1 Eso
1 Eso
1 Eso
40. forty 50. fifty 60. sixty 70. seventy 80. eighty 90. ninety 100. a
hundred (or one hundred)
1000. a thousand
1001. a thousand and one
1002. a thousand and two
1003. a thousand and three
1004. a thousand and four, etc.
2000. two thousand 3000. three thousand 4000. four thousand, etc.
8=eight
48=forty-eight
248=two hundred and forty-eight
6 248=six thousand two hundred and forty-eight
26 248=twenty-six thousand two hundred and forty-eight
126 248=one hundred and twenty-six thousand two hundred and forty-eight
2 126 248=two million one hundred and twenty-six thousand two hundred and forty-eight
When the division between two numbers is exact, we say there is a relation of
divisibility between them.
D d D is divisible by d.
112. q
36 4 36 5
0 9 1 7
The division is exact The division isn't exact
The multiple of a number is the product generated when that number is multiplied by
a natural number.
The first multiples of a number are obtained by mulyipling the number by each of
the natural numbers: 1, 2, 3, 4,
a={a 1, a 2, a 3, a 4, a 5, ...}
Factors of a number:
48 6 48 7
0 8 6 6
The division is exact The division isn't exact
If a number can be expresed as a product of two natural numbers, then the natural
numbers are factors of the first number.
A factors is any number that will divide into another number exactly (with no part left
over): 8 can be divide by 2 (the factor in this example) 4 times. However, in the total
number 8 has several factors: 1, 2, 4 and 8.
Numbers that are greater than 1 and have only two factors, 1 and itself are called prime numbers.
A prime number (or a prime) is a natural number wich has exactly two distint natural
numbers divisors: 1 and itself.
If a numbers has more than two divisors, it is called composite
number. The number 1 is by definition not a prime number.
A simple ancient algorithm for finding all prime numbers up to a specified natural
number, n ( we are going to do it with n=100), is the Sieve of Erastosthenes:
1. Write down the numbers 1, 2, 3, 4, 5, , n. (Remember n=100 for us). We will
eliminate composites by marking them: Inicially all numbers are unmarked.
2. Mark the number 1 as special (it is neither prime nor composite).
3. Cross out all numbers >2 wich are divisible by 2 (every even number).
4. Find the smallest remainder number >2. It is 3. So cross out all numbers >3
wich are divisible by 3.
5. Find the smallest remainder number >3. It is 5. So cross out all numbers >5
wich are divisible by 5.
Continue untill you have crossed out all number divisible by (in our case ).
n 100
Prime numbers 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,
Factoring numbers:
Prime factoring is to factor and then continue factoring a number untill you can no
longer reduce the factors into constituent factors any further.
Any number can be written as a product of prime numbers in a unique way (except for the order).
Examples:
24 2 24 is divisible by 2
12 2 12 is divisible by 2
6 2 6 is divisible by 2
3 3 3 is divisible by 3
1
24=233
Greatest common factor (GCF):
There are different ways to find the GCF of numbers. Look at them and choose the one you prefer!
Method 1
First list all factors of each number, then list the common factors and choose the largest one:
Although the numbers in bold are all common factors of both 12 and 18, 6 is the greatest
common factor. We write GCF (12, 18)=6.
Find the Greatest Common Factor (GCF) of the numbers 24 and 36 with this method.
Method 2
To find the GCF of a set of numbers, you must factor each of the numbers into primes.
Then for each different prime number in all of the factorizations, do the following
1. Count the number of times each prime number appears in all the factorizations.
2. For each prime number, take the lowest of these counts and write the result.
3. The greatest common factor is the product of all the prime numbers written down.
If GCF(a,b) =1, it is said that a and b are relative primes (they don't have any
common factors except 1.
Method 2: Find GCF(72, 90, 120)
72 2 90 2 120 2
36 2 45 3 60 2
18 2 15 3 30 2
9 3 5 5 15 3
3 3 1 5 5
1 1
72=2332 90=2325 120=2335
2. Take the prime numbers that appears in all the factorizations (Remember taking
the lowest number of times they appear).
Prime numbers selected: 2 and 3.
There are different ways to find the LCM of numbers. Look at them and choose the one you prefer!
Method 1
List the multiples of the larger number and stop when you find a multiple of the other
number. This is the LCM.
The multiples of 6 are: 6, 12, 18, 24, 30, 36, 42, 48, 54
The multiples of 9 are: 9, 18, 27, 36, 45, 54, 63, 72, 81 ...
Method 2
To find the LCM of a set of numbers, you must factor each number into primes. Then
for each different prime number in all of the factorizations, do the following
1. Count the number of times each number appears in each of the factorizations.
2. For each prime number, take the largest of these counts and write the result.
3. The least common multiple is the product of all the prime numbers written down.
Example: LCM (4,6) = 12, because 4= 22 and 6=23, so LCM (4,6)= 223=12 .
Method 2: Find LCM (16, 24, 40)
16 2 24 2 40 2
8 2 12 2 20 2
4 2 6 2 10 2
2 2 3 3 5 5
1 1 1
16= 24 24=233 40=235
2. Take all the prime numbers that appears in all the factorizations (Remember taking
the highest number of times they appear).
Primes number selected: 24 , 3 and 5.
CALCULATIONS
4. Write the missing words. Then, write the answers in numbers and symbols:
3___7___4 = 14 _________________________________________
5+48=37 5+4+37=54
The words in all the statements are jumbled up. Rewrite them so that they make sense.
11. In each statements, the words are in the correct order, but the letters of each word
have been jumbled up. Rewrite each sentence.
Integers:
The first set of number we knew was the set of Natural Numbers (also called whole numbers):
= {0, 1, 2, 3, 4, 5, 6, 7, }
There are many situations in which you need to use numbers below zero, one of these is
temperature, others are money that you can deposit (positive) or withdraw (negative) in a
bank, steps that you can take forwards (positive) or backwards (negative).
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,
Integers allow us to count and order below and above zero. The set of all Integers is
represented by the letter = { -4, -3, -2, -1, 0, 1, 2, 3, 4, }
6
The natural numbers are included in the set of Integers. This fact is represented by the symbol
' ' ' ' .
is read is a subset of .
The number line is a line labelled with the integers in increasing order from left to
right, that extends in both directions:
For any two different places on the number line, the integer on the right is greater than
the integer on the left.
Examples:
Opposite of an integer:
The opposite of an integer is the same number with the other sign. The distance from a
number to zero is the same as the distance from its opposite to zero.
The absolute value of an integer is the number of units is from zero on the number line.
The absolute value of an integer is always a positive number (or zero). We specify
the absolute value of a number n in between two vertical bars: n .
Examples:
7
1. Plot on the number line and after order them from less to great.
-2 +8 0 -5 3
Write the opposite and the absolute value of all these numbers.
When adding integers with the same sign: We add their absolute values, and give the
result with the same sign.
4 6 = 10 3 6 =9 5 2 =7
When adding integers with the opposite signs: We subtract their absolute values (we
subtract the smaller absolute value from the larger), and give the result with the sign of
the integer with the larger absolute value.
7 9 =2 8 5 = 3 6 1 =5
We convert the subtracted integer to its opposite, and add the two integers:
3 7 = 3 7 =4 2 8 = 2 8= 6
8
You can use a number line to help you to add or subtract integers:
Calculate 4-6:
Calculate -3+7:
The answer is 4.
1. Calculate:
a) 5 2 b) 913 c)15 12
d) 7 13 e) 2 10 f)4 5
g) 9 6 h) 8 3 i)12 3
Example: -3 + 8 4 + 2 5 =
9
3. Calculate:
If both numbers have the same sign (positive or negative), their product is the
product of their absolute values (their product is positive).
If the numbers have opposite signs, their product is the opposite of the product
of their absolute values (their product is negative).
If a number is 0, the product is 0.
5 4 = 20 5 4 =20
5 4 =20 5 4 = 20
1. Count the number of negative integers in the product. If this numbers is even, the
product is positive, but if the number is odd, the product is negative.
2. Take the product of their absolute values.
4 6 2 = 48 2 3 5 =30
0 5 . 7 =0 5 2 2 4 = 80
1 3 2 10 =60 1 2 1 3 5 = 30
11
Rules for Division:
To divide a pair of integers the rules are the same than for the product:
If both numbers have the same sign (positive or negative), divide the absolute values of
the first integer by the absolute value of the second integer (the result is positive).
If the number have opposite signs, divide the absolute value of the first integer by
the absolute value of the second integer, and give the result a negative sign.
12 : 3 = 4 12 : 3 =4
12 : 3 =4 12 : 3 = 4
1. Calculate:
a) 3 4
b) 5 4
c) 10 3
d) 15 : 3
e) 40 : 8
f) 56 : 7
2. Calculate:
a) 5 2 1
b) 2 1 4
c) 18 : 2 : 3
d) 20 : 2 : 1
e) 9 2 1 2
f) 5 2 1 8
g) 36 : 9 2
h) 15 3 : 5
12
Powers of Integers:
Examples:
4 2= 4 4 = 16 4 3= 4 4
4 = 64
3 4= 3 3 3 3 = 81 3
3
= 3 3 3 =27
Sign of the power of an integer:
Example:
2 1=2
2 2= 2 2 = 4 2 3 =
2 2 2 =8
2 4= 2 2 2 2 = 16 2 5= 2
2 2 2 2 =32
2 6 = 2 2 2 2 2 2 = 64
Multiplying powers: You can multiply powers with the same base by adding the exponents.
aman=am n
Dividing powers:You can divide powers with the same base by subtracting the exponents.
am :an =amn
13
Power of a power:You can simplify the power of a power by multiplying the exponents.
am n=amn
Multiplying powers with the same exponent:You can multiply powers with the same
exponent by multiplying the bases.
ab n=anbn
Dividing powers with the same exponent:You can divide powers with the same
exponent by dividing the bases.
a : b n=an :bn
a) [ 2 2 ]3 : 2 4
b) 24 23 : 25
c) 32 35 36 : 34 35
d) 84 :82 :82
e) 76 3 72 3
f) 64 :[ 28 :27 3]3
Square root:
Examples:
=1 because 12=1 =11 because 112=121
1 121
2
4=2 because 2 =4 144=12 because 122=144
9=3 because 32=9 169=13 because 132=169
16=4 because 42=16 196=14 because 142 =196
25=5 because 52=25 225=15 because 152=196
36=6 because 62 =36 256=16 because 162=256
49=7 because 72 =49 289=17 because 172=289
64=8 because 82=64 324=18 because 182=324
81=9 because 92 =81 361=19 because 192=361
100=10 because 102 =100 400=20 because 202=400
But, be careful! If we are working in the set of integers, a number can have two square roots:
Example:
If a radicand is not a perfect square, the square root is not exact. In this case, we talk
about integer square root.
The integer square root of a number a is the greater number b whose squared is less than a.
The remainder of the integer square root is the difference between the radicand a and
the squared of the integer root b.
Examples:
The multiples of a number are obtained multiplying the number by each integer. Usually,
the set of multiples of a number a is written a .
Example: Multiples of 2: 2 ={... ,6,4,2,0, 2, 4, 6,...}
The factors of a number are the numbers that divide exactly into it, with no remainder.
Prime Numbers:
If a number has only two different factors, 1 and itself, then the number is said to be
a prime number.
Remember, we have already studied the Sieve of Erastothenes that gives us the list
of the prime numbers. It starts as follows:
Divisible by 2
A number is divisible by 2 if the last digit is 0, 2, 4, 6 or 8.
Divisible by 3
A number is divisible by 3 if the sum of the digits is divisible by 3.
Divisible by 4
A number is divisible by 4 if the number formed by the last two digits is either 00 or divisible by 4.
Divisible by 6
Divisible by 10
Divisible by 11
To check if a number is divisible by 11, sum the digits in the odd positions counting
from the left (the first, the third, ) and then sum the remainder digits. If the difference
between the sums is either 0 or divisible by 11, then so is the original number.
72 2
36 2
18 2
9 3
3 3
1
72=23 32
Factors that are common to two or more numbers are said to be common
factors.
Example:
The largest common factor of two or more numbers is called the highest common
factor (HCF).
In general, there are two methods for finding the Highest common factor of two or more
numbers:
List the factor of each number, and find the common factors. The largest of
them is the highest common factor.
Example:
Factors of 8: 1, 2, 4, 8.
Factors of 12: 1, 2, 3, 4, 6, 12.
So, HCF(8,12)=4.
Method II (general):
Example:
Find HCF (360,300):
360 2 300 2
180 2 150 2
90 2 75 3
45 3 25 5
15 3 5 5
5 5 1
1 300=22 3 52
360=23 32 5
So, HCF (360,300) = 22 3 5=60
Lowest Common Multiple (LCM) or Least Common Multiple (LCM):
Multiples that are common to two numbers are said to be common multiples.
Example:
Multiples of 2 are 2, 4, 6, 8, 10, 12, 14, 16, 18,
Multiples of 3 are 3, 6, 9, 12, 15, 18,
So, common multiples of 2 and 3 are 6, 12, 18,
The smallest common multiple of two or more numbers is called the lowest
common multiple (LCM).
In general, there are two methods for finding the lowest common multiple of two or
more numbers:
List the multiple of the largest number and stop when you find a multiple of the
other number. This is the LCM.
Example:
Method II (general):
22
Roco Carmona
1 ESO MATEMTICAS
9 3 12 2
3 3 6 2
1 3 3
1
18=2 32 24=23 3
So, LCM (18,24)= 23 32=72 .
23
Roco Carmona