chapters summary (x class)

WHOLE NUMBER:-whole numbers are closed under addition and multiplication but not under subtraction and division.

INTEGER:- integers are closed under addition , subtraction and multiplication but not under division.

RATIONAL NUMBER:- A number written in the form p/q ,where p and q are integers and q is not equal to 0 is called a rational number. Since the number 0,-2, 4 can be written in the form p/q, they are also rational numbers. Any rational number a, a/0is not defined. So rational numbers are not closed under division. We exclude zero then the collection of, all other rational numbers is closed under division.

Addition and multiplication is communicative for rational numbers.

Subtraction and division is not communicative for rational numbers. When we multiply the rational number with 1, you get back that

rational number as the product. Check this for a few more rational number a.

Also, 2+ (-2) = (-2) +2=0, so we say 2 is the negative or additive inverse of -2 and vice – versa. In general , for an integer a, we have , a +(-a)=(-a)+a=0 ; so , as is the negative of –a and –a is the negative of a.

NUMBER SYSTEM

Zero has no reciprocal. A rational number c / d is called the reciprocal or multiplicative inverse of another rational number a/b if a/b x c / d=1. 1. Rational numbers are closed under the operations of addition, subtraction and multiplication.

2. The operations addition and multiplication are:-Communicative for rational numbers. (ii)Associative for rational numbers3. The rational number 0 is the additive identity for rational numbers.4. The rational number 1 is the multiplicative identity for rational

numbers.5. The additive inverse of the rational number a/b is –a/b and vice –

versa.6. The reciprocal or multiplicative inverse of the rational number a/b is

c / d if a / b x c / d =1.7. Distributive of rational numbers: For all rational numbers a, b and

c,a (b + c) =a b + ac and a (b – c ) =a b – a c8. Rational numbers can be represented on a number line.9. Between any two given rational numbers there are countless

rational numbers. The idea of mean helps us to find rational numbers between two rational numbers.

SOME PROBLEMS:- .Find: (i) 3 / 7 + (- 6 / 11) + (-8 / 21) + (5 / 22)

Solution : 3/7 + (-6 / 11) + (-8 / 21) + (5 / 22) = 198 / 462 + (-252 / 462) + (-176 / 462) + (105 / 462) (Note

that 462 is the L.C.M of 7, 11, 21 and 22)

=198 - 252 - 176 + 105/462 = -125 / 462 (ii) -4 / 5 x 3 / 7 x 15 / 16 x (-14 / 9)Solution: -4 / 5 x 3 / 7 x 15 / 16 x (-14 / 9) = (-4 x 3 / 5 x 7) x {15 x (-14) /16 x 9} = -12 / 35 x (-35 / 24) = -12 x (-35) / 35 x 24 = 1 / 22. Write the additive inverse of the following: (i) -7 / 19Solution: 7 / 9 is the additive inverse of -7 / 19 because -7 / 19 + 7 / 19

= -7 + 7 / 19 = 0 / 19 = 0. (ii) 21 / 112Solution: The additive inverse of 21 / 112 is -21 / 112.

3.Verify that – (-x) is the same as x for (i) x=13 / 17 Solution: We have, x = 13 / 17 =The additive inverse of x = 13 / 17 is –x =-13 / 17 since 13 / 17 +

(-13 / 17) = 0. The same equality 13 / 17 (-13 / 17) = 0, shows that the additive inverse of

-13 / 17 is 13 / 17 or (-13 / 17) = 13 / 17, i.e., (-x) =x. (ii) X= -21 / 31 Solution: Additive inverse of x = -21 / 31 is –x = 21 / 31 since -21 / 31 + 21 /

31 = 0 The same equality -21 / 31 + 21 / 31 = 0 , shows that the additive inverse of

21 / 31 is -21 / 31 , i.e., -(-x) = x . 4. Write any 3 rational numbers between -2 and 0. Solution: -2 can be written as -20 / 10 and 0 as 0 / 10. Thus we have -19 / 10, -18 / 10, -17 / 10, -16 / 10, -15 / 10, … , -1 /

10 between -2 and 0 . We can take any three of these. 5. Find any ten rational numbers between -5 / 6 and 5 / 8. Solution: We first convert -5 / 6 and 5 / 8 to rational numbers with the same

denominators. -5 x 4 / 6 x 4 = -20 / 24 and 5 x 3 / 8 x 3 =15 / 24 Thus we have -19 / 24, -18 / 24, -17 / 24, … ,14 / 24 as the rational numbers

between -20 / 24 and 15 / 24 .

LINEAR EQUATIONS IN ONE VARIABLEMany expressions have more than one variable. For example, 2xy + 5 has two variables. We however, restrict to expressions with only one variable when we form equations. Moreover, the expressions we use to form equations are linear. This means that the highest power of the variable appearing in the expression is 1.

Equations with linear expressions in one variable only. Such equations are known as linear equations in one variable.

(a) An algebraic equation is an equality involving variables. It has an equality sign. The expression on the left of the equality sign is the Left Hand Side (L.H.S). The expression on the right of the equality sign is the Right Hand Side (R.H.S).

(b)In an equations the values of the expressions on the L.H.S and R.H.S are equal. This happens to be true only for certain values of the variable .These values are the solutions of the equation.

(c) How to find the solution of an equation?

We assume that the two sides of the equation are balanced. We perform the same mathematical operations on both sides of the equation, so that the balance is not distributed. A few such steps give the solution.

1. An algebraic equation is an equality involving variables. It says that the value of the expression on one side of the equality sign is equal to the value of the expression on the other side.

2. The expressions which form the equation contain only one variable. Further, the equations are linear i.e, the highest power of the variable appearing in the equation is 1.

3. A linear equation may have for its solution any rational number.

4. An equation may have linear expressions on both sides.5. Just as numbers, variables can, also, be transposed from

one side of the equation to the other.6. Occasionally, the expressions forming equations have to be

simplified before we can solve them by usual methods. Some equations may not even be linear to begin with, but they can be brought to a linear form by multiplying both sides of the equation by a suitable expression.

SOME PROBLEMS:- 1. Solve 15 / 4 -7x = 9 Solution: We have 15 / 4 -7x = 9 -7x = 9 – 15 / 4 (transposing 15 / 4 to R.H.S) -7x = 21 / 4 x = 21 / 4 x (-7) (dividing both sides by -7) x = (- 3 x 7 / 4 x 7) x = -3 / 4 CHECK: L.H.S = 15 / 4 -7 (-3 / 4) = 15 / 4 + 21 / 4 = 36 / 4 = 9 = R.H.S 2. What should be added to twice the rational number -7 / 3 to get 3 / 7? Solution: Twice the rational number -7 / 3 is 2 x (-7 / 3) = -14 / 3. Suppose x Added to this number gives 3 / 7; i.e. x + (-14 / 3) = 3 / 7 x – 14 / 3 = 3 / 7 x = 3 / 7 + 14 / 3 (transporting 14 / 3 to R.H.S) = (3 x 3) + (14 x 7) / 21 = 9 + 98 / 21 = 107 / 21. Thus 107 / 21 should be added to 2 x (-7 / 3) to give 3 / 7. 3. The perimeter of a rectangle is 13 cm and its width is 2 x 3 / 4 cm. Find its length. Solution: Assume the length of the rectangle to be x cm. The perimeter of the rectangle = 2 x (length + width) = 2 x (x + 2 x 3 / 4) = 2 (x + 11 / 4) The perimeter is given to be 13 cm. Therefore, 2 (x + 11 / 4) = 13 x + 11 / 4 = 13 / 2 (dividing both sides by 2) x = 13 / 2 - 11 / 4 = 26 / 4 - 11 / 4 = 15 / 4 = 3 x 3 / 4 The length of the rectangle is 3 x 3 / 4 cm.

QUADRILATERALS Quadrilateral just means "four sides" (quad means four, lateral means side).

Any four-sided shape is a Quadrilateral. But the sides have to be straight, and it has to be 2-dimensional.

Properties Four sides (or edges) Four vertices (or corners).

The interior angles add up to 360 degrees: Try drawing a quadrilateral, and measure the angles. They should

add to 360°

Types of Quadrilaterals There are special types of quadrilateral:

The Rectangle

means "right angle" & show equal sides

A rectangle is a four-sided shape where every angle is a right angle (90°).

Also opposite sides are parallel and of equal length.

The Rhombus

A rhombus is a four-sided shape where all sides have equal length.

Also opposite sides are parallel and opposite angles are equal.

Another interesting thing is that the diagonals (dashed lines in second figure) of a rhombus bisect each other at right angles.

The Square means "right angle" show equal sides

A square has equal sides and every angle is a right angle (90°)

Also opposite sides are parallel.A square also fits the definition of

a rectangle (all angles are 90°), and a rhombus (all sides are equal length).

The Parallelogram

A parallelogram's opposite sides are parallel and equal in length. Also opposite angles are equal (angles "a" are the same, and angles "b" are the same)

The Kite

Hey, it looks like a kite. It has two pairs of sides. Each pair is made up of adjacent sides that are equal in length. The angles are equal where the pairs meet. Diagonals (dashed lines) meet at a right angle, and one of the diagonal bisects (cuts equally in half) the other.

The Trapezoid (UK: Trapezium)

Trapezoid Isosceles Trapezoid

A trapezoid (called a trapezium in the UK) has a pair of opposite sides parallel.

It is called an Isosceles trapezoid if the sides that aren't parallel are equal in length and both angles coming from a parallel side are equal, as shown.

A trapezium (UK: trapezoid) is a quadrilateral with NO parallel sides:

NOTE: Squares, Rectangles and Rhombuses are all Parallelograms!

  Trapezoid Trapezium

US: a pair of parallel sides NO parallel sides

UK: NO parallel sides a pair of parallel sides

Complex QuadrilateralsOh Yes! when two sides cross over, you call it

a "Complex" or "Self-Intersecting" quadrilateral like these:

They still have 4 sides, but two sides cross over.

The "Family Tree" Chart

Squares and Square RootsExample: What is 3 squared?

3 Squared= 3 × 3 = 9"Squared" is often written as a little 2 like this:

This says "4 Squared equals 16"(the little 2 says the number appears twice in multiplying)

Multiplication Table:-You can also find the squares on the Multiplication Table:-

Squares From 12 to 62

1 Squared=12=1 × 1=12 Squared=22=2 × 2=43 Squared=32=3 × 3=94 Squared=42=4 × 4=165 Squared=52=5 × 5=256 Squared=62=6 × 6=36

A square root goes the other way:A square root of a number is ...... a value that can be multiplied by itself to give

the original number.A square root of 9 is ...... 3, because when 3 is multiplied by itself you

get 9.The Square Root Symbol

This is the special symbol that means "square root", it is sort of like a tick, and actually started hundreds of years ago as a dot with a flick upwards.

It is called the radical, and always makes math look important!

You can use it like this: you would say "square root of 9 equals 3"

Example: What is √25?Well, we just happen to know that 25 = 5 × 5,

so if you multiply 5 by itself (5 × 5) you will get 25.

So the answer is:√25 = 5Example: What is √36 ?Answer: 6 × 6 = 36, so √36 = 6

Numbers  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 etc

Perfect Squares:

1 4 9 16 25 36 49 64 81 100 121 144 169 196 225 ...

When you square a negative number you get a positive result.

(because a negative times a negative gives a positive)

Just the same as if you had squared a positive number!

4   16

5   25

6   36

Here are some more squares and square roots:-

How to GuessWhat if you have to guess the square root for a difficult number such as "82,163" ... ?

In that case I would think to myself "82,163" has 5 digits, so the square root might have 3 digits (100x100=10,000), and the square root of 8 (the first digit) is about 3 (3x3=9), so 300 would be a good start.

Example: what is the cube root of 30?Well, 3 × 3 × 3 = 27 and 4 × 4 × 4 = 64, so we

can guess the answer is between 3 and 4.Let's try 3.5: 3.5 × 3.5 × 3.5 = 42.875Let's try 3.2: 3.2 × 3.2 × 3.2 = 32.768Let's try 3.1: 3.1 × 3.1 × 3.1 = 29.791We are getting closer, but very slowly ... at this

point, I get out my calculator and it says:3.1072325059538588668776624275224... but the digits just go on and on, without any

pattern. So even the calculator's answer is only an approximation ! 

A Fun Way to Calculate a Square RootThere is a fun method for calculating a square root that gets more and more accurate each time around:

a) start with a guess (let's guess 4 is the square root of 10)

b) divide by the guess (10/4 = 2.5)c) add that to the guess (4 + 2.5 = 6.5)d) then divide that result by 2, in other words halve it. (6.5/2 = 3.25)e) now, set that as the new guess, and start at b) again.

Cubes and Cube RootsTo cube a number, just use it in a multiplication 3 times ...

3 Cubed=3 × 3 × 3=27Note: we write down "3 Cubed" as 33

(the little "3" means the number appears three times in multiplying)

Some More Cubes:-4 cubed=43=4 × 4 × 4=645 cubed=53=5 × 5 × 5=1256 cubed=63=6 × 6 × 6=216

A cube root goes the other direction:3 cubed is 27, so the cube root of 27 is 3

3 27

The cube root of a number is ...... a special value that when cubed gives the original number.

The cube root of 27 is …... 3, because when 3 is cubed you get 27.

This is the special symbol that means“ cube root", it is the "radical"symbol (used for square roots) with a little three to mean cube root.

You can use it like this: "the cube root of 27 equals 3“Perfect Cubes

Have a look at this:If you cube 5 you get 125:5 × 5 × 5 = 125 If you cube -5 you get -125:-5 × -5 × -5 = -125So the cube root of -125 is -5

 NUMBERS 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 etc

Perfect Cubes:

1 8 27 64 125 216 343 512 729 1000 1331 1728 2197 2744 3375 ...

Here are some more cubes and cube roots:

4   64

5   125

6   216