Algebra Chapter 2 Concept Of Directed Numbers And Numbers Line
Algebra Chapter 2 Directed Numbers
- The numbers which have both magnitudes and directions are called Directed Numbers.
- For example, + 2, + 3, + 6, , – 1, – 2, – 3, etc. are directed numbers.
- The directed number (+2) has magnitude 2 and the direction is from 0 towards the right; (+2) is a directed number.
- Similarly (-3) is also a directed number because its magnitude is 3 and its direction is from 0 towards the left.
Algebra Chapter 2 Absolute Value Of A Number
Absolute Value Of A Number:
- If we omit the sign of a directed number, then only the magnitude of the number is called its absolute value.
- The absolute value of a directed number is a pure number which is always positive.
- We write the absolute value of a number x as |x| (Modulus x).
- By definition,
- \(|x|=\left\{\begin{array}{c}
x, \text { if } x>0 \\
-x, \text { if } x<0 \\
0 \text { if } x=0
\end{array}\right.\) - |+2| = 2 (∵ +2 > 0)
- |0| = 0
- |- 3| = – (- 3) = 3 (∵ – 3 < 0)
- |- 5| = – (- 5) = 5.
- Absolute value is always positive.
WBBSE Class 6 Directed Numbers Notes
Algebra Chapter 2 Opposite Number
Opposite Number:
- In our daily life, we use an infinite number of opposite words.
- For example, the opposite word of Long is Short, and the opposite word of More than is Less than.
- Similarly, the opposite of Small is Large
1. The opposite of Income is Expenditure
2. The opposite of Deposit is Sepnd
3. The opposite of Up is Down
4. The opposite of an Increase is a Decrease
5. The opposite of East is West
6. The opposite of North is South
7. The opposite of the Right side is the Left side
8. The opposite of Credit is Debit, etc. - If one of all these words is taken as positive, then the other will be taken as negative.
- Generally to make out more than or increase ‘+’ (positive or plus sign or addition) sign is used and less than or decrease (negative or minus sign or subtraction sign) sign is used.
- In the case of numbers,
1. The positive of + 2 is – 2
2. The opposite of -2 is + 2
3. The opposite of – 3 is + 3
4. The opposite of + 8 is – 8, etc. - So if we keep the magnitude of a directed number but take the direction of the number in the opposite sense, then we get the opposite directed number.
- The two numbers with opposite signs whose absolute values are the same then one is called the opposite of the other number.
- In general, we can determine the opposite number by putting a ‘+’ sign in place of the sign and by putting a sign in place of the ‘+’ sign.
- Since 0 is not with a positive or negative sign, the opposite number of 0 is 0.
Short Questions on Directed Numbers
Algebra Chapter 2 Opposite Quality
Opposite Quality:
- When we express any number of any value with a unit or by using any sign or symbol, then it is called a quantity.
- For example, x, y, z, a, b, c, ………………., p, q, r,……………., etc. are quantities.
- So ₹ 4, -7 kg, 10 meters, x, y,………etc. are quantities.
- Now we want to determine the opposite quantity of a quantity.
- + ₹ 8 is a quantity.
- Since the opposite number of (+8) is (-8).
- So the opposite quantity of (+ ₹ 8) is (- ₹ 8).
- Again the opposite quantity of ₹ 10 more is ₹ 10 less.
- The opposite quantity of 20 metres North is 20 metres South
- The opposite quantity of 12 metres above is 12 metres down
- The opposite quantity of 14 metres East is 14 metres West
- The opposite quantity of ₹ 6 profit is ₹ 6 loss.
Algebra Chapter 2 Synonymous Quality
Synonymous Quality:
- The synonymous quantity of 50 metres above is (- 50) metres down
- Similarly, the synonymous quantity of 5 metres long is (- 5) metres short
- The synonymous quantity of ₹ 25 profit is (- ₹ 25) loss.
Algebra Chapter 2 General Rule of Addition and Subtraction of Directed Numbers
General Rule of Addition and Subtraction of Directed Numbers:
- Rule 1: If the signs of two given directed numbers are the same, then first add the absolute values of the directed numbers and then put the same sign as that of the given directed numbers before the obtained sum.
- Rule 2: If the signs of two given directed numbers are opposite, then first subtract the absolute values of the given directed numbers and then put the sign of that directed number whose absolute value is greater and that will be the required sum. If the absolute values are the same, then the sum will be 0.
- Rule 3: When a given directed number is to be subtracted from another given directed number, then the result of subtraction will be obtained by the addition of the second number and the absolute value of the first directed number.
Algebra Chapter 2 Number Line
Number Line:
- The numbers 1, 2, 3, 4, etc. are positive integers and -1, -2, -3, -4, etc. are negative integers.
- We also take 0 (zero) as an integer (even integers).
- We take a point O on the straight line XX’ and point O is taken as the origin or 0 (zero).
- With O as the centre, we have to place the respective positive integers at equal intervals on the right side of O on the line OX and the respective negative integers are also to be placed at equal intervals on the left side of O on the line OX’.
- Line XX’ is called the Number Line.
- Now, place the points A, B, C, D, etc. on the right side of 0 on the XX’ line
- such that OA = AB = BC = CD = OA is taken as the. unit length.
- Denote the points A, B, C, D, etc. by the positive integers 1, 2, 3, 4, …….
- Again, in the same way, point A’ is placed on the left side of 0 such that OA’ = OA. A’ is denoted by (- 1).
- Now place the points B’, C’, D’, on OX’ such that OA’ = A’B’ = B’C’ = C’D’ =……….., and they are denoted by – 2, – 3, – 4, …… etc., the negative integers.
Properties Of Number Line:
- The number Line is a straight line
- A point O is marked on the number line, generally, it is placed in the middle of the number line and it is taken as 0 (zero)
- All the positive numbers are written on the right side of the O
- All the negative numbers are written on the left side of the O
- The number 0 (zero) is neither positive nor negative
- As we proceed from O towards the right, the magnitudes of the numbers are increasing
- As we proceed from O towards the left, the magnitudes of the numbers are decreasing
- The value of any number towards the right of a number on the number line is always greater than the number and the value of any number towards the left of a number on the number line is less than the number
- Two numbers which are indicated by two equidistant points from O on both sides of it on the number line are equal in magnitude (absolute values) but with opposite signs.
- These numbers are called opposite numbers.
- The density of real numbers on the number line is so large that between any two numbers on the number line within a small distance, there exist an infinite number of real numbers.
Algebra Chapter 2 Addition And Subtraction Of Directed Numbers With The Help Of Number Line
You have already learned how to place a directed number on the number line. Now we shall discuss the addition and subtraction of the directed numbers with the help of a number line.
Addition with the help of a number line :
The addition of directed numbers may be of four types:
- Addition of a positive number to a positive number ;
- Addition of a negative number to a positive number ;
- Addition of a positive number to a negative number ;
- Addition of a negative number to a negative number.
Common Questions About Positive and Negative Numbers
1. Addition of a positive number to a positive number :
Example : (+4) + (+6) =?
Here we shall find the sum of two directed numbers (+4) and (+6).
Place 0 which indicates the number 0 (zero) on the number line XX’ as shown in the figure.
Moving 4 units to the right of O (i.e., in the positive direction), we get the place of the directed number (+4).
Let this point be denoted by A.
Then move further 6 units in the same direction. So total units of movement from O is 10 units in the positive direction and this is the position of the directed number (+10).
Let this point be denoted by B.
OA = + 4 and AB = + 16
(+ 4) + (+ 6) = OA + AB = OB (according to the figure) = + 10
∴ (+ 4) + (+ 6) = (+ 10).
So the required sum = +10.
(+4) + (+6) = +10.
2. Addition of a negative number to a positive number :
Example : (+ 7) + (- 3) = ?
Here we have to add a negative directed number (- 3) to a positive directed number (+ 7).
Place O which indicates the number 0 (zero) on the number line XX’ as shown in the figure.
Moving 7 units to the right of O (i.e., in the positive direction), we get the position of the directed number (+ 7).
Let this point be denoted by A.
hen moving 3 units to the left of this point A (here we have to move back 3 units because the addition of (- 3) implies subtraction of (+ 3) or in other words come back 3 units in the negative direction), we get a point B (in the figure) which is 4 units from O in the positive direction and this is the position of the directed number (+4).
(+ 7) + (- 3) = (+ 4) = 4.
(+ 7) + (- 3) = OA + (- AB)
= OA – AB = OB
= (+ 4) = 4.
So the required sum = 4.
(+ 7) + (- 3) = 4.
Practice Questions on Directed Numbers and Number Line
3. Addition of a positive number to a negative number
Example : (- 7) + (+ 4) =?
Here we have to add a positive directed number (+4) to a negative directed number (-7).
Place O which indicates the number 0 (zero) on the number line XX’ as shown in the figure.
Moving 7 units to the left of O (i.e., the negative direction), we get the position of the directed number (- 7).
Let this point be denoted by A.
Then we move back 4 units to the right of this point (Here we have to move back 4 units because the addition of (+ 4) implies the movement of 4 units towards the right i.e., in the positive direction), and we get a point which is 3 units from O in the negative direction.
This is the position of the directed number (- 3). Let this point.
∴ (- 7) + (+ 4) = [- (7 – 4)]
= (- 3)
= – 3.
(- 7) + (+ 4) = – OA + AB
= – (OA – AB)
= – OB = – 3
The required sum = – 3.
(- 7) + (+ 4) = – 3.
4. Addition of a negative number to a negative number :
Example : (- 3) + (- 2) = ?
Place O which indicates the number 0 (zero) on the number line XX’ as shown in the figure.
Moving 3 units to the left of O (i.e., in the negative direction), we get the position of the directed number (- 3).
Let this point be denoted by A.
Then we move 2 units from point A on the number line in the same direction i.e., in the negative direction.
Let this point be denoted by point B. So total units of movement from O is 5 in the negative direction and this is the position of the directed number (- 5).
(-3) +(-2) = [-(3.+ 2)]
= (-5)
=-5
∴ (- 3) + (- 2) = – [OA + AB]
= (- OB)
= (- 5)
= – 5
The required sum = – 5.
(- 3) + (- 2) = – 5
Conceptual Questions on Number Line Operations
2. Subtraction of Directed Numbers
The subtraction of directed numbers may be of 4 types :
- Subtraction of a positive number from a positive number
- Subtraction of a negative number from a positive number
- Subtraction of a positive number from a negative number
- Subtraction of a negative number from a negative number.
Since the operation of subtraction is a reverse operation of addition, the operation of the subtraction can be performed through the operation of addition by changing the sign only.
1. Subtraction of a positive number from a positive number :
Example : (+ 10) – (+ 3) = ?
Solution : (+ 10) – (+ 3) = (+ 10) + (- 3)
[∵ The operation of subtraction is a reverse operation of addition and opposite number of (+ 3) = (- 3)]
= [+ (10 – 3)]
= (+ 7)
= 7.
(+ 10) – (+ 3) = 7.
2. Subtraction of a negative number from a positive number:
Example : (+ 3) – (- 9) =?
Solution : (+ 3) – (- 9) = (+ 3) + (+ 9)
(∵ Opposite number of – 9 = + 9)
= [+ (3 + 9)]
= (+ 12)
= 12.
(+ 3) – (- 9) =12.
Examples of Operations with Directed Numbers
3. Subtraction of a positive number from a negative number :
Example : (- 5) – (+ 7) = ?
Solution : (- 5) – (+ 7)
= (- 5) + (- 7) [∵ opposite number of + 7 is – 7].
= [- (5 + 7)] (∵ according to the rule of addition)
= (-12)
= – 12
(- 5) – (+ 7) = – 12
4. Subtraction of a negative number from a negative number
Example : (- 10) – (- 8) =?
Solution : (- 10) – (- 8) = (- 10) + (+ 8) [∵ opposite number of – 8 is + 8]
= [- (10 – 8)] [∵ according to the rule of addition]
= (-2)
= -2.
(- 10) – (- 8) = -2.
Algebra Chapter 2 Natural Numbers Positive Integers Negative Integers Integers
- The numbers which are positive integers i.e., the numbers 1, 2, 3, 4, …… up to infinity is called Natural Numbers.
- The natural numbers consecutively are placed equidistantly on the number line to the right side of the number 0 (zero).
- The integers which are greater than zero (0) are called Positive Integers.
- The positive integers are 1, 2, 3, to infinity.
- In fact, natural numbers and positive integers are the same.
- So the positive integers are also placed on the number line after the number zero.
- The integers which are less or smaller than zero (0) are called Negative Integers.
- The negative integers are written by putting a minus sign (-) towards the left side of the positive integers.
- So the negative integers are:
- -∞ (minus infinity), – 3, – 2, – 1.
- The negative integers start from (-1) and are going towards the left side of zero up to infinity.
- These numbers are placed on the number line towards the left side of the number 0 (zero).
- The number 0 (zero) is neither positive nor negative.
- The negative integers, zero, and positive integers together are called Integers.
- All the natural numbers including zero together are also Integers.
- Zero (0) is called an even integer.
Important Definitions Related to Directed Numbers
Algebra Chapter 2 Verification of Associative law and Commutative law of addition
Associative Law of Addition :
- If a, b, and c are integers (positive or negative), then (a + b) + c = a + (b + c).
- This is the Associative law of addition.
- For example (+ 4), (- 5) and (+ 2) are 3 integers, then we get,
- {(+ 4) + (- 5)} + (+ 2)
= (4 – 5) + 2
= – 1 + 2
= 1 ; and
(+ 4) + {(- 5) + (+ 2)}
= (+ 4) + (- 5 + 2)
= 4 + (- 3)
= 4 – 3
= 1.
∴ {(+ 4) + (- 5)} + (+ 2) = (+ 4) + {(- 5) + (+ 2)} - So, (+ 4), (- 5) and (+ 2) be 3 given integers and they obey the associative law of addition.
- In the same way, we can prove that any three integers obey the additive associative law.
Commutative law of addition:
- If a and b be any two integers, then a + b = b + a.
- This is the commutative law of addition.
- For example, (+ 4) and (- 7) be two integers.
- Then we get, (+ 4) + (- 7) = 4 – 7 = – 3 and
(- 7) + (+ 4)
= – 7 + 4
= – 3.
∴ (+ 4) + (- 7) = (- 7) + (+ 4). - So the given two integers obey the commutative law of addition.
- In the same way, we can prove that any two integers always obey the commutative law of addition.