Colegio Herma. Maths. Bilingual Departament 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 3 rd 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. 1 st = first 2 nd = second 3 rd = third 4 th = fourth 5 th fifth 6 th = 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: ___________ Now, do the exercises 1 to 4 from page 8

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 = = 16 CLV = = 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, Numbers less than 4000 a) 551 Separate the numbers in their addends 511 = Express each addends in roman numbers D + L + I = DLI b) = XL + IX = XLIX c) = DCCC + XX + VII = DCCCXXVII

Numbers greater than 4000 d) Separate Units, Tens and Hundreds from the rest of digits Express these amounts in roman numbers applying the rule of multiplying LXV + CCCVI = LCV CCCVI Now do the exercises 5 to 8 from page 9, and exercise 5 from page 21.

OPERATIONS WITH NATURAL NUMBERS MULTIPLICATION Multiplying is doing an addition of equal addends = 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 = 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 x 3 5 x (2 - 6) = 5 x x 6 Now do the exercises 9 to 12 from page 10

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 (:) 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 Now do the exercises 13 to 15 from page 11

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 8 2 the "2" says to use 8 twice in a multiplication, so 8 2 = 8 × 8 = 64 Exponent is also called power or index. In words: 8 2 could be called "8 to the power 2” "8 to the second power" "8 squared“

Example: 5 3 = 5 × 5 × 5 = 125 In words: 5 3 could be called "5 to the power 3“ "5 to the third power“ "5 cubed“ Example: 2 6 = 2 × 2 × 2 × 2 x 2 x 2 = 64 In words: 2 6 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 = 10 x 10 x 10 x 10 x 10 x 10 = = 10 x 10 x 10 x 10 x 10 = = 10 x 10 x 10 x 10 = = 10 x 10 x 10 = = 10 x 10 = = 10 = = 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: 7 5 x 7 3 = (7x7x7x7x7)x(7x7x7) = 7 8 So 7 5 x 7 3 = = 7 8 Now do the exercises 20 to 22 from page 13

Division: When powers with the same base are divided, the base remains unchanged and the exponents are subtracted. Example: 6 7 : 6 4 = 6 3 So 6 7 : 6 4 = 6 3 Now do the exercises 24 to 27 from page 14

Power with exponent 1 or 0 A power with exponent 1 is equal to the base a 1 = a A power with exponent 0 is equal to 1 a 0 = 1 Now do the exercises 24 to 27 from page 14

Power of a power: The exponents or indices must be multiplied (a m ) n = a m x n Example: (2 3 ) 5 = (2 3 ) x (2 3 ) x (2 3 ) x (2 3 ) x (2 3 ) = = 2 3x5 = 2 15 So (2 3 ) 5 = 2 15

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. 2 5 x 7 5 = (2 x 7) 5 = 14 5 Division: When powers with the same exponent are divided, bases are divided and the exponent remains unchanged. 8 3 : 2 3 = (8 : 2) 3 = 4 3 or Now do the exercises 28 to 31 from page 14

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 3 2 =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 (6 × )... 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 × 2 2 =5 × 4=20 OK 5 × 2 2 =10 2 =100 WRONG

3. Multiply or Divide before you Add or Subtract. Example: × 3=2 + 15=17 OK × 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 Now, do the exercise number 41 from page 18 and exercise 106 from page 25

Rememeber Order of Operations BEDMAS: 1) BRACKETS BRACKETS 2) EXPONENTS EXPONENTS 3) DIVISION DIVISION 4) MULTIPLICATION MULTIPLICATION 5) ADITION ADITION 6) SUBTRACTION SUBTRACTION DO FROM LEFT TO THE RIGHGT

BEDMAS SONG