Maths Summary

Number System

SUMMARY

1). The numbers. -4, -3, -1, 0, 1, 2, 3, 4, etc. are integers.

2). 1, 2, 3, 4, 5... are positive integers and -1-2-3 are negative integers.

3).0 is an integer which is neither positive nor negative.

4). On an integer number line, all numbers to the right of 0 are positive integers and all numbers to the left of O are negative integers.  

5). O is less than every positive integer and greater than every negative integer.

6). Every positive integer is greater than every negative integer.

7). Two integers that are at the same distance from 0, but on opposite sides of it are called opposite numbers.

8). The greater the number, the lesser is its opposite.

9). The sum of an integer and its opposite is zero.

10). The absolute value of an integer is the numerical value of the integer without regard to its sign. The absolute value of an integer a is denoted by a and is given by lala, if a is positive or 0 -a, if a is negative 

11). The sum of two integers of the same sign is an integer of the same sign whose absolute value is equal to the sum of the absolute values of the given integers.

12). The sum of two integers of opposite signs is an integer whose absolute value is the difference of the absolute values of addend and whose sign is the sign of the addend having greater absolute value.

13). To subtract an integer b from another integer a, we change the sign of b and add it to a. Thus, a-ba+(-b) 

14). All properties of operations on whole numbers are satisfied by these operations on integers.

15). If a and b are two integers, then (ab) is also an integer.

16).a and a are negative or additive inverses of each other.

17). To find the product of two integers, we multiply their absolute values and give the result a plus sign if both the numbers have the same sign or a minus sign otherwise.

18). To find the quotient of one integer divided by another non-zero integer, we divide their absolute values and give the result a plus sign if both the numbers have the same sign or a minus sign otherwise.

19). All the properties applicable to whole numbers are applicable to integers in addition, the subtraction operation has the closure property.

20). Any integer when multiplied or divided by 1 gives itself and when multiplied or divided by -1 gives its opposite.

Fractions

2.39

Fraction

SUMMARY

1). A fraction is a nomber representing a part of a whole.

2). A fraction can be expressed in the form a/b where a, b are whole numbers and b ne0

3). In a fraction a/b we call a' as numerator and 'b' as denominator.

4). A fraction whose momerator is less than the denominator is called a proper fraction.

5). A fraction whose numerator is more than or equal to the denominator is called an improper fraction.

6). A combination of a whole number and a proper fraction is called a mixed fraction.

7). To get a fraction equivalent to a given fraction, we multiply (or divide) its numerator and denominator ny the same non-zero number.

8). Fractions having the same denominators are called like fractions. Otherwise, they are called unlike fractions.

9). A fraction is said to be in its lowest terms if its numerator and denominator have no common factor other than 1.

10). To compare fractions, we use the following steps:

Step I Find the LCM of the denominators of the given fractions.

Step II Convert each fraction to its equivalent fraction with denominator equal to the LCM obtained in step 1.

Step III Arrange the fractions in ascending or descending order by arranging numerators in ascending or descending order.

11). To convert unlike fractions into like fractions, we use the following steps:

Step I Find the LCM of the denominators of the given fractions.

Step II Convert each of the given fractions into an equivalent fraction having denominator equal to the LCM obtained in step 1.

12). To add (or subtract) fractions, we may use the following steps:

Step I Obtain the fractions and their denominators.

Step II Find the LCM of the denominators.

Step III Convert each fraction into an equivalent fraction having its denominator equal to the LCM obtained in step II.

Step IV Add (or subtract) like fractions obtained in Step III.

Product of their numerators 13. Product of two fractions = Product of their denominators

14). Two fractions are said to be reciprocal of each other, if their product is 1. The reciprocal of a non-zero fraction is equal to b/a a/b

15). The division of a fraction a/b by a non-zero fraction c/d is the product a/b with the reciprocal of c/4

Decimals

angle to its op line from the Altitude of a meeting at night

3.16

Mathematics for Cless V

SUMMARY

1. Decimals are an extension of our number system.

2. Decimals are fractions whose denominators are 10, 100, 1000 etc.

3. A deconal has tue parts, namely, the whole number part and decimal part

4. The manber of digits contained in the decimal part of a decimal number is known at the number of decimat places

5. Decimals having the same mumber of places are called like decimals, otherwise they are known as unlis decimals

We have, 0.1 -0.10 -0.100 etc. 0.5-0.50 -0.500 etc and so on. That is by annexing zeros on the right side of the extreme right digit of the decimal part of a number does not alter the value of the number.

7. Unlike decimals may be converted into like decimals by annexing the requisite number of seras in the right side of the extreme right digit in the decimal part.

5. Decimal numbers may be compared by using the following steps.

Step 1

Obtain the decimal numbers.

Step

Compare the whole parts of the numbers. The number with greater whole part will be greater. If the whole parts are equal, go to next step.

Step III

Compare the extreme left digits of the decimal parts of two numbers. The number with greater extreme left digit twill be greater. If the extreme left digits of decimal parts are equal, then compar the next digits and so on.

9. A decimal can be converted into a fraction by using the following steps

Step 1

Obtain the decimal.

Step It

Take the numerator as the number obtained by removing the decimal point from the given decimal Take the denominator as the number obtained by inserting as many zeros with 1 leg. 10, 100 1000 etc.) as there are number of places in the decimal part.

Step

10. Fractions can be converted into decimals by using the following steps:

Step 1

Obtain the fraction and convert it into an equivalent fraction with denominator 10 or 100 or 1000 if it is not so.

Step U

Write its numerator and mark decimal point after one place or two places or three places from riglu towards left if the denominator is 10 or 100 or 1000 respectively. If the numerator is short of digas, insert zeros at the left of the numerator.

11. Decimals can be added or subtracted by using the following steps:

Step T

Convert the given decimals to like decimals.

Step

Write the decimals in columns with their decimal points directly below each other so that tenths come wnder tenths, hundredths come under hundredths and so on.

Step

Add or subtract as we add or subtract whole numbers.

Step IV

Place the decimal point, in the answer, directly below the other decimal points.

12. In order to multiply a decimal by 10, 100, 1000 etc., we use the following rules:

RULEI

On multiplying a decimal by 10, the decimal point is shifted to the right by one place.

RULE

It On multiplying a decimal by 100, the decimal point is shifted to the right by two places.

RULE II On multiplying a decimal by 1000, the decimal point is shifted to the right by three places, and so on

13. A decimal can be multiplied by a whole number by using following steps:

Step 1

Multiply the decimal without the decimal point by the given whole member.

Mark the decimal post in the product to have as many places of decimal as are there in the given decimal.

ils

3.17

To multiply a decimal by another decimal, we follow following steps:

Step 1

Multiply the two decimals without decimal point just like whole numbers

Step

Insert the decimal point in the product by counting as many places from the right to left as the sum of the number of decimal places of the given decimals.

A decimal can be divided by 10, 100, 1000 etc by using the following rules:

RULEI

When a decimal is divided by 10, the decimal point is shifted to the left by one place.

RULE II

When a decimal is divided by 100, the decimal point is shifted to the left by two places.

RULE

III When a decimal is divided by 1000, the decimal point is shifted to the left by three places.

Step 1

A decimal can be divided by a whole number by using the following steps:

Step II

Check the whole number part of the dividend.

If the whole number part of the dividend is less than the divisor, then place a 0 in the ones place in the quotient. Otherwise, go to step III.

Step III

Divide the whole number part of the dividend.

Step IV

Place the decimal point to the right of ones place in the quotient obtained in step 1.

Step V

Divide the decimal part of the dividend by the divisor. If the digits of the dividend are exhausted, then place zeros to the right of dividend and remainder each time and continue the process.

A decimal can be divided by a decimal by using the following steps:

Step 1

Multiple the dividend and divisor by 10 or 100 or 1000 etc. to convert the divisor into a whole number.

Step II

Divide the new dividend by the whole number obtained in step 1.

Rational Numbers

SUMMARY

1. For any two rational numbers p/q and r/q we define: P/q + r/q = (p + r)/q

to equivalent rational 2. For any two rational numbers and r/s to find p/q + r/s first we convert p/q and numbers having denominator equal to the LCM of q and s and then they are added. p/q r/s

3. For any two rational numbers we have and negative of r/s ) p/q r/5 P/q - r/s = p q +(

4. For any two rational numbers and we have p/q r/s p/q * r/s = (pr)/(qs)

5. The reciprocal of a non-zero rational number p/q is and we write (p/q) ^ - 1 = q/p q/p

6. For any two rational numbers and r/s ( ne0) we have p/q p/q / (r/s) = p/q * s/r

23.09

24. (b)

25. (d)

SUMMARY

721

1. Za ta non-com-damál minder anul x is a natural sumber, then the product exaxax to denoted by a and is read or is rated to the power of el. Rational number is called the bone number on to known as the exponent. Alen, &'is known as the exponential forms of xxx

2. Baranowe rational number, we have a ^ 2 = 1 x ^ 2 = a

3. Babamo national numbers and on and n are natural numbers, then following are the love of asponents

a ^ m * a ^ m = a ^ (m - n)

where > (a ^ m)/(a ^ n) = a ^ (m - n)

( x ^ prime prime )^ prime prime =x^ prime prime =(a^ prime prime )^ prime prime

(v) (a + b) ^ prime prime =a^ prime prime b^ prime prime

(a/b) ^ x = (a ^ n)/(b ^ n)

7.20

SUMMARY

Mathematics for Class

1. The letters which are used to represent numbers are called literal numbers or literals.

2. The literal members themselves as well as the combinations of literal numbers and mumbers obey att the rules (and signs) of addition, subtraction, multiplication and division of numbers along with the properties of these operations.

3. xxyxy, 5 xx=5x, 1xxx, xx44x.

* axaxaxx 12 times = a", yxyxyxx 15 times = y

5. In x, 9 is called the index or exponent and x is called the base. In a³, the index or exponent is 5 and the base is a.

6. A symbol having a fixed numerical value is called a constant.

7. A symbol which takes various numerical values is called a variable.

8. A combination of constants and variables connected by the signs of fundamental operations of addition, subtraction, multiplication and division is called an algebraic expression.

9. Various parts of an algebraic expression which are separated by the signs of'+' or are called the terms of the expression.

10. An algebraic expression is called a monomial, a binomial, a trinomial, a quadrinomial according as contains one term, two terms, three terms and four terms respectively.

11. Each term in an algebraic expression is a product of one or more number(s) and/or literal number(s). These number(s) and/or literal mumber(s) are known as the factors of that term.

12. A term of the expression having no literal factor is called a constant term.

13. In a term of an algebraic expression any of the factors with the sign of the term is called the coefficient of the product of the factors.

14. The terms having the same literal factors are called like or similar terms.

15. The terms not having same literal factors are called unlike or dissimilar terms.

16. The sum or difference of several like terms is another like term whose coefficient is the sum or difference of those like terms.

17. In adding or subtracting algebraic expressions, we collect different groups of like terms and find the sum or difference of like terms in each group.

18. To subtract an expression from another, we change the sign (from + to and from to of each term of the expression to be subtracted and then add the two expressions.

19. When a grouping symbol preceded by sign is removed or inserted, then the sign of each term of the corresponding expression is changed (from + to and from-to'+')

Summary 

1. Per cent means per hundred or for every hundred.

2. By a certain per cent, we mean that many hundredths.

3. A fraction with its denominator as 100 is called a per cent and is equal to that per cent as is the numerator.

4. A ratio with its second term 100 is also called a per cent.

5. To convert a fraction into a per cent we multiply the fraction by 100.

6. To convert a ratio into a per cent, we write it as a fraction and multiply it by 100.

7. To convert a decimal into a per cent, we shift the decimal point two places to the right.

8. To convert a per cent into a fraction, we drop per cent sign (%) and divide the remainder by 100.

9. To convert a per cent into a ratio, we drop per cent sign (%) and form a ratio with the remaining number as the first term and 100 as the second term.

10. To convert a per cent into a decimal, we drop per cent sign (%) and shift the decimal point two places to the left.

11. Increase% = Increase Original value ×100% 0)% Decrease% = ×100% Original value Decrease

SUMMARY

1. The money paid by the shopkeeper to buy the goods from a manufacturer or a wholesaler is called the cost price of the shopkeeper. The cost price is abbreviated as C.P.

11

2. The price at which a shopkeeper sells the goods is called the selling price of the shopkeeper. The selling price is abbreviated as S.P.

3. Effective cost = Cost price + Overhead charges.

4. If S.P. > C.P., then there is gain given by Gain = S.P. - C.P.

5. If S.P. <C.P., then there is loss given by Loss = C.P. - S.P.

6. Gain or loss is calculated on the cost price.

7. Gain percent = ( -(Gain x100) -x100) and Loss percent = = (x100). C.P.

SUMMARY

1. The person who borrows money is called the borrower.

2. The person who lends money is called the money lender.

3. The money borrowed from a lender is called the principal.

4. The additional money paid by the borrower to the lender for having used his money is called the interest.

5. The total money which the borrower pays back to the lender at the end of the specified period is called the amount.

6. Interest is said to be simple if it is calculated on the original principal throughout the loan period.

7. In calculating the number of days, we do not count the day on which the money is deposited but the date of withdrawal is counted.

8. Number of years = Number of days 365

9. If P = Principal, R = Rate of interest per annum and T = time, then the simple interest is given by S.I. = PRT 100

SUMMARY

1. Aline which intersects two or more given lines at distinct points is called a transversal to the given lines.

1. Lines in a plane are parallel if they do not intersect when produced indefinitely in either direction.

3. The distance between two intersecting lines is zero.

4. The distance between two parallel lines is the same everywhere and is equal to the perpendicular distance between them.

5. If two parallel lines are intersected by a transversal then

(1) pairs of alternate (interior or exterior) angles are equal.

(ii) pairs of corresponding angles are equal.

(i) interior angles on the same side of the transversal are supplementary.

6. If two non-parallel lines are intersected by transversal then none of (i), (ii) and (iii) hold true

in 5. 7. If two lines are intersected by a transversal, then they are parallel if any one of the following is true:

(i) The angles of a pair of corresponding angles are equal.

(ii) The angles of a pair of alternate interior angles are equal.

(ii) The angles of a pair of interior angles on the same side of the transversal are supplementary.

Properties of Triangles

13.33

SUMMARY

1. A triangle is a figure made up by three line segments joining, in pairs, three non-collinear points. That is, if A, B, C are three non-collinear points, the figure formed by three line segments AB, BC and CA is called a triangle with vertices A, B, C

2. The three line segments forming a triangle are called the sides of the triangle

3. The three sides and three angles of a triangle are together called the six parts or elements of the triangle

4. A triangle whose two sides are equal, is called an isosceles triangle.

5. A triangle whose all sides are equal, is called an equilateral triangle.

6. A triangle whose no two sides are equal, is called a scalene triangle.

7. A triangle whose all the angles are acute is called an acute triangle

8. A triangle whose one of the angles is a right angle is called a right triangle.

4. A triangle whose one of the angles is an obtuse angle is called an obtuse triangle.

10. The interior of a triangle is made up of all such points P of the plane, as are enclosed by the triangle

11. The exterior of a triangle is that part of the plane which consists of those points Q, which are neither on the triangle nor in its interior.

12. The interior of a triangle together with the triangle itself is called the triangular region.

13. The sum of the angles of a triangle is two right angles or 180".

14 . If a side of a triangle is produced, the exterior angle so formed is equal to the sum of the interior opposite angles.

15. In any triangle, an exterior angle is greater than either of the interior opposite angles.

16. The sum of any two sides of a triangle is greater than the third side.

17. In a right triangle, if a, b are the lengths of the sides and e that of the hypotenuse, then c²= a²

18. If the sides of a triangle are of lengths a, b and e such that c²= a² + b², then the triangle is right-angind and the side of length c is the hypotenuse.

19. Three positive numbers a, b, c in this order are said to form a Pythagorean triplet, if Triplets (3, 4, 5) (5, 12, 13), (8, 15, 17), (7, 24, 25) and (12, 35, 37) are some Pythagorean triplets.

Class VI

Symmetry

SUMMARY

14.33

1. If a figure is made up of parts that repeat in a definite pattern, then we say that the figure has symmetry. Such a figure is called a symmetrical figure.

2. If a line divides a figure into two identical halves, then we say that the given figure is symmetrical about that line and the line is called the axis of symmetry or line of symmetry. 3. A figure may have multiple lines of symmetry. A circle has infinitely many lines of symmetry.

4. A figure is said to have rotational symmetry if it fits onto itself more than once during a full turn ie rotation through 360°.

5. The point about which the figure turns is called the centre of symmetry or centre of rotational symmetry.

6. An angle through which a figure can be rotated about a point on it, to look exactly the same is called an angle of rotational symmetry or simply an angle of symmetry.

7. A figure that has an angle of symmetry strictly between 0° and 360° is said to have rotational symmetry.

8. A figure may have multiple angles of symmetry.

9. The number of times a figure fits onto itself in one full-turn is called the order of rotational symmetry.

360° Order of rotational symmetry = Angle of rotation 360° Angle of rotation Order of rotational symmetry 11

A regular polygon with n sides has an order of rotation n and the angle of rotation = 360°

10. The order of rotation of a circle is infinite.

11. Some figures may have a line of symmetry but no angle of symmetry while others may have angles of symmetry but no lines of symmetry. Some figures may have both lines of symmetry as well as angles of symmetry.

16.20

SUMMARY

Mathematics for Class Vii

1. A square centimetre is the area of the region formed by a square of side 1 cm.

I = z ^ (mu / 3) * 1 deg * z ^ (mu / 2) * I = z^ mu aax 001 dm²

2. Standard units of area relations are:

I=z^ mu c3 0000I^ * z^ mu L=z^ mu*mu*p * 0I m²

I=z^ mu*x * 0I are I = suv * 0l hectare

LEARNING OBJECTIVES

100 hectares 1 sq. km.

3. Perimeter of a rectangle = 2 (Length + Breadth) or, P = 2(1 + b)

Perimeter of a square IF= d' * 10 ( partial pHS)* f=

Area of a rectangle = ql = V Also, length of a q V = 1 ^ 4 i0 ippaag pa * 4V = 2p

Breadth of a rectangle 1/V = q' * 40 * (iq * 8u*beta*gamma)/(v24gamma) = a

Area of a square (Side) or, A=1xL 4. Area of a parallelogram = Base Height or, A = bx h

Area Also, Base of parallelogram = height or, b=

Area Height of a parallelogram = base or, h=

1 A=xbxh 5. Area of a triangle = 2 x Base x Height or, A=

2x Area Height of a triangle Base b or, h=2A h = 2A

2x Area Base of a triangle = height or, b=2

1 6. Area of a rhombus 2 x Product of diagonals

7. Area of a trapezium = (Sum of the parallel sides) x (Distance between the parallel sides)

Pesmeter and Area of Rectilinear Figures

16.3

Perimeter and area of a rechangle: Letland & denote the length and breadth respectively of a rectangle. Then

(1) Perimeter =2(1+b)

(Area=1xb

Area (i) Length Breadth

D

C

B

Fig. 16.1

[1 dm³-100 cm³]

called a hectare, written as

(iv) Breadth= Area Length

(v) Diagonal+

Perimeter and area of a square: Let a be the length of each side of a square. Then,

(1) Perimeter = 44

D

C

Perimeter (Area-a, Area 4

a

()Side of the square = Area

(iv) Diagonal -√24

(v) Area-(Diagonal)

(Area of four walls of a room: Let 1, b and h denote respectively the length, breadth and height of a room. Then,

Area of the 4 walls =2(1+b) x h

(i) Diagonal of the room = √²+b²+h²

ILLUSTRATIVE EXAMPLES