ICSE Class 6 Maths Chapter 02 Integers

Chapter 2: Integers

Including Absolute Values and Rules for four fundamental operations on Integers

2.1 Integers

Integers consist of:

(i) the negatives of all natural numbers, i.e. -4, -3, -2, -1

(ii) zero (0).

(iii) natural numbers, i.e. 1, 2, 3, 4

Thus, collection of integers

I = {..., -4, -3, -2, -1, 0, 1, 2, 3, 4, ...}

Negatives of natural numbers | Zero | Natural numbers

Clearly,

(a) 1, 2, 3, 4, 5, etc. are positive integers.

(b) -1, -2, -3, -4, -5, etc. are negative integers.

(c) Zero (0) is neither a positive integer nor a negative integer.

(d) 0, 1, 2, 3, 4, 5, etc. are non-negative integers.

2.2 Absolute Value Of An Integer

The absolute value of an integer is its numerical value regardless of its sign.

For example:

The absolute value of -8 = 8, the absolute value of 4 = 4 and so on.

To represent the absolute value of an integer, write the integer in between two vertical line segments (bars).

Thus, absolute value of -68 = |-68| = 68,

absolute value of +47 = |+47| = 47 and so on.

Therefore, if 'a' represents an integer, its absolute value is represented by |a| and is always non-negative.

1. The absolute value of a negative number is always greater than the number e.g. Absolute value of -10 = |-10| = 10, which is greater than -10.

2. The absolute value of a positive number is always equal to the number itself. e.g. Absolute value of 10 = |10| = 10

3. When the absolute value of a number is the number itself, the number under consideration is either zero or positive. e.g. |0| = 0, |5| = 5, |16| = 16 and so on.

Teacher's Note

Understanding absolute value helps students compare debts and assets in real life - whether you owe 10 rupees or have 10 rupees, the amount is the same when considering magnitude alone.

Example 1:

Evaluate:

(i) |-15| (ii) -|15| (iii) -|-15| (iv) |7| + |-3| (v) |8| - |-6|

Solution:

(i) |-15| = 15 (Ans.)

(ii) Since |15| = 15 .. -|15| = -15 (Ans.)

(iii) Since |-15| = 15 .. -|-15| = -15 (Ans.)

(iv) Since |7| = 7 and |-3| = 3

.. |7| + |-3| = 7 + 3 = 10 (Ans.)

(v) Since |8| = 8 and |-6| = 6

.. |8| - |-6| = 8 - 6 = 2 (Ans.)

2.3 Comparing Integers

1. Every positive integer is greater than every negative integer and 0.

2. Every negative integer is smaller than every positive integer and 0.

3. Zero (0) is greater than every negative integer and smaller than every positive integer.

4. Out of two negative integers, the one with the smaller absolute value is greater and the one with greater absolute value is smaller.

For example:

Since absolute value of -15 = |-15| = 15

and absolute value of -23 = |-23| = 23

.. -15 > -23 as |-15| < |-23|

Similarly, -65 < -47 as |-65| > |-47|

Example 2:

Compare the integers:

(i) (-12) and (-15) (ii) (+12) and (-15) (iii) (-12) and (+15)

(iv) (+12) and 0 (v) 0 and (-12).

Solution:

(i) Since (-12) and (-15) are both negative and |-12| < |-15|, as 12 < 15

.. (-12) > (-15) (Ans.)

(ii) Since every positive integer is greater than every negative integer,

therefore (+12) > (-15) (Ans.)

(iii) Since every negative integer is smaller than every positive integer,

therefore (-12) < (+15) (Ans.)

(iv) Since every positive integer is greater than zero (0),

therefore (+12) > 0 (Ans.)

(v) Since 0 is greater than every negative integer,

therefore 0 > (-12) (Ans.)

Teacher's Note

Comparing integers is like comparing temperature changes - understanding which direction is "greater" helps interpret weather forecasts and body temperature readings correctly.

2.4 Use Of Integers As Directed Numbers

Positive and negative integers are together called directed numbers. Directed numbers can be used in several ways.

For example:

1. If moving 5 m towards the East is represented by +5, then -5 represents moving 5 m towards the West, i.e. in the opposite direction of East.

2. If +4 represents 4 m towards the North, then -6 represents 6 m towards its opposite direction i.e. towards the South.

Thus, if a positive integer represents a particular direction, then the negative integer represents its opposite direction.

Conversely, if a negative integer represents a particular direction, the positive integer represents its opposite direction.

In the same way:

Integers are used to express our day-to-day situations in mathematical terms:

(i) If profits are represented by positive integers, then losses are shown by negative integers.

(ii) If heights above the sea level are represented by positive integers, then depths below the sea level are shown by negative integers.

(iii) If a rise in prices is represented by positive integers, then a fall in prices is shown by negative integers and so on.

(iv) If 5 m above the earth's surface is represented by +5, then 15 m below the earth's surface is represented by -15.

(v) If +10 represents a profit of ₹ 10, then -25 represents a loss of ₹ 25.

(vi) If -56 indicates a giving of ₹ 56, then taking of ₹ 85 is denoted by +85.

(vii) If a rise in temperature by 32° C is denoted by +32, then -15 indicates a fall in temperature by 15° C.

Example 3:

A man moves 20 m due East and then 16 m due West. Find his position with respect to his starting point.

Solution:

If 20 m due East is represented by +20, then -16 represents 16 m due West.

On adding +20 and -16, we get:

(+20) + (-16) = +20 - 16

= +4, which is positive

.. The position of the man with respect to his starting point is 4 m due East (Ans.)

Teacher's Note

Using directed numbers to track position helps when following GPS navigation or understanding relative movement in sports and transportation.

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