Transcript N1 Integers - Haringeymath's Blog

KS4 Mathematics
N1 Integers
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Contents
N1 Integers
A N1.1 Classifying numbers
A N1.2 Calculating with integers
A N1.3 Multiples, factors and primes
A N1.4 Prime factor decomposition
A N1.5 LCM and HCF
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Classifying numbers
Natural numbers
Positive whole numbers 0, 1, 2, 3, 4 …
Integers
Positive and negative whole numbers … –3, –2, 1, 0, 1, 2, 3, …
Rational numbers
Numbers that can be expressed in the form n/m, where n and m
are integers. All fractions and all terminating and recurring
.
decimals are rational numbers, for example, ¾, –0.63, 0.2.
Irrational numbers
Numbers that cannot be expressed in the form n/m, where n and
m are integers. Examples of irrational numbers are  and 2.
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Even numbers
Even numbers are numbers that are exactly divisible by 2.
For example, 48 is an even number. It can be written as
48 = 2 × 24.
All even numbers end in 0, 2, 4, 6 or 8.
Even numbers can be illustrated using dots or counters
arranged as follows:
E(1) = 2
E(2) = 4
E(3) = 6
E(4) = 8
E(5) = 10
The nth even number can be written as E(n) = 2n.
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Odd numbers
Odd numbers leave a remainder of 1 when divided by 2.
For example, 17 is an odd number. It can be written as
17 = 2 × 8 + 1
All odd numbers end in 1, 3, 5, 7 or 9.
Odd numbers can be illustrated using dots or counters
arranged as follows:
U(1) = 1
U(2) = 3
U(3) = 5
U(4) = 7
U(5) = 9
The nth odd number can be written as U(n) = 2n – 1.
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Triangular numbers
Triangular numbers are numbers that can be written as the
sum of consecutive whole numbers starting with 1.
For example, 15 is a triangular number. It can be written as
15 = 1 + 2 + 3 + 4 + 5
Triangular numbers can be illustrated using dots or counters
arranged in triangles:
T(1) = 1
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T(2) = 3
T(3) = 6
T(4) = 10
T(5) = 15
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Triangular numbers
Suppose we want to know the value of T(50), the 50th
triangular number.
We could either add together all the numbers from 1 to 50 or
we could find a rule for the nth term, T(n).
If we double the number of counters in each triangular
arrangement we can make rectangular arrangements:
T(1) = 1
2T(1) = 2
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T(2) = 3
2T(2) = 6
T(3) = 6
2T(3) = 12
T(4) = 10
2T(4) = 20
T(5) = 15
2T(5) = 30
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Triangular numbers
Any rectangular arrangement of counters can be written as
the product of two whole numbers:
T(1) = 1
T(2) = 3
T(3) = 6
T(4) = 10
T(5) = 15
2T(1) = 2
2T(2) = 6
2T(3) = 12
2T(4) = 20
2T(5) = 30
2T(1) = 1 × 2 2T(2) = 2 × 3 2T(3) = 3 × 4 2T(4) = 4 × 5 2T(5) = 5 × 6
From these arrangements we can see that 2T(n) = n(n + 1)
So, for any triangular number T(n)
T(n) = n(n + 1)
2
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Triangular numbers
We can now use this rule to find the value of the 50th
triangular number.
T(n) =
n(n + 1)
2
T(50) = 50(50 + 1)
2
T(50) = 50 × 51
2
T(50) = 2550
2
T(50) = 1275
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Gauss’ method for adding consecutive numbers
There is a story that when the famous mathematician Karl
Friedrich Gauss was a young boy at school, his teacher asked
the class to add up the numbers from one to a hundred.
The teacher expected this activity to keep the class quiet for
some time and so he was amazed when Gauss put up his
hand and gave the answer, 5050, almost immediately!
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Gauss’ method for adding consecutive numbers
Gauss worked the answer out by noticing that you can quickly
add together consecutive numbers by writing the numbers
once in order and once in reverse order and adding them
together.
For example, to add the numbers from 1 to 10:
1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10
10 + 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1
11 + 11 + 11 + 11 + 11 + 11 + 11 + 11 + 11 + 11
= 110
Sum of the numbers from 1 to 10 = 110 ÷ 2 = 55
Use this method to show that the nth triangular number is:
T(n) = n(n + 1)
2
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Square numbers
Square numbers are obtained when a whole number is
multiplied by itself. They are sometimes called perfect squares.
For example, 49 is a square number. It can be written as
49 = 7 × 7 or 49 = 72.
Square numbers can be illustrated using dots or counters
arranged in squares:
S(1) = 1
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S(2) = 4
S(3) = 9
S(4) = 16
S(5) = 25
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Making square numbers
The nth square number S(n) can be written as S(n) = n2.
There are several ways to generate a sequence of square
numbers.
We can multiply a whole number by itself.
For example, 25 = 5 × 5 or 25 = 52.
We can add consecutive odd numbers starting from 1.
For example, 25 = 1 + 3 + 5 + 7 + 9.
We can add together two consecutive triangular numbers.
For example, 25 = 10 + 15
We can find the product of two consecutive even or odd
numbers and add 1.
For example, 25 = 4 × 6 + 1.
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Difference between consecutive squares
Show that the difference between two consecutive square
numbers is always an odd number.
If we use the general form for a square number n2, where n
is a whole number, we can write the square number
following it as (n + 1)2.
The difference between two consecutive square numbers can
therefore be written as
(n + 1)2 – n2 = (n + 1)(n + 1) – n2
= n2 + n + n + 1 – n2
= 2n + 1
2n + 1 is always an odd number for any whole number n.
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Cube numbers
Cube numbers are obtained when a whole number is
multiplied by itself and then by itself again.
For example, 64 is a cube number. It can be written as
64 = 4 × 4 × 4 or 64 = 43.
Cube numbers can be illustrated using spheres arranged
in cubes:
C(1) = 1
C(2) = 8
C(3) = 27
C(4) = 64
C(5) = 125
The nth cube number C(n) can be written as C(n) = n3.
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Squares, triangles and primes
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Contents
N1 Integers
A N1.1 Classifying numbers
A N1.2 Calculating with integers
A N1.3 Multiples, factors and primes
A N1.4 Prime factor decomposition
A N1.5 LCM and HCF
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Negative numbers
A positive or negative whole number, including zero, is
called an integer.
For example, –3 is an integer.
–3 is read as ‘negative three’.
This can also be written as –3.
It is 3 less than 0.
0 – 3 = –3
Here the ‘–’ sign means
minus 3 or subtract 3.
Here the ‘–’ sign means
negative 3.
We say, ‘zero minus three equals negative three’.
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Integers on a number line
Positive and negative integers can be shown on a number line.
–8
–3
Negative integers
Positive integers
We can use the number line to compare integers.
For example,
–3 > –8
–3 ‘is greater than’ –8
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Adding integers
We can use a number line to help us add positive and
negative integers.
–2 + 5 = 3
-2
3
To add a positive integer we move forwards up the
number line.
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Adding integers
We can use a number line to help us add positive and
negative integers.
–3 + –4 == –7
-7
-3
To add a negative integer we move backwards down the
number line.
–3 + –4 is the same as
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–3 – 4
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Subtracting integers
We can use a number line to help us subtract positive and
negative integers.
5 – 8 == –3
-3
5
To subtract a positive integer we move backwards down
the number line.
5–8
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is the same as
5 – +8
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Subtracting integers
We can use a number line to help us subtract positive and
negative integers.
3 – –6 = 9
3
9
To subtract a negative integer we move forwards up the
number line.
3 – –6
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is the same as
3+6
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Subtracting integers
We can use a number line to help us subtract positive and
negative integers.
–4 – –7 = 3
-4
3
To subtract a negative integer we move forwards up the
number line.
–4 – –7
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is the same as
–4 + 7
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Adding and subtracting integers
To add a positive integer we move forwards up the
number line.
To add a negative integer we move backwards down the
number line.
a + –b is the same as a – b.
To subtract a positive integer we move backwards down
the number line.
To subtract a negative integer we move forwards up the
number line.
a – –b is the same as a + b.
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Integer circle sums
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Rules for multiplying and dividing
When multiplying negative numbers remember:
+ × + = +
+ × – = –
– × + = –
– × – = +
Dividing is the inverse operation to multiplying.
When we are dividing negative numbers similar rules apply:
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+ ÷
+ = +
+ ÷
– = –
– ÷
+ = –
– ÷
–
= +
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Multiplying and dividing integers
Complete the following:
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–3 × 8 = –24
–36 ÷
42 ÷
–7 = –6
540 ÷ –90 = –6
–12 × –8 = 96
–7 × –25 = 175
47 × –3 = –141
–4 × –5 × –8 = –160
–72 ÷ –6 =
3 × –8 ÷ –16 = 1.5
12
9
= –4
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Using a calculator
We can enter negative numbers into a calculator by using the
sign change key: (–)
For example:
–456 ÷ –6 can be entered as:
(–)
4
5
6
÷
(–)
6
=
The answer will be displayed as 76.
Always make sure that answers given by a calculator are
sensible.
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Sums and products
What two integers have a sum of 2 and a product of –8?
Start by writing down all of the pairs of numbers that multiply
together to make –8.
Since –8 is negative, one of the numbers must be positive
and one of the numbers must be negative.
We can have:
–1 × 8 = –8
–1 + 8 = 7
1 × –8 = –8
1 + –8 = –7
–2 × 4 = –8
–2 + 4 = 2
or
2 × –4 = –8
2 + –4 = –2
The two integers are –2 and 4.
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Sums and products
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Contents
N1 Integers
A N1.1 Classifying numbers
A N1.2 Calculating with integers
A N1.3 Multiples, factors and primes
A N1.4 Prime factor decomposition
A N1.5 LCM and HCF
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Multiples
A multiple of a number is found by multiplying the number
by any whole number.
What are the first six multiples of 7?
To find the first six multiples of 7 multiply 7 by 1, 2, 3, 4, 5
and 6 in turn to get:
7,
14, 21,
28,
35
and 42.
Any given number has infinitely many multiples.
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Factors
A factor (or divisor) of a number is a whole number that
divides into it exactly.
Factors come in pairs. For example,
What are the factors of 30?
1 and 30, 2 and 15, 3 and 10, 5 and 6.
So, in order, the factors of 30 are:
1, 2, 3, 5, 6, 10, 15 and 30.
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Prime numbers
If a whole number has two, and only two, factors it is called
a prime number.
For example, the number 17 has only two factors, 1 and 17.
Therefore, 17 is a prime number.
The number 1 has only one factor, 1.
Therefore, 1 is not a prime number.
There is only one even prime number. What is it?
2 is the only even prime number.
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Prime numbers
There are 25 prime numbers less than 100.
These are:
2
3
5
7
11
13
17
19
23
29
31
37
41
43
47
53
59
61
67
71
73
79
83
89
97
What if we go above 100? Around 400 BC the Greek
mathematician, Euclid, proved that there are infinitely many
prime numbers.
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Contents
N1 Integers
A N1.1 Classifying numbers
A N1.2 Calculating with integers
A N1.3 Multiples, factors and primes
A N1.4 Prime factor decomposition
A N1.5 LCM and HCF
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Prime factors
A prime factor is a factor that is a prime number.
For example,
What are the prime factors of 70?
The factors of 70 are:
1
2
5
7
10
14
35
70
The prime factors of 70 are 2, 5, and 7.
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Products of prime factors
70 = 2 × 5 × 7
56 = 2 × 2 × 2 × 7
This can be written as 56 = 23 × 7
99 = 3 × 3 × 11
This can be written as 99 = 32 × 11
Every whole number greater than 1 is either a
prime number or can be written as a product of
two or more prime numbers.
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The prime factor decomposition
When we write a number as a product of prime factors it is
called the prime factor decomposition or prime factor
form.
For example,
The prime factor decomposition of 100 is:
100 = 2 × 2 × 5 × 5
= 22 × 52
There are two methods of finding the prime factor
decomposition of a number.
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Factor trees
36
4
2
9
2
3
3
36 = 2 × 2 × 3 × 3
= 22 × 32
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Factor trees
36
3
12
4
2
3
2
36 = 2 × 2 × 3 × 3
= 22 × 32
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Factor trees
2100
30
6
2
70
5
3
10
2
7
5
2100 = 2 × 2 × 3 × 5 × 5 × 7
= 22 × 3 × 52 × 7
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Factor trees
780
78
2
10
39
3
5
2
13
780 = 2 × 2 × 3 × 5 × 13
= 22 × 3 × 5 × 13
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Dividing by prime numbers
2
96
2
48
2
24
2
12
2
6
3
3
96 = 2 × 2 × 2 × 2 × 2 × 3
= 25 × 3
1
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Dividing by prime numbers
3
315
3
105
315 = 3 × 3 × 5 × 7
5
35
= 32 × 5 × 7
7
7
1
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Dividing by prime numbers
2
702
3
351
3
117
3
39
13
13
702 = 2 × 3 × 3 × 3 × 13
= 2 × 33 × 13
1
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Using the prime factor decomposition
Use the prime factor form of 324 to show that it is a square
number.
2
324
2
162
3
81
3
3
3
= 22 × 34
This can be written as:
(2 × 32) × (2 × 32)
27
9
3
1
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324 = 2 × 2 × 3 × 3 × 3 × 3
or
(2 × 32)2
If all the indices in the prime factor
decomposition of a number are even,
then the number is a square number.
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Using the prime factor decomposition
Use the prime factor form of 3375 to show that it is a cube
number.
3
3
3375
= 33 × 53
1125
3
375
5
125
5
25
5
5
1
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3375 = 3 × 3 × 3 × 5 × 5 × 5
This can be written as:
(3 × 5) × (3 × 5) × (3 × 5)
or
(3 × 5)3
If all the indices in the prime factor
decomposition of a number are
multiples of 3, then the number is a
cube number.
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Using the prime factor decomposition
168 = 23 × 3 × 7
4116 = 22 × 3 × 73
294 = 2 × 3 × 72
Use the prime factor decompositions of the numbers given
above to answer the following questions.
1) What is 168 × 294 as a product of prime factors?
168 × 294 = (23 × 3 × 7) × (2 × 3 × 72 )
= 23 × 2 × 3 × 3 × 7 × 72
= 24 × 32 × 73
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Using the prime factor decomposition
168 = 23 × 3 × 7
4116 = 22 × 3 × 73
294 = 2 × 3 × 72
Use the prime factor decompositions of the numbers given
above to answer the following questions.
2) What is 4116 ÷ 294?
4116 ÷ 294 =
22 × 3 × 73
2 × 3 × 72
2×2×3×7×7×7
=
2×3×7×7
=2×7
= 14
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Using the prime factor decomposition
168 = 23 × 3 × 7
4116 = 22 × 3 × 73
294 = 2 × 3 × 72
Use the prime factor decompositions of the numbers given
above to answer the following questions.
3) Is 4116 divisible by 168?
If we divide 4116 by 168 we have:
22 × 3 × 73
4116 ÷ 168 =
23 × 3 × 7
2×2×3×7×7×7
=
2×2×2×3×7
There is a 2 left in
the denominator
No, 4116 is not divisible by 168.
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Using the prime factor decomposition
168 = 23 × 3 × 7
4116 = 22 × 3 × 73
294 = 2 × 3 × 72
Use the prime factor decompositions of the numbers given
above to answer the following questions.
4) Show that 294 × 6 is a square number.
We can write 6 as 2 × 3
294 × 6 = 2 × 3 × 72 × 2 × 3
Rearranging,
294 × 6 = 2 × 2 × 3 × 3 × 72
= 22 × 32 × 72
= (2 × 3 × 7)2
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Using the prime factor decomposition
4116 = 22 × 3 × 73
168 = 23 × 3 × 7
294 = 2 × 3 × 72
Use the prime factor decompositions of the numbers given
above to answer the following questions.
5) Write the fraction 168 in its simplest form.
294
168
23 × 3 × 7
=
294
2 × 3 × 72
2×2×2×3×7
=
2 × 3 × 7× 7
4
=
7
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Contents
N1 Integers
A N1.1 Classifying numbers
A N1.2 Calculating with integers
A N1.3 Multiples, factors and primes
A N1.4 Prime factor decomposition
A N1.5 LCM and HCF
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The lowest common multiple
The lowest common multiple (or LCM) of two numbers is
the smallest number that is a multiple of both the numbers.
For small numbers we can find this by writing down the first
few multiples for both numbers until we find a number that is
in both lists.
For example,
Multiples of 20 are :
20,
40,
60,
80,
100,
Multiples of 25 are :
25,
50,
75,
100,
125, . . .
120, . . .
The LCM of 20 and 25 is 100.
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The lowest common multiple
We use the lowest common multiple when adding and
subtracting fractions.
For example,
Add together 4
9
and 5
12
The LCM of 9 and 12 is 36.
4
9
+
×4
×3
5
16
12
×4
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=
36
+
15
36
=
31
36
×3
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The highest common factor
The highest common factor (or HCF) of two numbers is the
highest number that is a factor of both numbers.
We can find the highest common factor of two numbers by
writing down all their factors and finding the largest factor in
both lists.
For example,
Factors of 36 are : 1,
2,
3,
4,
6,
Factors of 45 are : 1,
3,
5,
9,
15,
9,
12,
18,
36.
45.
The HCF of 36 and 45 is 9.
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The highest common factor
We use the highest common factor when cancelling fractions.
For example,
Cancel the fraction 36
48
The HCF of 36 and 48 is 12, so we need to divide the
numerator and the denominator by 12.
÷12
36
48
=
3
4
÷12
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Using prime factors to find the HCF and LCM
We can use the prime factor decomposition to find the HCF
and LCM of larger numbers.
For example,
Find the HCF and the LCM of 60 and 125.
2
2
3
5
60
30
15
5
1
60 = 2 × 2 × 3 × 5
60 of 64
2
3
7
7
294
147
49
7
1
294 = 2 × 3 × 7 × 7
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Using prime factors to find the HCF and LCM
60 = 2 × 2 × 3 × 5
294 = 2 × 3 × 7 × 7
60
294
2
7
2
5
3
7
HCF of 60 and 294 = 2 × 3 = 6
LCM of 60 and 294 = 2 × 5 × 2 × 3 × 7 × 7 = 2940
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Using prime factors to find the HCF and LCM
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The LCM of co-prime numbers
If two numbers have a highest common factor (or HCF) of 1
then they are called co-prime or relatively prime numbers.
For two whole numbers a and b we can write:
a and b are co-prime if HCF(a, b) = 1
If two whole numbers a and b are co-prime then:
LCM(a, b) = ab
For example, the numbers 8 and 9 do not share any common
multiples other than 1. They are co-prime.
Therefore,
63 of 64
LCM(8, 9) = 8 × 9 = 72
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The LCM of numbers that are not co-prime
If two numbers are not co-prime then their highest common
factor is greater than 1.
If two numbers a and b are not co-prime then their lowest
common multiple is equal to the product of the two numbers
divided by their highest common factor.
We can write this as:
LCM(a, b) =
ab
HCF(a, b)
For example,
8 × 12
96
LCM(8, 12) =
=
= 24
HCF(8, 12)
4
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