Properties of Whole Numbers

Properties of Whole Numbers A whole number added to 0 remains unchanged.

A whole number added to 0 remains unchanged. Thus, 0 is called the additive identity in whole numbers.  The product of two whole numbers is the same, no matter in which order they are multiplied. This is called the commutative property of multiplication.  A whole number multiplied by 1 remains unchanged. Thus, 1 is called the multiplicative identity in whole numbers.  Whole numbers are closed under addition and multiplication.  Subtraction and division are not commutative in whole numbers. Whole numbers are not closed under subtraction and division.  While adding whole numbers, we can group the numbers in any order. This is called the associative property of addition.  While multiplying whole numbers, we can group them in any order. This is called the associative property of multiplication.  The sum of the products of a whole number with two other whole numbers is equal to the product of the whole number with the sum of the two other whole numbers. This is called the distributive property of multiplication over addition.

Whole Numbers

Whole Numbers The numbers used for counting are called natural numbers.

The numbers used for counting are called natural numbers.  The number that comes immediately before another number in counting is called its predecessor.  The number that comes immediately after another number in counting is called its successor.  To find the successor of any given natural number, just add 1 to the given number.  The value of nothing is represented by the number zero. 

 Eg: 3 – 3 = 0
Natural numbers together with the number zero are called whole numbers.  When comparing two whole numbers, the number that lies to the right on the number line is greater.  When comparing two whole numbers, the smaller number lies to the left on the number line.

Importance of Brackets

Importance of Brackets Brackets help in simplifying an expression that has more than one mathematical operation.

Using brackets: Brackets help in simplifying an expression that has more than one mathematical operation. If an expression that includes brackets is given, then turn everything inside the bracket into a single number, and then carry out the operation that lies outside. Eg: 1. (6 + 8) x 10 = 14 x 10 = 140 2. (8 + 3) (9 - 4) = 11 x 5 = 55 Expanding brackets: The use of brackets allows us to follow a certain procedure to expand the brackets systematically. For example: 8 x 109 = 8 x (100 + 9) = 8 x 100 + 8 x 9 = 800 + 72 = 872 105 x 108 = (100 + 5) x (100 + 8)

= (100+5)x100+(100+5)x8 =100 x 100 + 5 x 100 + 100 x 8 + 5 x 8 =10000 + 500+ 800 + 40 =1134

Roman Numerals

Roman Numerals Many years ago, Hindus and Arabs developed a number system called the Hindu–Arabic number system.

Hindu–Arabic number system: Many years ago, Hindus and Arabs developed a number system called the Hindu–Arabic number system.  It is the name given to the number system that we use today. Roman numerals: It is the numeral system that originated in ancient Rome.  This numeral system is based on certain letters, which are given values and are used as numerals.  The following are the seven number symbols used in the Roman numeral system, and their values:  I V X L C D M 1 5 10 50 100 500 1000 Seven letters of English alphabet, i.e. I, V, X, L, C, D and M, are used to represent Roman numerals.  Roman numerals do not have a symbol for zero.  Roman numerals are read from left to right, and are arranged from the largest to the smallest.  Multiplication, division and other complex operations were difficult to perform on Roman numerals.  So Arabic numerals were used.  The Roman numerals for the numbers 1 - 15 are shown below: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 I II III IV V VI VII VIII IX X XI XII XIII XIV XV We can find these roman numerals in some clocks. Rules for Roman numerals:

In Roman numerals, a symbol is not repeated more than thrice.   If a symbol is repeated, its value is added as many times as it occurs. For example, if the letter I is repeated thrice, then its value is three. The symbols V, L and D are never repeated. If a symbol of smaller value is written to the right of a symbol of greater value, then its value gets added to the symbol of greater value.   For example, in case of VI, I is written to the right of V.  It means that 1 should be added to 5.  Hence, its value is 6.If a symbol of smaller value is written to the left of a symbol of greater value, then its value is subtracted from the symbol of greater value. For example, in case of IV, I is written to the left of V.  It means that 1 should be subtracted from 5.  Hence, its value is 4.  The symbols V, L and D are never written to the left of a symbol of greater value, so V, L and D are never subtracted.  For example, we write 15 as XV and not VX. The symbol I can be subtracted from V and X only. For example, the value of IV is four and the value of VI is six.  The symbol X can be subtracted from L, M and C only.  For example, X is subtracted from L to arrive at 40, which is represented by XL

Estimation of the Numbers

Estimation of the Numbers The estimation of a number is a reasonable guess of the actual value.

The estimation of a number is a reasonable guess of the actual value.  Estimation means approximating a quantity to the accuracy required.  This is done by rounding off the numbers involved and getting a quick, rough answer.  The numbers 1, 2, 3 and 4 are nearer to 0.  So, these numbers are rounded off to the lower ten. The numbers 6, 7, 8 and 9 are nearer to 10.   So, these numbers are rounded off to the higher ten. The number 5 is equidistant from both 0 and 10, so it is rounded off to the higher ten.  Eg: (i)  We round off 31 to the nearest ten as 30 (ii)   We round off 57 to the nearest ten as 60 (iii)  We round off 45 to the nearest ten as 50 The numbers 1 to 49 are closer to 0.   So, these numbers are rounded off to the nearest hundred. The numbers 51 to 99 are closer to the lower hundred. So, these numbers are rounded off to the higher hundred. The number 50 is rounded off to the higher hundred. Eg: (i)  We round off 578 to the nearest 100 as 600. (ii)  We round off 310 to the nearest 100 as 300. Similarly, 1 to 499 are rounded off to the lower thousand, and 501 to 999 to the higher thousand. The number 500 is equidistant from both 0 and 1000, and so it is rounded off to the higher thousand.  Eg: (i)We round off 2574 to the nearest thousand as 3000. (ii)We round off 7105 to the nearest thousand as 7000. Estimation of sum or difference: When we estimate sum or difference, we should have an idea of the place to which the rounding is needed.

Eg: (i)  Estimate 4689 + 19316 We can say that 19316 > 4689 We shall round off the numbers to the nearest thousands.  is rounded off to 19000 4689 is rounded off to 5000 Estimated sum: 19000 + 5000=24000 (ii)  Estimate 1398-526 We shall round off these numbers to the nearest hundreds.  1398  is rounded off to 1400 526 is rounded off to 500 Estimated difference: 1400-500=900 Estimation of the product: To estimate the product, round off each factor to its greatest place, then multiply the rounded off factors.  Eg: Estimate 92 x 578 The first number, 92, can be rounded off to the nearest ten as 90. The second number, 578, can be rounded off to the nearest hundred as 600. Hence, the estimated product =90 x 600 = 54,000 Estimating the outcome of number operations is useful in checking the answer.

Comparing Numbers

Comparing Numbers The arrangement of numbers from the smallest to the greatest is called ascending order.

The arrangement of numbers from the smallest to the greatest is called ascending order.   Ex: 2789, 3560, 4567, 7662, 7665 The arrangement of numbers from the greatest to the smallest is called descending order.  Ex: 7665, 7662, 4567, 3560, 2789 If two numbers have an unequal number of digits, then the number with the greater number of digits is greater. If two numbers have an equal number of digits, then the number with the greater digit is greater. The greatest single-digit number is 9.  When we add 1 to this single-digit number, we get 10, which is the smallest two-digit number.  Therefore, the greatest single-digit number +1=the smallest two-digit number. The greatest two digit-number is 99. When we add 1 to this two-digit number, we get 100, which is the smallest three-digit number.  Therefore, the greatest two-digit number +1=the smallest three-digit number. The greatest three-digit number is 999.  When we add 1 to this three-digit number, we get 1000, which is the smallest four-digit number.  Therefore, the greatest three-digit number +1=the smallest four-digit number. The greatest four-digit number is 9999.  When we add 1 to this four-digit number, we get 10,000, which is the smallest five-digit number.  Therefore, the greatest four-digit number +1=the smallest five-digit number. The greatest five-digit number is 99999.  When we add 1 to this five-digit number, we get 1,00,000, which is the smallest six digit number.  Therefore, the greatest five-digit number +1=the smallest six-digit number. The number, that is, one with five zeroes (100000), is called one lakh.

The greatest six-digit number is 999999.  When we add 1 to this six-digit number, we get 10,00,000, which is the smallest seven-digit number.  Therefore, the greatest six-digit number +1=the smallest seven-digit number. The number, that is, one with six zeroes (1000000), is called ten lakh. The greatest seven-digit number is 9999999.  When we add 1 to this seven-digit number, we get 10000000, which is the smallest eight-digit number.  Therefore, the greatest seven-digit number +1=the smallest eight-digit number. The number, that is, one with seven zeroes (10000000),is called one crore. Commas are placed to the numbers to help us read and write large numbers easily.  Commas in Indian numeration: As per Indian numeration, the first comma is placed after the hundreds place. Commas are then placed after every two digits. Ex: (i)   88,76,547 The number can be read as eighty-eight lakh seventy-six thousand five hundred and forty-seven.  (ii)5 , 67, 89, 056 The number can be read as five crore sixty-seven lakh eighty-nine thousand and fifty-six.  Use the following place value chart to identify the digit in any place in the Indian system.

Crores Lakhs Thousands Ones Tens Ones Tens Ones Tens Ones Hundreds Tens Ones Commas in international system: As per international numeration, the first comma is placed after the hundreds place. Commas are then placed after every three digits. Ex: (i)   8,876,547 The number can be read as eight million eight hundred seventy-six thousand five hundred and forty-seven.  (ii)56,789, 056 The number can be read as fifty-six million seven hundred eighty-nine thousand and fifty-six. Billions Millions Thousands Ones Hundreds Tens Ones Hundreds Tens Ones Hundreds Tens Ones Hundreds Tens Ones Use the following place value chart to identify the digit in any place in the international system.  Comparison of the Indian and the international numeration systems: Indian Numeration  Crore Ten Lakh Lakh Ten Thousand Thousand Hundred Tens Ones Numbers 10000000 1000000 100000 10000 1000 100 10 0 International Numeration Ten Million Million Hundred Thousand Ten Thousand Thousand Hundred Tens Ones Units of measurement: 1 metre=100  centimetres 1 kilogram = 1,000 grams 1 kilometre = 1000 metres 1 litre=1,000 millilitres

Recycling Paper and Plastics

Recycling Paper and Plastics It is important to reuse things than discarding them as waste.

It is important to reuse things than discarding them as waste. Many nice articles can be made out of waste. Industries use recycled or waste paper to regenerate paper. Paper that is suitable for recycling is called "scrap paper". You can recycle old newspapers, magazines, notebooks and used envelopes, but not waxed paper, oil-soaked paper, paper contaminated with food, carbon paper, thermal fax paper, plastic laminated paper, stickers, and sanitary products or tissues. Steps involved in recycling paper Tear paper into small pieces. Soak these pieces in water for a day. Make a thick paste and spread it on a net or sieve. Let water drain off completely. Use an old cloth or newspaper to remove the extra water from the paste and dry it. Use this paste to get beautiful patterns. Disadvantages of using of polythene bags We use many plastic items such as tooth brush, combs, containers, bottles, shoes, toys, wires, frames and bags every day. Certain parts of vehicles like cars and buses, and electronic goods like radios, televisions and refrigerators, are all made of plastic. All these are useful to us in many ways, but using plastic is very harmful in terms of health and as well as the environment. Plastics are not suitable for storing cooked food because they emit harmful chemicals when they are exposed to high temperatures. Using plastics causes health problems such as heart disease, diabetes and reproductive dysfunction. Harmful gases are emitted from burning plastics, which cause cancer and they kill living beings.

That is why plastics should be disposed in the right way. Plastics that do not contain the chemical, BPA, are usually licensed for storing food items. Plastics thrown casually get into drains and sewages, often blocking the way and causing water-logging. A major cause of the floods in Mumbai, India, in August 2005 was the choking of the drainage system by plastic waste. So polythene bags should not be used for garbage disposal. Measures taken to prevent from the dangerous effects of plastics: Adopt healthy practices such as: Reduce, reuse and recycle plastics. Carry  jute or cloth bags for shopping Do not store food items in plastic bags. Do not burn plastic items. Recycle plastics so that new plastic items can be made. Waste that does not decompose should be put in blue dustbins, while waste that decomposes easily should be put in green dustbins. Vermicomposting Conversion of the waste generated in the kitchen into organic matter with the combined action of earthworms and micro-organisms is called vermicomposting. The organic matter so formed is used as manure for plants.