colegio herma. maths. bilingual department by isabel ... · there are four terms in a division:...
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Colegio Herma. Maths. Bilingual Department by Isabel Martos Martínez. 2013
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NATURAL NUMBERS Natural numbers are the
numbers used for counting things.
Natural numbers are POSITIVE (numbers that are more than 0). They are 1, 2, 3, 4, 5…and so on until infinity. The natural numbers set is an unlimited set.
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They are the first numbers that a child uses naturally for the first time, for this reason are named NATURAL NUMBERS.
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Natural numbers have two main purposes: You can use them for counting:
“there are three apples on the table”
You can use them for ordering: “this is the 3rd largest city in the country”
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Mathematicians use N to refer to the set of all natural numbers
Natural numbers are also called Counting Numbers.
In Spanish they are called Números Cardinales if they are used for counting
1 = ONE 2= TWO 3= THREE
or Números Ordinales if they are used for ordering.
1st = first 2nd = second 3rd = third
4th = fourth 5th fifth 6th = sixth
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THE DECIMAL NUMERAL SYSTEM A Numeral System is a set of rules and symbols that are
used to represent the numbers.
The Decimal Numeral System is a DECIMAL and POSITIONAL system.
It is DECIMAL because it is made up of 10 symbols: 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9.
It is POSITIONAL because the value of each digit depends on the position in the number.
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Exercise: Write a number with 8 digits over the lines and translate the following words:
___ ___ ___ ___ ___ ___ ___ ___
Units or Ones:__________
Tens : __________
Hundreds: ___________
Thousands:_____________
Ten thousands: __________
Hundred thousands: ______
Millions:_______________
Ten millions: ___________
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THE ROMAN NUMERAL SYSTEM This is another numeral
system that was used by Roman Civilization.
Nowadays this system is also frequently used.
To express different amounts by using the Roman numeral system seven different letters are used with different values each.
Each letter always has the same value.
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Basic Rules to write numbers by using the roman numeral system
Addition: A letter on the right of another letter adds its value to it.
XVI = 10 + 5 + 1 = 16
CLV = 100 + 50 + 5 = 155
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Repetition: The letters I, X, C and M can be repeated no more than three times. The rest of the letters cannot be repeated.
III = 3
XXX = 30
CCC = 300
MMM= 3000
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Subtraction:
The letter I on the left of V or X subtracts its value to them IV = 4
The letter X on the left of L or C subtracts its value to them XC =90
The letter C on the left of D or M subtracts its value to them CM =900
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Multiplication: A bar on the top of the letter or a group of letters multiply their value for 1000
VI = 6000
V I = 5001
XL = 40000
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Express as roman numbers: 551, 49, 827, 63306
Numbers less than 4000 a) 551
Separate the numbers in their addends 511 = 500 + 50 +1
Express each addends in roman numbers D + L + I = DLI
b) 49 49 = 40 + 9
XL + IX = XLIX
c) 827 827 = 800 + 20 + 7
DCCC + XX + VII = DCCCXXVII
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Numbers greater than 4000
d) 65306
Separate units, tens and hundreds from the rest of digits
65 306
Express these amounts in roman numbers applying the rule of multiplying
LXV + CCCVI = LCV CCCVI
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OPERATIONS WITH
NATURAL NUMBERS
MULTIPLICATION
Multiplying is doing an addition of equal addends
3 + 3 + 3 + 3 + 3 + 3 + 3 = 3 x 7 = 21
We read: seven times three is twenty one
The FACTORS are the numbers that are multiplied together. The PRODUCT is the result of multiplying.
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The properties of multiplication
Commutative property: The order of the factors doesn´t change the result.
5 x 7 = 7 x 5
35 = 35
We read:
• Five sevens are equal to seven fives.
• Five times seven is the same that seven times five.
• 5 multiplied by 7 is the same that 7 multiplied by 5
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Associative property: When three or more numbers are multiplied, the product is the same regardless of the grouping of the factors.
(4 x 7) x 5 = 4 x (7 x 5)
28 x 5 = 4 x 35
140 = 140
Multiplicative Identity Property: The product of any number and one is that number.
5 x 1 = 5
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Distributive property: The sum or subtraction of two numbers times a third number is equal to the sum or subtraction of each addend times the third number.
4 x (6 + 3) = 4 x 6 + 4 x 3
5 x (2 - 6) = 5 x 2 - 5 x 6
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DIVISION
Dividing is to share a quantity into equal groups. It is the inverse of multiplication. In Spanish we write 6 : 2 but in English it is always 6 ÷ 2 but never with
the colon (:) 6 2 0 3 colon: dos puntos
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We say:
270: 3 = 90
Two hundred and seventy divided by three is equal to ninety
8:2 = 4
In smallers calculations
Two into eight goes four
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There are four terms in a division: dividend, divisor, quotient and remainder.
The dividend is the number that is divided. In Spanish is dividendo (D)
The divisor is the number that divides the dividend. In Spanish is divisor (d)
The quotient is the number of times the divisor goes into the dividend. In Spanish is cociente (c)
The remainder is a number that is too small to be divided by the divisor and in Spanish is called resto (r)
D d always r < d
r c
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The division can be: a) Exact division: the reminder is zero
b) Inexact division: the reminder is different from zero
To check if the division is correct we do the division algorithm (prueba de la división):
Division Algorithm:
Dividend = Divisor x Quotient + Remainder
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POWERS OF NATURAL NUMBERS
It’s a number obtained by multiplying a number by itself a certain number of times.
8² = 8 x 8
5³ = 5 x 5 x 5
8 and 5 are the BASE
2 and 3 are the EXPONENT
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The exponent of a number says how many times to use the number in a multiplication.
In 82 the "2" says to use 8 twice in a multiplication, so
82 = 8 × 8 = 64
Exponent is also called power or index.
In words:
82 could be called
"8 to the power 2”
"8 to the second power"
"8 squared“
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Example: 53 = 5 × 5 × 5 = 125
In words: 53 could be called
"5 to the power 3“
"5 to the third power“
"5 cubed“
Example: 26 = 2 × 2 × 2 × 2 x 2 x 2 = 64
In words: 26 could be called
"2 to the power 6"
"2 to the sixth power"
"2 to the 6th"
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Power of base 10 You only need to put the number 1 followed by the number of
zeros which is indicated in the exponent.
106= 10 x 10 x 10 x 10 x 10 x 10 = 1.000.000 105= 10 x 10 x 10 x 10 x 10 = 100.000
104= 10 x 10 x 10 x 10 = 10.000
103= 10 x 10 x 10 = 1.000 102 = 10 x 10 = 100
101 = 10 = 10
100 = 1
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Operations with powers To manipulate expressions with powers we use some
rules that are called laws of powers.
Multiplication:
When powers with the same base are multiplied, the base remains unchanged and the exponents are added
Example:
75 x 73 = (7x7x7x7x7)x(7x7x7) = 78
So
75 x 73 = 75+3 = 78
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Division:
When powers with the same base are divided, the base remains unchanged and the exponents are subtracted.
Example:
67 : 64 = 67-4 = 63
So 67 : 64 = 63
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Power with exponent 1 or 0
A power with exponent 1 is equal to the base
a1 = a
A power with exponent 0 is equal to 1
a0 = 1
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Power of a power:
The exponents or indices must be multiplied
(am)n = amxn
Example:
(23)5 = (23) x (23) x (23) x (23) x (23) = 23+3+3+3+3 = 23x5 = 215
So
(23)5 = 215
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Powers with different base but the same exponent
Multiplication: When powers with the same exponent are multiplied, multiply the bases and keep the same exponent.
25 x 75 = (2 x 7)5 = 145
Division: When powers with the same exponent are divided, bases are divided and the exponent remains unchanged.
83 : 23 = (8 : 2)3 = 43 or
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SQUARE ROOTS
The inverse operation of power is root.
The inverse of a square is a square root, that is:
If we say that
that means that 32 =9
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Exact Square roots and Inexact Square roots
Exact Square Numbers with an exact
square root are called “perfect squares“, in Spanish “cuadrados perfectos”
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Inexact square Numbers without an
exact square root.
The radicand is not a perfect square.
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ORDER OF OPERATIONS When you see something like...
7 + (6 × 52 + 3)
... what part should you calculate first? Start at the left and go to the right? Or go from right to left?
Calculate them in the wrong order, and you will get a wrong answer!
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So, long ago people agreed to follow rules when doing calculations, and they are ORDER OF OPERATIONS
1. Do things in Brackets First. Example:
6 × (5 + 3)=6 × 8=48 OK
6 × (5 + 3)=30 + 3=33 WRONG
2. Exponents (Powers, Roots) before Multiply, Divide, Add or Subtract. Example:
5 × 22=5 × 4=20 OK
5 × 22=102=100 WRONG
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3. Multiply or Divide before you Add or Subtract. Example:
2 + 5 × 3=2 + 15=17 OK
2 + 5 × 3=7 × 3=21 WRONG
4. Otherwise just go left to right. Example:
30 ÷ 5 × 3=6 × 3=18 OK
30 ÷ 5 × 3=30 ÷ 15=2 WRONG
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Rememeber Order of Operations
BEDMAS: 1)BRACKETS
2)EXPONENTS
3)DIVISION
4)MULTIPLICATION
5)ADITION
6)SUBTRACTION
DO FROM LEFT TO THE RIGHGT