Integers

Integers:

( Learning outcome )

After studying this post, Readers Will be able to Know that ;

1-  The natural numbers(قدرتی اعداد) 1, 2, 3, 4,5,6,......., are also called positive(مثبت) integers and the corresponding negative numbers (منفی) —1, —2, —3, -4, -5,...... ..., are called negative integers, and zero `0' is an integer which is neither positive(نہ ہی مثبت) nor negative(نہ ہی منفی). 

2- Recognize integers. (صحیح اعداد کی پہچان)

3- Represent integers on a number line (نمبر لاین)

4-  Know that on the number line any number lying to the right of zero (0) is positive (مثبت)and to the left of zero(0) is negative(منفی). Readers also know that to the right of another number is greater(بڑا), to the left of another number is smaller(چھوٹا). 

5- Readers know that every positive integer is greater than a negative integer.

6- Readers Know that every negative integer is less than a positive integer. 

7- Arrange a given list of integers in ascending (ترتیب صعودی)and descending order(ترتیب نزولی). 

8- Define absolute or numerical value (عددی قیمت) of a number as its distance from zero on the number line which is always positive. 

9- Arrange the absolute or numerical values of the given integers in ascending (تریب صعودی) and descending order. 

10- Use number line to display: 

(A) sum(جمع) of two or more given negative integers(منفی صحیح اعداد), 

(B) Difference (فرق)of two given positive integers(مثبت صحیح اعداد),

(C)  Sum of two given integers(صحیح اعداد).

11-  Add two integers (with like signsایک جیسی علامت والے ) in the following three steps:

 a) Take absolute values of given integers(صحیح اعداد), 

b) Add the absolute values,

 c) Give the result(نتیجہ) with the common sign, 

12- Add two integers (with unlike signs مختلف علامت والے) in the following three steps: 

a) Take absolute values of given integers,

 b) Subtract (تفریق) the smaller absolute value from the larger,

 c) Give the result (نتیجہ) with the sign of the integer with the larger absolute value. 

13- Recognize subtraction as the inverse(الٹ) process of addition. 

14- Subtract one integer from the other by changing (تبدیل کرنا) the sign of the integer being subtracted and adding according to the rules for addition of integers. 

15- Readers recognize that :

The product (حاصل ضرب)of two integers of like signs is a positive integer.

 The product (حاحل ضرب) of two integers of unlike signs is a negative integer. 

Recognize that division (تقسیم) is the inverse process of multiplication. 

Recognize (پہچان) that on dividing(تقسیم) one integer by another.

-if both the integers have like signs the quotient( حاصل تقسیم) is positive.

-if both the integers have unlike signs(مختلف علامات) the quotient is negative. 

16- Know that division of an integer by zero '0 is not possible(نا ممکن). 

Integers:

Positive Integers: مثبت صحیح اعداد

Every one is familiar with the numbers like 1,2,3,4,5,.... These numbers can be added(جمع), subtracted(تفریق), multiplied(ضرب) and divided (تقسیم) as the rules expressed in previous classes. These numbers are known as natural numbers(قدرتی اعداد). These numbers are also called or known as positive Integers.

Negative Integers: منفی صحیح اعداد 

All of us know that the natural numbers which are starting form 1 , All the natural numbers are greater than zero (0). If we think is there any number which is less than zero (0)? We understand this by an example. If we subtract 4 from 7 then the result is 3 i.e.  7-4= 3. But what is the result if 4 - 7 = ???

So remember that there is an other type of numbers which are called negative numbers because if we subtract 4 - 7 then the result is -3. So there are two types of numbers ( by sign) positive numbers and negative numbers, further we can understand these numbers by a number line

Hance zero (0) is known as starting point(نقطہ آغاز), so these positive and negative numbers with zero are called integers.

Examples: 

5-8 = -3  , 3-4 = -1  , 3-5 = -2 and so...

These numbers -1, -2 ,-3 are called negative integers(منفی صحیح اعداد).

We know that 1 is less than 2 and 0 is less then 1. In the same way 0 is greater than -1 and -1 is greater than -2.

So again we understand this, the whole numbers(مکمل اعداد) 0,1,2,3,4.... together with negative numbers ( -1,-2,-3,-4......) are called integers.

We can write them as;

Note:

Zero (0) is also an integer but it is neither positive integer nor a negative Integer.

Integers are also known as directed numbers because these numbers also represent the direction(سمت) as well as measurement(پمائش).

Ordering of Integers: صحیح اعداد کی ترتیب

We note that on a number line, the integers that lies to the left of another Integers is always(ہمیشہ) smaller and the integer that lies to the right of the same integer is always greater. For example zero (0) lies to the left of all positive Integers on the number line. So 

0 ≺ 1,2,3,4,5,....

And the integer zero (0) lies to the right of all negative integers. So

0 > -1,-2-3,-4,......

We can write the above statements as

.....-5 < -4 < -3< -2< -1 < 0 < +1< +2 < +3< +4........

Absolute or Numerical value of an Integer: صحیح اعداد کی عددی قیمت

Numerical or absolute (عددی قیمت) value of a number is distance from zero (0) to that number on the number line. If we find the absolute or Numerical value of 4 and -4 ,we note that 4 and -4 both the numbers are at equal distance(فاصلہ) from zero (0). So both have equal absolute or Numerical value that is 4. We write this as

| +4 | = 4  and | -4 | = - ( -4 ) = 4

Now we solve some examples:

Find numerical values of following

-6 ,-8 , 12

So | -6 | = - ( - 6 ) = 6  numerical value of -6 is 6

 | -8 | = - ( -8 ) = 8  numerical value of -8 is 8

| 12 | = 12  numerical value of 12 is 12

Addition of Integers: جمع

The Integers with like sign and unlike sign are added in the following way

Subtraction of Integers: تفریق

Subtraction is the opposite process of addition, the integers are subtract from each other as following,

Multiplication of Integers: ضرب 

We should know that when signs (+,-,×,÷) are multiply each other then what will be the result

Two like signs multiply each other then

(+) × (+) = +

(-) × (-) = +

When opposite signs multiply each other then

(+) × (-) = -

(-) × (+) = -

Now we multiply the integers as following

Division of Integers: تقسیم 

In division we follow the same rule of multiplication for signs.

Examples, 

Division of an integer by zero (0) is impossible(ناممکن).

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