Colegio Herma. Maths. Bilingual Department by Isabel Martos Martínez. 2013

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.

They are the first numbers that a child uses naturally for the first time, for this reason are named NATURAL NUMBERS.

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”

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

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.

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: ___________

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.

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

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

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

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

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

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

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.

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

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

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

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

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

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

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

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

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“

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"

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

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

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

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

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

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

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

Exact Square roots and Inexact Square roots

Exact Square Numbers with an exact

square root are called “perfect squares“, in Spanish “cuadrados perfectos”

Inexact square Numbers without an

exact square root.

The radicand is not a perfect square.

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!

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

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

Rememeber Order of Operations

BEDMAS: 1)BRACKETS

2)EXPONENTS

3)DIVISION

4)MULTIPLICATION

5)ADITION

6)SUBTRACTION

DO FROM LEFT TO THE RIGHGT

BEDMAS SONG