The document discusses various number systems, focusing on whole numbers, integers, and rational numbers, detailing their properties and operations, such as closure under addition, subtraction, and multiplication, along with the concepts of identities and inverses. It covers linear equations in one variable, the structure of quadrilaterals, and the concepts of squaring and square roots, providing examples and problems for clarification. Additionally, it explains the significance of operations and identities in rational numbers and equations, while illustrating different quadrilateral types and mathematical properties.

1. NUMBER SYSTEM
 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.

2.  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 numbers
 3. 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 c
 8. 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.

3.  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 / 2
 2. Write the additive inverse of the following:
 (i) -7 / 19
 Solution: 7 / 9 is the additive inverse of -7 / 19 because -7 / 19 + 7 / 19 =
-7 + 7 / 19 = 0 / 19 = 0.
 (ii) 21 / 112
 Solution: The additive inverse of 21 / 112 is -21 / 112.

4.  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 .

5. LINEAR EQUATIONS IN ONE
VARIABLE We however, restrict to expressions with
 Many expressions have more than one variable. For example, 2xy
+ 5 has two variables.
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?

6.  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.

7.  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.

8. 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:

9.  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.

10.  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.

11.  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).

12.  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.

13.  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.

14.  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

15. Complex Quadrilaterals
 Oh 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.

16. The "Family Tree" Chart

17. Squares and Square Roots
 Example: 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)

18. Multiplication Table:-
 You can also find the squares on the Multiplication
Table:-

19.  Squares From 12 to 62
 1 Squared=12=1 × 1=1
 2 Squared=22=2 × 2=4
 3 Squared=32=3 × 3=9
 4 Squared=42=4 × 4=16
 5 Squared=52=5 × 5=25
 6 Squared=62=6 × 6=36

20.  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"

21. Numbers 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 etc
Perfect
1 4 9 16 25 36 49 64 81 100 121 144 169 196 225 ...
Squares:
 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 = 5
 Example: What is √36 ?
 Answer: 6 × 6 = 36, so √36 = 6

22.  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!

23. Here are some more squares and square roots:
4 16
5 25
6 36

24. How to Guess
 What 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.

25.  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 = 42.875
 Let's try 3.2: 3.2 × 3.2 × 3.2 = 32.768
 Let's try 3.1: 3.1 × 3.1 × 3.1 = 29.791
 We 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 !

26. A Fun Way to Calculate a Square
Root
 There 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.

27. Cubes and Cube Roots
 To cube a number, just use it in a multiplication 3
times ...
 3 Cubed=3 × 3 × 3=27
Note: 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=64
 5 cubed=53=5 × 5 × 5=125
 6 cubed=63=6 × 6 × 6=216

28.  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.

29.  You can use it like this:
 "the cube root of 27 equals 3“
 Perfect Cubes
NUMBERS 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 etc
Perfect
1 8 27 64 125 216 343 512 729 1000 1331 1728 2197 2744 3375 ...
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 = -125
 So the cube root of -125 is -5

30.  Here are some more cubes and cube roots:
4 64
5 125
6 216