UNIT 1: NATURAL NUMBERS.

The set of Natural Numbers: UNIT 1: NATURAL NUMBERS. The set of Natural Numbers ( they are also called whole numbers) is N={0,1,2,3,4,5...}. Natural have two purposes: Counting: There are three apples on the table CARDINAL NUMBERS Ordering: Barcelone is the second largest city in our country ORDINAL NUMBERS Cardinal Numbers 1. one 2. two 3. three 4. four 5. five 6. six 7. seven 8. eight 9. nine 10. ten 11. eleven 12. twelve 13. thirteen 14. fourteen 15. fifteen 16. sixteen 17. seventeen 18. eighteen 19. nineteen 20. twenty 21. twenty-one 22. twenty-two 23. twenty-three 24. twenty-four 30. thirty 40. forty 50. fifty 60. sixty 70. seventy 80. eighty 90. ninety 100. a/one hundred 101. a/one hundred 200. two hundred 1 000. a/one thousand 10 000. ten thousand 100 000. a/one hundred thousand 1 000 000. a/one million Ordinal Numbers 1 st. First 2 nd. Second 3 rd. Third 4 th. Fourth 5 th. Fifth 6 th. Sixth 7 th. Seventh 8 th. Eighth 9 th. Ninth 10 th. Tenth 11 th. Eleventh 12 th. Twelfth 13 th. Thirteenth 14 th. Fourteenth 15 th. Fifteenth 16 th. Sixteenth 17 th. Seventeenth 18 th. Eighteenth 19 th. Nineteenth 20 th. Twentieth 21 st. Twenty-first 22 nd. Twenty-second 23 rd. Twenty-third 24 th. Twenty-fourth 30 th. Thirtieth 40 th. Fortieth 50 th. Fiftieth 60 th. Sixtieth 70 th. Seventieth 80 th. Eigthtieth 90 th. Ninetieth 100 th. Hundreth 101 st. Hundred and first 200 th. Two hundreth 1 000 th. One thousandth 10 000 th. Ten thousandth 100 000. One hundred thousandth 1 000 000 One millionth 1

Ordering Natural Numbers: We are going to learnt symbols are used to show the size of one number compares to another. These symbols are: a b a is less than b a b a is less or equal than b a b a is greater than b a b a is greater or equal than b Place the following numbers in order: 14, 21, 9 : < < ( in ascending order) 52, 40, 32: > > ( in descending order) A number line can be used to represented the set of the Natural Numbers in order: 2

Activities. 1. Draw the following numbers on the number line: 3 6 0 2 8 7 2. Complete, using <, > or = : a) 2 3 c) 3 3 b) 4 7 d) 6 1 e) 8 2 f) 4 4 3. Write this numbers in ascending order: 1 5 7 2 3 4 4. Find the missing number: a) 4< <6 b) 8> >6 c) 2< <4 d) 7> >5 5. Write three more numbers that follow the same patterns. a) 3, 6, 9, 12, 15, b) 4, 7, 10, 13, 16, c) 15, 13, 11, 6. Write the odd numbers greater than 10 and less than 24. 7. Write the even numbers greater than 31 and less or equal than 48. 3

Addition of Natural Numbers: Addition is the process of finding the total of two o more numbers. We first learnt addition through counting: But it is not necessary to draw the picture. You can say: 2+1=3. 5+7=12 132+335=467 12+15+8=35 142+265+146=553 Addition properties: If a, b, and c are natural numbers: 1. Associative property: a b c=a b c 2. Commutative property: a b=b a 3. Existence of an identity element (zero): a 0=0 a=a Subtraction of Natural Numbers: If you have a set of numbers and extract some of them, how many of them are left? Let's see the picture: But is not necessary to draw the picture. You can say: 3-1=2. 245-129=116 3456-2391=1056 4

Activities. 1. Write out all the numbers pairs that add up to 12 using these cards. You will end five pairs of numbers. 6 7 10 4 8 5 3 6 9 2 2. Work out these addition problems in your head: 75p 90p 2 95p 5

3. Add the score from the three dice. Substract the total from 20 in your head: 4. Substract 20 from each number below. Do the calculation in your head. a) 33 b) 23 c) 45 d) 22 e) 29 f) 28 g) 36 h) 67 5. Calculate: a) 12+4+43 b) 32+11+10 c) 21-12-1 d) 23-12+10 6. Calculate: a) 24-(11-5) b) 21+(10-8) c) 35-(13-7) e) 23+11+5 f) 33-1-30 g) 9+13+21 h) 31+21-30 d) 21+(10-8) e) 14-(13-1) f) 25+(11-7) 7. A box full of cardboard weighs 11 kg. It is taken to the recycling centre. The cardboard is thrown into the cardboard bank. The box now weighs 2 kg. Hom much did the cardboard weigh? 8. The life expentacy of a woman is 83 years. The life expentacy of a man is 77 years. How much longer is the life expentacy of a woman? 6

9. Julian works for a parcel delivery company. The picture shows the distances between the tows in his area. One day he has deliver some parcels from town A to town B. What is the shortest route? 15 km B D 9 km 20 km 12 km C 13 km A E 10. Robert's family are going to visit Seville this weekend. They expect to expend 100 on the hotel, 50 on a meal and 45 on petrol. At the moment they have 200. Do they have enough money for their trip? What is the difference between the money they have and the money they need? 7

Multiplication of Natural Numbers: The product of two Natural Numbers, m and n is: m n=m m... m (n times) This is just a shorthand for saying: Add m to itself n times Example: 7 5=7 7 7 7 7=35 but you only write 7 5=35 Properties of multiplicacion 1. Associative property: For any three natural numbers a, b and c: a b c = a b c. Example: 5 7 2=35 2=70 5 7 2 =5 14=70 2. Conmutative property: For any two natural numbers a and b: a b=b a. Example: 12 7=7 12=84 3. Existence of an identity element (one): For any natural numbers a: a 1=1 a=a 4. Distributive property: For any three natural numbers a, b and c: a b c =a b a c a b c =a b a c 5 7 3 =5 10=50 5 7 5 3=35 15=50 5 7 3 =5 4=20 5 7 5 3=35 15=20 Activities. 1. Write the following multiplications as repeat additions as in the example: 3 5=3 3 3 3 3=15 a) 2 4 b) 3 6 c) 4 5 d) 5 7 2. Calculate: a) 21 2 b) 9 8 c) 11 11 d) 123 2 e) 7 13 f) 13 13 8

3. Calculate: a) 14 10 b) 14 100 c) 14 1000 d) 333 10 e) 333 100 f) 333 1000 g) 5 10 h) 5 100 i) 5 1000 4. Calculate and compare the results: a) 4 3 b) 6 5 a') 3 4 b') 5 6 c) 12 9 c') 9 12 Name the property that explains these results. 5. Calculate and compare the results: a) 5 3 2 b) 4 10 9 c) 10 11 1 d) 21 4 2 a') 5 3 5 2 b') 4 10 4 9 c') 10 11 10 1 d') 21 4 21 2 Name the property that explains these results. 6. Calculate using the distributive property and doing the calculations in parenthesis first. Do you get the same results? a) 5 2 6 b) 10 12 2 c) 11 11 1 d) 26 9 1 7. Calculate: a) The number of seconds in one hour. b) The number of seconds in a day. c) The numbers of seconds in the month of October. 8. Paul has 16 heartbeats for 15 seconds. Calculate his number of hearthbeats for one minute. Determine the number of heartbeats for one hour, a day and a year. 9

9. There are 5 280 feet in a mile. How many feet are there in ten miles? And in one hundred miles? 10. Find the area of the following shape: 8 cm 8 cm 18 cm 11. Cordoba's Mosque is 174 m long and 128 m wide. Calculate its total area in metres squared. 10

Division of Natural Numbers: In a division of natural numbers, we find four elements: D=dividend, d=divisor, q=quotient and r=remainder. D d r q 15 2 1 7 The dividend is 15, the divisor is 2, the quotient is 7 and the remainder is 1. This division isn't exact ( its remainder isn't 0). But in this division: 15 3 0 5 the remainder is zero, so we say this division is exact. If we want to know the division is correct, we can use the formula: In the examples: 15=2 7 1 and 15=3 5 0. D=d q r Activities. 1. Work out these calculations: a) 45:5 b) 235:5 c) 1476:12 2. Complete these division problems: a) 20:10 b) 30:10 3. Give a remainder in these division problems: a) 20:3 b) 30:7 d) 5175:15 e) 9913:23 f) 7062:22 c) 150:10 d) 1000:10 c) 150:11 d) 1000:12 11

4. Work out these calculations and check the results: a) (200:4):2 a') 200:(4:2) b) (150:30):5 b') 150:(30:5) c) (24:6):2 c') 24:(6:2) Can you explain these results? 5. A carpenters needs to cut a rod wich is 39 cm long into three equals lengths. How long will the new pieces be? 6. Eggs are packed in boxes of twelve. How many boxes do you need to pack 1 476 eggs? 7. Lucas and his two friends earn 65 working as waiters. Can the money be shared out exactly between the three of them? 8. Convert these mesaurements of times to the units indicates in parenthesis: a) 200 minutes (into hours and minutes) b) 200 hours (into days and hours) c) 300 seconds (into minutes and seconds) 9. DVD's are packed in bundles of 25. How many bundles are needed to pack 1 028 DVD's? How many DVD's will be left over? 10. Susan drinks 21 litres of water every week. How much does she drink each day? 11. Richard buys 150 text messages for 5. What is the price of one text messages? 12

Powers of Natural Numbers: Maybe, you will already be familiar with the notation for squares and cubes: a 2 =a a ( we read a squared ) and a 3 =a a a (we read a cube ) and this is generalised by defining: a n =a a... a (n times a) and we read a raised to the power of n-th, or easier a to the n-th. Four squared: 4 2 =4 4=16 Five cube: 5 3 =5 5 5=125 Two to the fifth: 2 5 =2 2 2 2 2=32 Three to the sixth: 3 6 =3 3 3 3 3 3=729 That is, powers are used to describe the result of repeatedtly multipling a number by itself. The number that is successively multiplied by itself is called the based. A small raised number called an exponent follows the base and indicates the number of times the base is to be multiplied. And, remember: a 1 =a and a 0 =1, for any natural number a. 2 2 2 2=2 4 3 3 3 3 3 3 3=3 7 5 5 5 5 5 5=5 6 Write the followings numbers in exponencial form. How do they read?: 7 7 7 7 7 10 10 10 20 20 11 11 11 11 11 11 11 Multiplying powers: You can multiply powers with the same base by adding the exponents. a m a n =a m n 3 4 3 7 =3 4 7 =3 11 2 5 2 4 2=2 5 4 1 =2 10 13

Dividing powers: You can divide powers with the same base by subtracting the exponents. a m :a n =a m n 3 12 :3 7 =3 12 7 =3 5 2 5 :2=2 5 1 =2 4 Power of a power: You can simplify the power of a power by multipling the exponents. a m n =a m n 4 2 5 =4 2 5 =4 10 3 4 2 =3 4 2 =3 8 Multiplying powers with the same exponent: You can multiply powers with the same exponent by multipling the bases. a n b n = a b n 3 4 5 4 = 3 5 4 =15 4 2 3 5 =2 5 3 5 Dividing powers with the same exponent: You can divide powers with the same exponent by dividing the bases. a n :b n = a:b n 15 4 :3 4 = 15:3 4 =5 4 6:3 5 =6 5 :3 5 14

Activities. 1. Write the result in exponential form: a) 2 5 2 6 2 2 b) 5 3 5 5 4 c) 7 6 :7 2 d) 3 5 3 3 3 e) 6 10 : 6 4 f) 2 8 3 g) 4 7 : 4 h) 5 2 10 i) 10 3 10 2 10 2 j) 7 3 3 2. Calculate. Remember you have to do first, the operations inside the parenthesis: a) 3 4 3 5 :3 6 b) 7 8 : 7 2 7 4 c) 5 2 4 5 5 3 d) 2 3 4 2 7 : 2 5 3. Fill in the missing exponent: a) 2 2 5 =2 7 b) 3 5 3 4 =3 c) 6 :6 2 =6 5 d) 2 6 :2 =2 4 4. If in 10 towns of a region there are 10 football teams in each town with 10 players in every team, how many players are there in this region? 15

Square root: The square root of a number a is another number b such that b squared is a, a number b whose squared is a. a=b when b 2 =a The number a is called radicand, the symbol is called radical and b is called the squared root of a. The numbers with an exact square root are called perfect squares. 1=1 because 1 2 =1 4=2 because 2 2 =4 9=3 because 3 2 =9 16=4 because 4 2 =16 25=5 because 5 2 =25 36=6 because 6 2 =36 49=7 because 7 2 =49 64=8 because 8 2 =64 81=9 because 9 2 =81 100=10 because 10 2 =100 Example: The area of a square is 100 cm 2 121=11 because 11 2 =121 144=12 because 12 2 =144 169=13 because 13 2 =169 196=14 because 14 2 =196 225=15 because 15 2 =196 256=16 because 16 2 =256 289=17 because 17 2 =289 324=18 because 18 2 =324 361=19 because 19 2 =361 400=20 because 20 2 =400. What is the length of his side? A= 100 l l A= 100 cm 2 l 2 =100 l= 100=10cm Integer square root: If a radicand isn't a perfect square, the square root isn't 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. 11 3 Remainder= 11 3 2 =11 9=2 29 5 Remainder= 29 5 2 =29 25=4 37 6 Remainder= 37 6 2 =37 36=1 16

Activities. 1. Verify if the following square roots are true or false: a) 196=14 c) 10 000=100 b) 300=30 d) 100 000=1000 2. Find with your calculator: a) 900 b) 1600 c) 2116 d) 9801 3. Calculate the length of the side of a square whose area is 81 m 2. A=81 4. Find the integer square root and the remainder of the numbers: a) 15 b) 39 c) 57 d) 95 5. Complete: a) 50 7 Remainder= b) 5 Remainder=7 c) 8 2 Remainder= d) 9 Remainder=2 17

Order of Operations: Remember the rules for order of operations: 1. First perform any calculation inside square brackets. 2. Next perforn any operations inside parenthesis. 3. Next perfom all exponents and roots. 4. Next perform all multiplicacions and divisions, working from the left to right. 5. Next perform all additions and subtractions, working from the left to right. a) 6 2 3 2 4 =6 2 6 4 =6 2 2=36 2=34 b) [15 9 2 ] 6 2= 15 11 6 2=4 6 2=4 12=16 c) 4 8 5 2 4 =4 8 25 4 =4 8 21=4 168=172 d) 100 : 3 2 2 2 =10: 9 4 =10:5=2 e) 25 1 : 9 1 = 5 1 : 3 1 =6: 2=3 Before we begin our journey to completing an order of operation: Here s a sentence you don t want to leave behind. Please Excuse My Dear Aunt Sally. This sentence will be your guide: 1. P- Parenthesis- we must do all operations inside first. 2. E- Exponents- Simplify all exponents and find any square roots. 3. M-D- Multiply or Divide- Procceding from left to right. 4. A-S- Add and Subtract, from left to right. Here is one problem more: 8 9 : 36 4 :2 14 8 PLEASE Parentheses first 14-8=6 (operation in) 8 9: 36 4 :2 6 EXCUSE Our operation don't have an exponent, so let's do our square root. The square root of 36=6. 8 9:6 4 : 2 6 MY Multiply from left to right, 8 9=72 72 :6 4 :2 6 DEAR Divide from left to right, 72:6=12 There is another division operation 4:2=2 12-2+6 AUNT SALLY Add or subtract form left to right. Because subtration is our first order of operation, let's start there. 12-2=10 Finally let's add 10+6=16 16 That's our final answer! 18

Calculate: a) 7 3 6 4 3 3 b) 2 3 1 3 1 4 c) 3 3 36 4 d) 64 2 :2 e) 4 2 10 2 :5 2 f) 11 16 : 9 g) 16 4 16 4 Rouding Off Method: When we talk about quantities, sometimes we use approximations. For example, if the price of a car is 18 043 exactly, you could say that it costs 18 000 approximatly; or if the price of a subway's ticket is 0.95, you could say that it costs 1 approximatly. The method more usual in Mathematics to approximate a quantity is the roundig off method. It consists: We round off upward when the digit of the right is 5 or greater and we round off down when it is less than 5. Round off to the nearest hundreds the following numbers: 12 345 ROUNDING: 12 300 45 678 ROUNDING: 45 700 11 357 ROUNDING: 11 400 Round off to the nearest tens the following numbers: 1 267 ROUNDING: 1 270 13 473 ROUNDING: 13 470 12 945 ROUNDING: 12 950 19

Activities. 1. Round off to the nearest thousands the following numbers: a) 13 456 c) 43 987 b) 6 569 d) 11 287 2. Round off to the nearest ten thousands the following numbers: a) 10 938 c) 987 654 b) 1 234 567 d) 648 903 e) 127 654 f) 2 507 e) 43 567 f) 575 673 3. Calculate: a) 6 2 1 : 49 b) 144 2 4 4 c) 9 2 64: 4 d) 4 2 3 3 2 :6 2 Mathematics Glossary Write here new words you had learnt related with mathematics: 20