounces. An ounce is 28 grams. Jodie weighs 85kg. What is his weight in pounds?
7.A teaspoon contains 5ml. How many spoonfuls of medicine can be taken from a 2 litre bottle?
2.5A Arithmetic sequences:
A sequence is a number pattern – it can be generated by a rule
e.g. write the first 5 terms of the sequence given by:
(notice that is the symbol given to the nth term)
So means the 1st term and would mean the 27th term
The sequence 5, 7, 9, 11, 13… is an arithmetic sequence. It has a common difference between terms of 2. We can always find the common difference by taking a term and subtracting the term before.
We require two things to define an arithmetic sequence: - the first term and d – the common difference.
Finding a formula for an arithmetic sequence:
In your formula book you are given:
e.g. Find a formula for the general term (this means ‘express as a function of n) of 6, 2, -2…and calculate the 20th term.
1.We check that the sequence is arithmetic.
The sequence is arithmetic as the common difference is -4. We subtract 4 every step we take. The first term is 6.
2.Substitute into the formula.
3.Simplify.
4.Check by substituting n=1, 2, 3
5.Substitute n=20 to find the 20th term
If you are really stuck in the exam:
1.Write down the values of and d as you may get marks for this.
2.On your calculator press ‘6 Enter’
3.Now type ‘-4 Enter’
4.Repeatedly press Enter until you arrive at the 20th term.
Finding other information using the formula:
In the above formula there are 4 unknowns: . In many questions you will be given three of them and asked to work out the other one.
e.g. The first term of an arithmetic sequence is 120. The 10th term is 57. Find:
1)The value of the common difference.
2)Which term is the first negative term.
1)a)Write down the known information and what is unknown:
b)Substitute:
c)Solve:
2)a)Rewrite the general formula for the sequence:
b)Set up an inequality
c)Solve
d)Give the first whole number solution
The 19th term is the first negative one
Using the GDC:
1.Set up a list with the position (n) in L1 and the term () in L2
2.Perform a linear regression on the data (STAT, CALC, LinRegL1, L2)
3.This shows that as shown before
4.Enter
5.Use TBLSET to start your table at 1 and go in steps of 1
6.Go to TABLE and scroll down to see the numbers
Which shows that the 19th term is the first negative one.
e.g. The common difference of an arithmetic sequence is 8. The 16th term is 192.
a)Find the first term
b)Determine whether 270 is a member of the sequence
1)a)Write down the known information and what is unknown:
b)Substitute:
c)Solve:
2)a)Rewrite the formula for the sequence:
b)Set up an equation
c)Solve
As n is not a whole number 208 is not in the sequence
Using the formula for simultaneous equations:
If you are given two terms of the sequence (not including the first term) you will have two unknowns (and d). This requires you to use simultaneous equations.
e.g. , find and d.
1.Subsitute the known information into two separate equations:
2.Simplify:
3.Solve either algebraically or using Polysmlt
This shows that the first term is -1 and the difference is -5.
Algebraic questions:
Most algebraic questions rely on you to understand that the common differences must be equal.
e.g. three terms of an arithmetic sequence are given by Find the value of k and of the three terms:
a)Form an equation showing that the differences are equal. Remember that the difference between terms is found by subtracting the previous term.
b)Simplify and solve
c)Substitute back to find the terms and check that they form an arithmetic sequence
Questions on 2.5A:
1.Find a formula for the general term and evaluate the 10th term for the following sequences:
a)3, 7, 11…
b)-2, -4, -6…
c)…
2.The common difference of a sequence is 2, the fifth term is 15. Find the first and twelfth terms.
3.The first term of a sequence is -5, the twelfth term is 0.5. When does the sequence first exceed 180?
4.The first term of a sequence is 13, the 14th term is 52. Is 312 a member of the sequence?
5.The fifth term of a sequence is -2 and the twelfth term is -12.5. What is the first term which is less than -20?
6.The eleventh term of a sequence is -16, the eighth term is -11.5. Find the 100th term.
7.For each of the following questions find the value of k and the three terms:
a)
b)
c)(assume k is positive)
2.5B Arithmetic Series:
An arithmetic series is the sum of an arithmetic sequence
e.g. 3, 5, 7, 9… is a sequence
3+5+7+9… is a series
You are given two formulae for the sum of terms of a sequence. The first one is used if you know d (the difference), the second one is used if you know the value of (the last term that you are adding.
e.g. Find the sum of 3, 7, 11, 15 and 19
i)By adding
3+7+11+15+19=55
ii)By using the first formula
iii)By using the second formula
You will probably notice that the second formula is a lot easier than the first one but you often will not have the choice of which to use.
e.g. Find the sum of -3, 1, 5…45
This one is harder than it looks. We have not been given the value of n so we will have to work it out.
1.Set up an equation to find n.
Using the formula for the general term:
So we know that 45 must be the 13th term.
2.Find the sum of the sequence.
We have enough information to use both formulae so we’ll use the second one as it is easier:
Finding other information using the formula:
e.g. The first term of a sequence is 6. The sum of the first 8 terms is 160. Find the 12th term.
1.Write out the information that has been given:
2.As we will need to find d we should use the first formula. Substitute in and solve:
3.We now need to find the twelfth term (note it does not ask for the sum of 12 terms).
e.g. The sum of the first n terms of a sequence is 24. The first term is -1, the common difference is 2. Find n.
1.Write out the information that has been given:
2.Substitute into the formula (note that you do not know the last term)
3.Notice we are left with a quadratic – rearrange them into decreasing powers and solve manually or using polysmlt.
And n must be positive so it is 6.
Questions on 2.5B
1.Find the sum of the first 50 terms of the sequence 17, 15, 13, 11…
2.An arithmetic sequence has 12 terms. The first term is 10, the last term is -16. Find the sum of terms.
3.The first term of an arithmetic sequence is 14, the sum of 12 terms is 264. What is the twelfth term?
4.Find the sum of 3, 10, 17…157
5.In a room there is 1 chair in the first row, 2 in the second, 3 in the third…How many rows are there if 153 chairs are used in total?
2.6 Geometric Sequences and Series:
2, 6, 18, 54… is an example of a geometric sequence.
The next term in the sequence can always be found by multiplying by a common ratio (r).
r can be calculated by dividing a term by the previous one:
so whichever two terms we use to find r it is always the same.
Notice that r can be a fraction:
e.g. 81, 27, 9, 3… -
e.g. 100, 40, 16, 6.4… -
r can also be negative:
e.g. 36, -24, 16… - Notice that when r is negative the sequence oscillates between positive and negative values.