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Class 10 Maths Revision Notes for Some Applications of Trigonometry of Chapter 9

It was exclaimed by L. Kroneckerthe reputed German Mathematician. This statement reveals in a nut shell the significant role of the universe of numbers ch 9 of maths class 10 notes in the evolution of human thought. N : The set of natural numbers, W : Ch 9 of maths class 10 notes set of whole numbers.

In our day to life we deal with different types of numbers which can be broady classified as follows. Natural Numbers N : The counting numbers cobia boats models design, 2, 3, Whole numbers W : The number '0' together with the natural numbers 1, 2, 3, Integers I or Z : All natural numbers, 0 and negative of natural numbers are called integers.

The collection of integers is denoted by Z or I. Real numbers :� The totality of rational numbers and irrational numbers is called the set of real number i.

Every real number is either a rational number or an irrational number. A 1st method : Find a rational number between x and y then, is a rational number lying between x and y. B 2nd method : Find n rational number between x and y when x and y is non fraction number then we use formula. C 3rd method : Find n rational number between x and y when x and y is fraction Number then we use formula. Terminating Decimal : The word "terminate" means "end". A mtahs that ends is a terminating decimal.

A terminating decial doesn't keep going. Ch 9 of maths class 10 notes terminating decimal will have a finite number of digits after the decimal point. Theorem-1 : Let x be a rational number whose decimal expansion terminates. Then, x has a decimal expansion motes terminates.

Then, x has a decimal expansion notee is nonterminating repeating. We have learnt how to represent integers on the number line. Draw any line. Take a point O on it. Call it 0 zero. Set of equal distances on the right as well as on the left of O. Such a distance is known claas a unit length.

Clearly, the points A, B, C, D represent the integers 1, 2, 3, 4 respectively and the point A', B', Matjs D' represent the integers �1, �2, �3, �4 respectively.

Thus, we may represent any integer ch 9 of maths class 10 notes a point on the number line. Clearly, every positive integer lies to the right of O and every negative integer lies to the left of O.

Similarly we can represent rational numbers. Draw a line. Let mqths represent 0. The, A represents the integer 1 and A' represents the integer �1. Now, divide OA into two equal parts.

Let OP be the first part out of these two parts. Again, divide OA' into two equal parts. Let OP' be the first part out of these 2 parts. Termentaing Decimal :- Suppose we want to represent 2. We know that it lies between 2 and 3. We therefore, look at the portion of the number line between 2 and 3. We divide this portion into 10 equal parts. This first mark will represent 2.

The process of visualization of numbers on the number ch 9 of maths class 10 notes through a magnifying glass is known as successive magnification. Sometimes, we are noes to check the numbers like 3. Represent 3. This number lies between 3 and 4. The distance 3 and 4 is mathz into 10 equal parts. Then the first mark to the right of 3 will represent 3. Now, 3. We divide the distance between 3.

Suppose we want to represent on the number line, up to 4 decimals places by magnifying glass. This number lies between 4 and 5. The distance between 4 and 5 is divided into 10 equal parts. Then the ch 9 of maths class 10 notes mark to the right of 4 will represent 4. Now 4. We divide the distance between 4.

Now, 4. Again we divide the distance between 4. The number 4. The distance between Ch 10 Maths Class 10 Notes Vol 4. Sixth mark from right to the 4. Terminating decimal numbers :- When the decimal number is of terminating nature. Algorithm : Setp-1 : Obtain the rational number. Setp-2 : Determine the number of digits in its decimal.

Setp-3 : Remove decimal point from the numerator. Write 1 in the denominator and put as many zeros on the right side of 1 as the number of digits in the decimal part ofthe given rational number. Setp-4 : Find a common divisor of the numerator and denominator and express the rational numbe to lowest terms by dividing its numerator and denominator by the common divisor.

Non-terminating repeating decimal numbers :- In a non terminating repeating decimal, cobia boats models design are Notes Of Ch 2 Maths Class 10 Reading two types of decimal representation. These type of decimals are known as pure recurring decimals. For example : are pure recurring decimals. This type of decimals are cobia boats models design as mixed recurring decimals.

For example, are mixed recurring decimals. Algorithm : Step-1 : Obtain the repeating decimal and put it equal to x say. Step-2 : Write the number in decimal form by removing bar from the top of repeating digit and listing repeating digits at least twice. Step Determine the number of digits having bar on their heads. Step-4 : If the repeating decimal has 1 place repetition, multiply by 10; a two place repetition, multiply by ; a three place repetition, multiply by and so on.

Step-5 : Subtract the number in step ii from the number obtained in step iv. Step-6 : Divide both sides of the equation by the coefficient of x. Step-7 : Write the rational number in its simplest form. We have, [Using the above rule, we. Step-2 : Determine the number of digits after the decimal point amths do not have bar on. Let there be n digits without bar just after the decimal point. Step-3 : Multiply both sides of x by 10n so that only the repeating decimal is on the right side of the decimal point.

So, we multiply both sides of x by 10 so that only the repeting decimal is left on the right side of the decimal point. Addition and subtraction of recurring decimals can be done in two ways � either by converting ch 9 of maths class 10 notes into vulgar fractions and then ch 9 of maths class 10 notes them or alternatively as follows :- Add Step-1 : Express the number without bar as 5.

Step-3 : Divide the number into two parts. In the first part i. In the right side write as many digits as the LCM of the number of recurring digits in the given decimal number e. Step-4 : Now add or subtract as usual. Step-5 : Put the bar over the digits which are on the right side in the resultant value. If can clss done as usual.

Just convert the decimals into vulgar notew and then operate as required. A non-terminating and non repeating decimal is called cclass irractional numbers 0. We cobia boats models design, in the previous section, that there may be numbers on the number line that are not rationals. In this section, we are going to investigate these numbers.

So, you may ask : are there numbers which are not of this form? There are indeed such numbers. The Pythagoreans in Greece, followers of the famous mathematician and philosopher Pythagoras, where the first to discover the numbers which were not rationals, around BC.

You can revise it again in future grades before you start with the Trigonometry chapter in upper classes as well. What is horizontal ray? All the contents are updated for new academic session What do you mean by Angle of Elevation? Revision notes of Chapter 9 will benefit the students in a number of ways such as it is going to help them in speeding up the problem-solving skills of the students and for this, we suggest practising and revising these notes regularly.
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