Digital Textbook

1. DIGITAL TEXT BOOK
MATHEMATICS
ANU KM
ROLL NO 119
REG NO 13304001

2. PREFACE
Through the years , from the time of kothari commision there
have been several committes looking at ways the school
currriculum meaningful and enjoyable for the leader. Based On
the understanding developed over the years , a National
curriculum Framework (NCF) was finalised in 2005, highlighted a
constructvist approach to the teaching and learning of
mathematics .

3. contents
1. Negetive numbers
2. The number line
3. Signed numbers
4. Addition
5. Subtraction
6. Multiplication
7. Division
8. Negation

4. Negative number
A negative number is a real number that is less than zero. Such numbers are often used to
represent the amount of a loss or absence. For example, a debt that is owed may be thought of as a
negative asset, or a decrease in some quantity may be thought of as a negative increase. Negative
numbers are used to describe values on a scale that goes below zero, such as the Celsius
and Fahrenheit scales for temperature.
Negative numbers are usually written with a minus sign in front. For example, −3 represents a
negative quantity with a magnitude of three, and is pronounced "minus three" or "negative three". To
help tell the difference between a subtraction operation and a negative number, occasionally the
negative sign is placed slightly higher than the minus sign . Conversely, a number that is greater
than zero is called positive; zero is usually thought of as neither positive nor negative.[1 The positivity
of a number may be emphasized by placing a plus sign before it, e.g. +3. In general, the negativity
or positivity of a number is referred to as its sign.
In mathematics, every real number other than zero is either positive or negative. The positive whole
numbers are referred to as natural numbers, while the positive and negative whole numbers
(together with zero) are referred to as integers.
As the result of subtraction
Negative numbers can be thought of as resulting from the subtraction of a larger number from a
smaller. For example, negative three is the result of subtracting three from zero:
0 − 3 = −3.
In general, the subtraction of a larger number from a smaller yields a negative result, with the
magnitude of the result being the difference between the two numbers. For example,
5 − 8 = −3
since 8 − 5 = 3.
The number line
The relationship between negative numbers, positive numbers, and zero is often expressed in the
form of a number line:

5. Numbers appearing farther to the right on this line are greater, while numbers appearing farther to
the left are less. Thus zero appears in the middle, with the positive numbers to the right and the
negative numbers to the left.
Note that a negative number with greater magnitude is considered less. For example, even
though (positive) 8 is greater than (positive) 5, written
8 > 5
negative 8 is considered to be less than negative 5:
−8 < −5.
(Because, for example, if you have £-8 you have less than if you have £-5.)
Therefore, any negative number is less than any positive number, so
−8 < 5 and −5 < 8.
Signed numbers
In the context of negative numbers, a number that is greater than zero is referred to as positive.
Thus every real number other than zero is either positive or negative, while zero itself is not
considered to have a sign. Positive numbers are sometimes written with a plus sign in front,
e.g. +3 denotes a positive three.
Because zero is neither positive nor negative, the term nonnegative is sometimes used to refer to a
number that is either positive or zero, while nonpositive is used to refer to a number that is either
negative or zero. Zero is a neutral number.
Arithmetic involving negative numbers
The minus sign "−" signifies the operator for both the binary (two-operand) operatio of subtraction
(as in y − z) and the unary (one-operand) operation of negation (as in −x, or twice in −(−x)). A
special case of unary negation occurs when it operates on a positive number, in which case the
result is a negative number (as in −5).
The ambiguity of the "−" symbol does not generally lead to ambiguity in arithmetical expressions,
because the order of operations makes only one interpretation or the other possible for each "−".
However, it can lead to confusion and be difficult for a person to understand an expression when
operator symbols appear adjacent to one another. A solution can be to parenthesize the unary "−"
along with its operand.

6. For example, the expression 7 + −5 may be clearer if written 7 + (−5) (even though they mean
exactly the same thing formally). The subtraction expression 7–5 is a different expression that
doesn't represent the same operations, but it evaluates to the same result.
Sometimes in elementary schools a number may be prefixed by a superscript minus sign or plus
sign to explicitly distinguish negative and positive numbers as in[9]
−2 + −5 gives −7.
Addition
A visual representation of the addition of positive and negative numbers. Larger balls represent numbers with
greater magnitude.
Addition of two negative numbers is very similar to addition of two positive numbers. For
example,
(−3) + (−5) = −8.
The idea is that two debts can be combined into a single debt of greater magnitude.
When adding together a mixture of positive and negative numbers, one can think of the
negative numbers as positive quantities being subtracted. For example:
8 + (−3) = 8 − 3 = 5 and (−2) + 7 = 7 − 2 = 5.
In the first example, a credit of 8 is combined with a debt of 3, which yields a total credit
of 5. If the negative number has greater magnitude, then the result is negative:
(−8) + 3 = 3 − 8 = −5 and 2 + (−7) = 2 − 7 = −5.

7. Here the credit is less than the debt, so the net result is a debt.
Subtraction
As discussed above, it is possible for the subtraction of two non-negative numbers
to yield a negative answer:
5 − 8 = −3
In general, subtraction of a positive number is the same thing as addition of a
negative. Thus
5 − 8 = 5 + (−8) = −3
and
(−3) − 5 = (−3) + (−5) = −8
On the other hand, subtracting a negative number is the same as adding a positive. (The idea is
that losing a debt is the same thing as gaining a credit.) Thus
3 − (−5) = 3 + 5 = 8
and
(−5) − (−8) = (−5) + 8 = 3.
Multiplication
When multiplying numbers, the magnitude of the product is always just the product of the two
magnitudes. The sign of the product is determined by the following rules:
The product of one positive number and one negative number is negative.
The product of two negative numbers is positive.
Thus
(−2) × 3 = −6
and
(−2) × (−3) = 6.
The reason behind the first example is simple: adding three −2's together yields −6:
(−2) × 3 = (−2) + (−2) + (−2) = −6.
The reasoning behind the second example is more complicated. The idea again is that losing a debt
is the same thing as gaining a credit. In this case, losing two debts of three each is the same as
gaining a credit of six:

8. (−2 debts ) × (−3 each) = +6 credit.
The convention that a product of two negative numbers is positive is also necessary for multiplication
to follow the distributive law. In this case, we know that
(−2) × (−3) + 2 × (−3) = (−2 + 2) × (−3) = 0 × (−3) = 0.
Since 2 × (−3) = −6, the product (−2) × (−3) must equal 6.
These rules lead to another (equivalent) rule—the sign of any product a × b depends on the sign
of a as follows:
if a is positive, then the sign of a × b is the same as the sign of b, and
if a is negative, then the sign of a × b is the opposite of the sign of b.
The justification for why the product of two negative numbers is a positive number can be observed
in the analysis of complex numbers.
Division
The sign rules for division are the same as for multiplication. For example,
8 ÷ (−2) = −4,
(−8) ÷ 2 = −4,
and
(−8) ÷ (−2) = 4.
If dividend and divisor have the same sign, the result is always positive
.Negation
The negative version of a positive number is referred to as its negation. For example, −3 is the
negation of the positive number 3. The sum of a number and its negation is equal to zero:
3 + −3 = 0.
That is, the negation of a positive number is the additive inverse of the number.
Using algebra, we may write this principle as an algebraic identity:
x + −x = 0.
This identity holds for any positive number x. It can be made to hold for all real numbers by
extending the definition of negation to include zero and negative numbers. Specifically:
The negation of 0 is 0, and

9. The negation of a negative number is the corresponding positive number.
For example, the negation of −3 is +3. In general,
−(−x) = x.
absolute value of a number is the non-negative number with the same magnitude. For example, the
absolute value of −3 and the absolute value of 3 are both equal to 3, and the absolute value
of 0 is 0.