1. Integers
1. Operations
Addition AND / PLUS
In small additions we say and for addition
and is/are for the result 2+6 = 8 - two and six are eight - two and six is eight In larger additions and in more formal style
(in maths) we use plus for +, and equals or is for the result.
234 + 25 = 259
two hundred and thirty four plus twenty five equals / is two hundred and fifty nine
Subtraction MINUS/TAKE AWAY/FROM
In conversational style, with small numbers 9!5=4
- Five from nine leaves/is four - Nine take away five leaves/is four In a more formal style, or with larger
numbers, we use minus and equals
510 - 302 = 208
Five hundred and ten minus three hundred and two equals/is two hundred and eight
Multiplication TIMES/MULTIPLIED BY
In small calculations
3 x 4 = 12 three fours are twelve
6 x 7 = 42 six sevens are forty-two In larger calculations 17 x 381 = 6,477
17 times 381 is/makes 6,477 In a more formal style 17 x 381 = 6,477
17 multiplied by 381 equal 6,477
Division DIVIDED BY
In small calculations 8:2 = 4
two into eight goes four (times)
In formal style 270:3 = 90
Two hundred and seventy divided by three equals ninety
Powers 65
6 is the base
5 is the index or exponent
- The fifth power of six - Six to the power of five - Six powered to five.
Squares
32 is read as: - Three squared
- Three to the power of two.
Cubes
53 Is read as:
2. Integers
Positive integers are all the whole numbers greater than zero: 1, 2, 3, 4, 5... . Negative integers are all the opposites of these whole numbers: -1, -2, -3, -4, -5 … . We do not consider zero to be a positive or negative number. For each positive integer, there is a negative integer, and these integers are called opposites. For example, -3 is the opposite of 3, -21 is the opposite of 21, and 8 is the opposite of -8. If an integer is greater than zero, we say that its sign is positive. If an integer is less than zero, we say that its sign is negative.
Absolute value of an integer
The number of units a number is from zero on the number line. The absolute value of a number is always a positive number (or zero). We specify the absolute value of a number n by writing n in between two vertical bars: |n|.
Examples: |6| = 6 |-12| = 12 |0| = 0
|1234| = 1234 |-1234| = 1234
Recognising negative numbers
You have to be able to use negative (or directed) numbers in many everyday contexts, for example with temperatures, bank balances etc.
A temperature of 6 degrees below zero is !6°.
If your bank account is £100 overdrawn, then your balance is !£100.
In your exam, you might be asked to put some numbers in order. You have to remember, for instance, that !10 is less than !5.
Sometimes it helps to see negative numbers on a number line.
NEGATIVE NUMBERS AND TEMPERATURE
You need to be able to work out rises and falls in temperature using negative numbers.
the temperature has gone up by 11°C.
Question
At 6pm, the temperature was 3°C. By midnight, it had dropped to !5°C. How great was the fall in temperature?
Question
At 7am, Joe recorded the temperature in his garden as being !4°C. He went back out outside at 1pm and found that the temperature had increased by 12°C. What was the temperature at 1pm?
Question
At 6pm, the outside temperature was 7°C. By 6am, the temperature had dropped by 13°C. What was the temperature at 6am?
Here is a table showing temperatures in cities worldwide:
City Algiers Barcelona London
New York Moscow Oslo Paris Sydney
Temp 16°C 17°C 5°C 7°C !9°C !5°C 12°C 23°C
You could be asked to find the difference in temperature between two cities, eg, Moscow and London.
Moscow is !9°C. London is 5°C.
Starting at !9, we count up 14 to get to 5. So the difference in temperatures is 14°C.
Question
What is the difference in temperature between Moscow and Oslo?
3. Negative number arithmetic
Adding
1. When adding integers of the same sign, we add their absolute values, and give the result the same sign.
!+"#+!+$#=+% !!&#'+!!(#=!%
2. When adding integers of the opposite signs, we take their absolute values, subtract the smaller from the larger, and give the result the sign of the integer with the larger absolute value.
(!6) +(+5) =!1
Subtracting
Subtracting an integer is the same as adding its opposite. (+5) !(+2) =(+5) +(!2) = +3
(!6) !(!1) =(!6) +(+1) = !5 Solved questions:
a) (+7) + (+2) = 7 + 2 = 9 b) (!4) + (!1) = !4 ! 1 = !5
c) First way: +(!5 + 3 !2 + 7) = !5 + 3 ! 2 + 7 = !7 + 10 = +3
Second way: +(!5 + 3 !2 + 7) = !5 + 3 ! 2 + 7 = !2 ! 2 + 7 = !4 + 7 = +3 d) First way: !(!5 + 3 !2 + 7) = +5 ! 3 + 2 ! 7 = 7 ! 10 = !3
Second way: !(!5 + 3 !2 + 7) = +5 ! 3 + 2 ! 7 = +2 + 2 ! 7 = + 4 ! 7 = !3 Try now with these:
1. a) (+7) + (+1) = d) (+10) ! (+2) = b) (!15) + (!4) = e) (!11) ! (!10) = c) (+9) ! (!5) = f) (!7) + (+1) =
2. a) 7 ! 5 = d) !3 + 8 = b) 11 ! 4 + 5 = e) !1 + 8 + 9 = c) !9 ! 7 = f) !10 + 3 + 7 =
3. a) 5 ! 7 + 19 ! 20 + 4 ! 3 + 10 = b) !(8 + 9 ! 11) =
c) 9 ! 11 + 13 + 2 ! 4 ! 5 + 9 = d) !(20 + 17) ! 16 + 7 ! 15 + 3 = 4. –33 – 22 + (-11)
5. 45 – 81 + 17
6. – 7 – (-12) + 8
7. 88 – 98 – 65
8. –3 – (-11)
9. –73 + (-21)
10. 99 – 78 – (-61)
11. –64 – (-64)
12. You enter the hotel elevator in the parking garage, 4 floors below the lobby level. You get off to go to your room on the 11th floor. How many floors have you traveled in the
elevator?
13. Last season the long jumpers on the track team averaged a distance of 15 feet per jump. In the first meet this season, Jason jumped 3 feet short of the average, Abdul jumped 2 feet over the average, Van jumped 4 feet over the average, and Maria jumped 1 foot short of the average. Write an expression to show each jumper’s distance in relation to the average from last year.
Multiplication
If the signs are the same, the answer is positive. If the signs are different, the answer is negative.
Here are some examples: • !4 " 2 = !8 • !3 " !2 = 6 • 5 " !3 = !15
Division
Division follows the same rules as multiplication. So, for example, !24 ÷ 6 = !4, because 24 ÷ 6 = 4 and the answer must be negative.
Questions
1. a) (+7) ! (+2) = d) (!5) ! (+8) = b) (+12) ! (!3) = e) (!1) ! (!1) = c) (!10) ! (+10) = f) (+5) ! (+20) =
2. a) (+16) : (+2) = c) (!25) : (+5) = e) (+12) : (!3) = b) (!8) : (!1) = d) (!100) : (+10) = f) (+45) : (+9) =
3. Complete the missing numbers
a) (+9) ! ... = !36 d) (!7) ! ... = +21 g) ... ! (!8 ) = !40 b) ... ! (+10) = !100 e) (!30) ! ... = +30 h) (+6) ! ... = 0 c) (+3) ! ... = !15 f) (!8) ! ... = +16 i) ... ! (!5 ) = +25
4. Complete the missing numbers
a) (+42) : ... = !7 d) (!8) : ... = +1 g) ... : (!9 ) = +6 b) (!20) : ... = !20 e) ... : (!6) = +5 h) (+9) : ... = !9 c) (+12) : ... = !4 f) (!64) : ... = +8 i) (!8) : ... = !2
5. Write the following as directed numbers. • 40 metres below sea level: m
• A temperature of 8 degrees below zero.: °C
6. Use the number line to help you to answer the following questions.
•
At 9pm, the temperature was 2°C. By midnight, the temperature had dropped to -5°C. By how many degrees had the temperature fallen?•
The temperature at midnight was -3°C. By 8am, it had risen by 6°C. What was the temperature at 8am?City Algiers Barcelona Berlin London New
York Moscow Oslo Paris Sydney
Temp 16°C 17°C 8ºC 5°C 7°C !9°C !5°C 12°C 23°C
7. What was the difference in temperature between Barcelona and Berlin?
8. How much higher was the temperature in Rome than the temperature in Oslo?
4. Order of operation
• Do all operations in brackets first.The easy way to remember is
This gives you BODMAS.
Worked example: Calculate: 3 x (7 - 3)
Solution
In this question, we have a bracket, a subtraction and a multiplication.
BODMAS tells us that brackets come first, so we calculate: 3 x (7 - 3) = 3 x 4 = 12
Calculate: 2 + 3 x 5 - 4
2 + 3 x 5 - 4 = 2 + 15 - 4 = 13
Don't use a calculator to find the value of the following expressions:
1. 5 – 8 · (- 3 – 2)
2. - 2 - (-3)+(-6) · 5
3. 3 + [2 + 3 · (- 2)] - (-3)2 · 6
4. - 32
5. (- 3)2
6. 42 - 2 · (-4)
7. ( - 4)2 - 2 · (-4)
8. 4 · (10 - 3 · 4) - 2 · (-3 + 15: 3)
9. 3 · 5 - ( 2 + 3 · (- 2))
10. 3 - ( - 2 - (- 1 - (- 6)) - 3) + 7
11. 1 + (2 + (- 7 - 4 - 1)) – 6
12. (10+ (32 - 2) · (- 2))2
5. Powers and roots
SQUARES, CUBES AND THEIR ROOTSSquaring a number
32 means ‘3 squared’, or 3 x 3.
The small 2 is an index number, or power. It tells us how many times we should multiply 3 by itself.
So, 12 = 1 x 1 = 1 22 = 2 x 2 = 4
32 = 3 x 3 = 9 42 = 4 x 4 = 16
52 = 5 x 5 = 25
1, 4, 9, 16, 25… are known as square numbers.
Square roots
The opposite of a square number is a square root. We use the symbol to mean square root.
So we can say that
4
=
2
and25
= 5.However, this is not the whole story, because -2 x -2 is also 4, and -5 x -5 is also 25.
So, in fact,
4
= 2 or -2. And25
= 5 or -5.Remember that every positive number has two square roots.
Cubing a number
2 x 2 x 2 means ‘2 cubed’, and is written as 23.
13 = 1 x 1 x 1 = 1 23 = 2 x 2 x 2 = 8
33 = 3 x 3 x 3 = 27 43 = 4 x 4 x 4 = 64
53 = 5 x 5 x 5 = 125
1, 8, 27, 64, 125… are known as cube numbers.
Cube roots
The opposite of a cube number is a cube root. We use the symbol3 to mean cube root.
So 3
8
is 2 and27
3 is 3.
Each number only has one cube root.
POWERS
You will already be familiar with the notation for squares and cubes: a2 = a x a and a3 = a x a x a
and this is generalised by defining: an = a x a x...x a (n times a)
That is, powers are used to describe the result of repeatedly multiplying a number by itself.
RULES OF INDICES
Multiplying
When multiplying add the powers.
23" 24 = (2·2·2) · (2·2·2·2) = 27
Dividing
When dividing subtract the powers.
25 ÷ 22 =
2·2·2·2·2
2·2
= 2·2·2 (Cancelling two of the 2s) = 2 3The power of a power
When taking the power of a number already raised to a power, multiply the powers.
For example this is how to find the square of
2
3.square of 23 = (23)2 = (2 · 2 · 2) · (2 · 2 · 2) =
2
6Notice that the answer has an index of 6, which comes from multiplying the powers at the beginning (3 · 2). Here is another example.
(22)4 = (2 · 2) · (2 · 2) · (2 · 2) · (2 · 2) =
2
8So you see that in both examples the powers have been multiplied (3·2 and 2·4) 1. Write in power form
a) 22! 24 ! 23 = 22+4+3 = d) 52 ! 53 =
b) 64 ! 6 ! 63! 62 = e) (!4)4 ! (!4)4 =
c) (!5)5 ! (!5)2 = f) (!10)3 ! (!10)3! (!10)4 =
2. Write the missing index
a) 22 ! 2.... ! 2.... = 26 e) 5....! 5.... = 55
b) (!2)4! (!2)....! (!2).... = (!2)8 f) 42 ! 4....! 4.... ! 4.... = 47
c) (!7).... ! (!7).... = (!7)5 g) 106 ! 10.... ! 10.... = 109
d) 3.... ! 3....! 3.... = 35 h) 10.... ! 10.... = 105
3. Write in power form
a) 36 : 32 = 36-2 = d) 44 : 43 =
b) 65 : 63 = e) (!4)4 : (!4)4 =
c) (!5)5 : (!5)2 = f) (!10)6 : (!10)3 =
4. Write in power form
a) [(4)5]2 = (4)5 ! 2 = 4.... d) [(5)2]4 =
b) [(!3)3]3 = e) [(6)0]2 =
c) [(!8)2]3 = f) [(10)3]4
'!
5. Write the missing index
a) [2....].... = 28 c) [3....].... = 310 e) [(!5)....].... = (!5)6
6. Calculate:
1 (!2)2 · (!2)3 · (!2)4 =
2 (!8) · (!2)2 · (!2)0 (!2) =
3 (!2)2 · (!2)3 · (!2)4 =
4 22 · 23 · 24 =
5 22 : 23 =
6 [(!2) 2] 3 · (!2)3 · (!2)4 =
7 [(!2)6 : (!2)3 ]3 · (!2) · (!2)4= 7. Work out:
1(!3)1 · (!3)3 · (!3)4 =
2 (!27) · (!3) · (!3)2 · (!3)0=
3 (!3)2 · (!3)3 · (!3)4 =
4 (!3)1 · [(!3)3]2 · (!3)4 =
5 [(!3)6 : (!3)3] 3 · (!3)0 · (!3)4 =
8. What is
4
?• Just 2
• 2 or -2
• 16
11. A Roman emperor was born in 63 B.C. and died in 14 A.D. How many years did he live?
12. A bomb displaces the oil in a well from 975 m deep and raises it to a deposit place 48 m above ground level. How far does the oil displaced?
13. What is the change in temperature a customer in a grocery store experiences when they walk from the chilled vegetable section at 4 ºC to the frozen fish to section which is set to to !18 ºC?
14. The air temperature in the atmosphere decreases at the rate of 9 ºC every 300 meters. What height would a plain have to fly to experience a temperature of
!81 ºC?
6. Factors and multiples
FACTORS
The factors of a number are any numbers that divide into it exactly. This includes 1 and the number itself.
For example, the factors of 6 are 1, 2, 3 and 6. The factors of 8 are 1, 2, 4 and 8.
For larger numbers it is sometimes easier to ‘pair’ the factors by writing them as multiplications. For example, 24 = 1 x 24 = 2 x 12 = 3 x 8 = 4 x 6
MULTIPLES
The multiples of a number are all the numbers that it will divide into. This includes the number itself.
• For example, the multiples of 2 are 2, 4, 6, 8, 10, 12, 14, 16… • The multiples of 5 are 5, 10, 15, 20, 25, 30, 35…
• The multiples of 20 are 20, 40, 60, 80, 100…
7. Tests of divisibility
One number is divisible by:2 If the last digit is 0 or is divisible by 2, (0, 2, 4, 8). 3 If the sum of the digits is divisible by 3.
4 If the last two digits are divisible by 4. 5 If the last digit is 0 or is divisible by 5, (0,5). 9 If the sum of the digits is divisible by 9. 8 If the half of it is divisible by 4.
6 If it is divisible by 2 and 3.
11 If the sum of the digits in the even position minus the sum of the digits in the uneven position is 0 or divisible by 11.
EXERCISES
1. Write down four consecutive multiples of: a) 7 greater than 100
b) 15 greater than 230 c) 9 greater than 1230
2. Write down all the multiples of 6 between 92 and 109. 3. Write down all the multiples of 6 between 1200 and 1250. 4. Write down all the factors of
a) 18 b) 90 c) 140 d) 80 e) 50
5. Find out the missing figure so the number (there can be more than one answer) a) 3[ ]1 is a multiple of 3
b) 57[ ] is a multiple of 2 c) 23[ ] is a multiple of 5 d) 52[ ]3 is a multiple of 11
6. Prime factors
PRIME FACTORSEvery number can be written as a product of prime numbers.
(Remember: ‘product’ means ‘times’ or ‘multiply’.)
For example 40 = 2 x 2 x 2 x 5 126 = 2 x 3 x 3 x 7 28 = 2 x 2 x 7
How do we write a number as a product of its prime factors?
Worked example
Write 24 as a product of its prime factors.
There are lots of ways to solve this. Here are two: Solution 1
Start with the smallest prime number that divides into 24, in this case 2. (2 divides into all even numbers.) So we can write 24 as:
24 = 2 x 12
Now think of the smallest prime number that divides into 12. Again we can use 2, and write the 12 as 2 x 6, to give.
24 = 2 x 2 x 6
6 also divides by 2 (6 = 2 x 3), so we have: 24 = 2 x 2 x 2 x 3 2, 2, 2 and 3 are all prime numbers, so we have our answer.
In short, we would write the solution as:
24 = 2 x 12 = 2 x 2 x 6 = 2 x 2 x 2 x 3
Solution 2
We can also use a factor tree:
This shows us that 24 = 2 x 2 x 2 x 3
Remember that we can write these factors in index form. In this case
24 = 2 x 2 x 2 x 3, so 24 = 2# x 3
EXERCISES
1. What number is described by 2# x 3$ x 5? 2. Write 36 as a product of its prime factors. 3. Factorise:
a) 46 b) 180 c) 60 d) 1500 e) 135
7. Highest common factor and Lowest commond multiple
HIGHEST COMMON FACTOR (HCF)
We have already seen that the factors of 24 are 1, 2, 3, 4, 6, 8, 12 and 24
and that the factors of 40 are 1, 2, 4, 5, 8, 10, 20 and 40.
The numbers 1, 2, 4 and 8 appear in both lists, so they are known as the common factors of 24 and 40.
The number 8 is the highest of them, and it is called the highest common factor (HCF).
Factors that are common to two or more numbers are said to be common factors.
Method I For small numbers
For example: 8 = 1x 8 = 2 x 4 12 = 1x12 = 2x6 = 3x4 - Factors of 8 are 1, 2, 4 and 8
- Factors of 12 are 1, 2, 3, 4, 6 and 12.
So, the common factors of 8 and 12 are 1, 2 and 4. HCF is 4
Method II (General) To find the Highest Common Factor of two or more higher numbers: - Find the prime factor decomposition.
- Choose only the common factors with the lowest exponents.
LOWEST COMMON MULTIPLES (LCM)
The multiples of 4 are 4, 8, 12, 16, 20, 24, 28, 32, 36, …
The multiples of 6 are 6, 12, 18, 24, 30, 36, 42, …
The numbers 12, 24 and 36 appear in both lists, so they are known as common multiples.
The number 12 is the lowest number to appear in both lists, and it is called the lowest common multiple (LCM).
Method I (For small numbers)
List the multiples of the largest number and stop when you find a multiple of the other number. This is the LCM.
Method II (General)
To find the lowest common multiple (LCM) of two or more higher numbers: - Find the prime factor decomposition.
- Choose the non common factors and the common factors with the highest exponents.
Example: Find the lowest common multiple of 18 and 24.
18 = 2⋅32 24=23⋅3 So, LCM(18,24)=23 ⋅32 =72.
EXERCISES
1. What is the HCF of 12 and 20?
2. What is the LCM of 6 and 10?
3. Find the HCF and LCM of 90 and 175.
4. Find the HCF and LCM of 60 and 150.
5. A beacon flashes its light every 12 seconds, another every 18 seconds and a third every minute. At 6.30 pm the three flash simultaneously.
Find out the times when the three flash simultaneously again in the next five minutes.
6. A businessman goes to Chicago every 18 days for one day and another businessman every 24 days, also for only one day. Today, both men are in Chicago.
Within how many days will the two business men be in Chicago again at the same time?
8. In a cellar there are 3 differently sized casks of wine, whose capacities are: 250 liters, 360 litres, and 540 litres. The owner of the cellar wants to package the wine in barrels with an equal amount of wine in each one. Calculate the maximum capacities of these barrels so that the owner can package equal amounts of wine in each cask, and determine the quantity of barrels he will need.
9. The floor of a room that needs to be tiled is 5 m long and 3 m wide.
Determine the ideal size of the tiles and the number of the tiles needed, such that the number of tiles that are placed is minimal and none of them are to be cut. Keep in mind that all tiles are to be the same size.
10. A trader wants to put 12,028 apples and 12,772 oranges into boxes. Each box is to contain the an equal number of apples and an equal number of oranges and also the greatest number of each. Find the ideal number of oranges and apples for each box and the number of boxes needed.
11. What is the size of the largest possible square tile that can fit an exact number of times into a room 8 m long and 6.4 meters wide? How many tiles are needed? 12. Read the text below and answer the questions:
Dionysius the Meagre was a monk that was born at the end of the V century and died in the middle of the VI century. Of Armenian origin, was an abbot in a convent in Rome and was intellectually regarded in his time, so he received the job from the Pope to unify in the calendar the celebration of Easter in the Christian
world.
While investigating and determining the date of Easter, which is the most important one in Christian Religion, he did not use as a reference the date of the foundation of Rome but the birth of Jesus Christ, that occurred in 754 a.u.c. (ab urbe condita), that is to say, 754 years after the foundation of Rome. This date and the beginning of Christian era, received definitive support by Charlemagno more than two centuries later when he dated official documents with this system.
The debate regarding year zero grew as authors affirmed that year zero did not exist because in Europe the people did not know the zero yet. In fact what did not really exist was not zero but negative numbers. So we do not talk of years before Christ as negative numbers until the XVII century. So the year 1 a. D. was not year – 1 but 753 a.u.c.
This criterion to order the dates it is important because Arabic System does not have year zero either
despite in the VIII century they used this number, which was borrowed from India by them.
QUESTIONS: Imagine that Dionysius the Meagre had used as a reference de date of the foundation of Rome instead the birth of Jesus Christ. Find the dates of the following events and rewrite them with this new system.
