numbers. multiples and factors UNIT 1 NUMBERS, POSITIONS AND COLUMNS Our number system is called the decimal system.it is based on tens. This is probably because we have ten fingers and thumbs. A digit is a 1,2,3,4,5,6,7,8,9, or 0 used as part of a number. In the decimal system, it is important that the digits in the number are in the correct places. 18 is not the same as 81. 18 has one ten and 8 units. 81 has 8 tens and 1 unit. The place of a digit is often called a column.this is because numbers are written underneath each other when you do sums. This is a table showing the names of the first seven columns. MILLIONS HUNDRED THOUSANDS TEN THOUSANDS THOUSANDS HUNDREDS TENS UNITS 2 3 0 0 0 0 0 For example: The 2 is in the hundred thousands column and the 3 is in the ten thousands column. All the other digits are zeros. So this number is two hundred and thirty thousand. DIGITS These are the correct names of numbers. 1 one 10 ten 100 one hundred 1000 one thousand 10 000 ten thousand 100 000 one hundred thousand 1 000 000 one million 1 000 000 000 one billion ( thousand million ) 1 000 000 000 000 one trillion ( million million ) 2
Digits mean different things depending on where they are in a number. Look at the number 676. The first 6 stands for 600 ( six hundred ) It is in the hundreds column. The second 6 stands for 6 (six) It is in the units column. Zeroes are important too.without them,you couldn t tell the difference between 63,630,603,6300,6030,6003, or even 603 000. R EMEMBER Figures numbers written using 1,2,3,4,5,6,7,8,9,and 0. Digit a 1,2,3,4,5,6,7,8,9,or 0 used as part of a number. Place/column the position of a digit in a number. Value the number that a digit is worth when it is in column. OPERATIONS INVOLVING NATURAL NUMBERS The natural numbers are the ordinary numbers, 1, 2, 3, etc., with which we count. The operations are : ADDITION Addition is finding the total, or sum, by combining two or more numbers. Example: 5 + 11 + 3 = 19 is an addition Subtraction Taking one number away from another. If you have 5 apples and you subtract 2, you will be left with 3. The symbol of subtraction is - Example: 5-2 = 3 Multiplication The basic idea of multiplication is repeated addition. For example: 5 3 = 5 + 5 + 5 = 15 Division Division is splitting into equal parts or groups. It is the result of fair sharing. Example: there are 12 chocolates, and 3 friends want to share them, how do they divide the chocolates? Answer: They should get 4 each. We use the symbol, or sometimes the / symbol to mean divide: 12 / 3 = 4 12 3 = 4 3
MULTIPLES The multiples of a given number are the product of all whole numbers and the given number.for example, the multiples of 6 are 6x1 =6, 6x2=12, 6x3=18..6x100=600 and so on. Multiples come from multiplying. Multiples of a number are the numbers in its multiplication table DIVISOR (FACTORS) Factors of a number are numbers that divide it exactly. The factors of 12 are ( 1,2,3,4,6,12) Factors come in pairs: the factor pairs of 12 are 1and 12, 2 and 6, 3 and 4. The factors of a number is any whole number that divides into it exactly,without leaving a remainder PRIME AND COMPOSITE NUMBERS A prime number can only be divided by 1 or itself. For example 5 only can be divided exactly by 1 and 5. Some prime numbers are 2,3,5,7,11,13,17,19,23.There are 25 prime numbers less than 100. Composite numbers have more than 2 divisor (factors). For example10 can be divided exactly by 1,10,2 and 5. The number 1 is special.it is not a prime number.it is not a composite number either. PRIME NUMBERS LESS THAN 100 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 67 71 73 79 83 89 93 97... 4
LOWEST COMMON MULTIPLE (L C M ) Common multiples of two numbers are those which appear in the list of multiples for both numbers For example : Some multiples of 3 are 3,6,9,12,15,18,21,24,27,30,33,36, Some multiples of 5 are 5, 10, 15, 20, 25, 30, 35, 40, 45 Some common multiples of 3 and 5 are 15, 30, 45 The smallest of the common multiples is called the lowest common multiple(lcm) The LCM of 3 and 5 is 15. HIGHEST COMMON FACTOR The common factors of two numbers appear in the list of factors for both numbers. Factors of 28 = (1, 2, 7, 14, 28) Factors of 42 = (1, 2, 3, 6, 7, 14, 21, 28) So common factors of 28 and 42 are 1, 2,7,14. The highest common factor (H C F) is the largest of these. For example the HCF of 28 and 42 are 14 R EMEMBER Multiples of a number are the numbers in its multiplication table. 1. Multiples of 2 : 2,4,6,8,.units digits is 2,4,6,8,or 0 2. Multiples of 3:3, 6, 9 the sum of the digits must be 3,6, or 9. 3. Multiples of 5: 5, 10, 15.units digit is 5 or 0. 4. Multiples of 9: 9, 18, 27,.the sum of the digits must be 9. 5. Multiples of 10: 10,20, 30,.units digit is 0. Divisor (factors) of a number are numbers that divide exactly. Square numbers are made by multiplying numbers by themselves. Squared (2 ) multiplied by itself Prime numbers have no factors except 1 and themselves. Numbers that are not prime are called composite. The highest common factor (HCF) is the largest of the common factors. The lowest common multiple (LCM) is the smallest of the common multiples. USEFUL WEBSITES http://edhelper.com/math/number_theory90.htm www.aaamath.com/fra.html 5
ACTIVITIES To find the last line of this knock-knock joke you must follow these directions. In each set of factors below there is a missing factor. A letter stands for that factor. When you figure out the missing factor in each set find it in the grid below and then put the letter from that set in the box above that. Keep working until you discover the missing line in the joke. Factors of 20 = ( 1, 5, 4, 10, 20, R ) Factors of 25= (25,1, S ) Factors of 21 = ( 1, 3, 21, T ) Factors of 36=(1,12,2,3,9,6,36,4,A) Factors of 24 = ( 1,2,12, 24, 8, 3, 4, M ) Factors of 40= (1,10,8,2,4,40,5,I ) Factors of 18 = ( 1, 18, 2, 6, 3, H) KNOCK-KNOCK WHO S THERE? YEW!!!... YEW WHO? Factors of 15 =(1,15,5,U) Factors of 32 = (1, 8, 32, 4, 2, O ) Factors of 41= (1,N ) Factors of 30 = (1, 5, 30, 10, 3, 2, 6, P) Factors of 50= ( 1,5,50,2,10, D ) 6
FIND OUT! Why do we need rules? Because they make our life easier; for example, they make ugly divisions easier,or they help us to to know if a number is PRIME or COMPOSITE DID YOU KNOW THAT... Mathematicians LOVE Prime Numbers because just about all of maths is made out of prime numbers. In the same way, Scientists LOVE atoms because just about everything we know is made out of atoms. But, what can we do with them? 1. Smash big sums to bits! 2. Secret codes! 3. A good trick! Here s an interesting piece of reading taken from THE CURIOUS INCIDENT OF THE DOG IN THE NIGHT-TIME Chapters in books are usually given the cardinal numbers 1, 2, 3, 4, 5, 6 and so on. But I have decided to give my chapters prime numbers 2,3, 5, 7, 11, 13 and so on because I like prime numbers. This is how you work out what prime numbers are. First, you write down all the positive whole numbers in the world. Then you take away all the numbers that are multiples of 2. Then you take away all the numbers that are multiples of 3. Then you take away all the numbers that are multiples of 4 and 5 and 6 and 7 and so on. The numbers that are left are the prime numbers. This is called Sieve of Eratosthenes 7