Integers

Category : 7th Class

INTEGERS

 

FUNDAMENTALS

  • In lower classes, you would have read about counting number. 1, 2, 3,.........
  • They are called natural numbers (N).

N= (1, 2, 3, 4,............)

 

Elementary Question - 1: Which is the smallest natural number?

Ans.: 1

 

  • Representation of natural numbers on a number line. To represent natural numbers on a number line, we should draw a line and write the number at equal distances on it as shown below:

 

 

 

  • Whole Number (W): The set of natural numbers together with zero is known as the set of whole numbers.

 

Elementary Question - 2: Which is the smallest whole number?

Ans.: 0

W= (0, 1, 2, 3,............)

  • In set notation, set of whole numbers (W) = set of natural numbers (N) + zero; { } is used for set notation.
  • Integers (Z): The set containing negatives of natural numbers along with whole numbers is called the set of integers.

That is, \[Z=\{........-4,\,\,-3,\,\,-2,\,\,-1\}\cup \{0,\text{ }1,\,\,2,\,\,3,\,\,4,\,\,5,.....\}\]

Where \[\cup \] denote "union" or combination of these two sets.

 

Concept of Infinity

We can go on adding more and more numbers to the right side of the number line (e.g., 100, 101, ......100000, .........1 crore, ............1000 crores............. m an unending manner upto plus infinity and similarly to the left side of the number line upto minus infinity.

 

 

\[\left( -\,\infty  \right)\]Minus infinity crore  Plus infinity \[\left( +\infty  \right)\]

This very, very large unending number on the right side and left side of number line are called plus infinity \[\left( +\,\infty  \right)\]and minus infinity \[\left( -\,\infty  \right)\] respectively.

\[\left( -\,\infty  \right)\] \[\left( +\,\infty  \right)\]

 

Note:

  1. Usually, negative numbers are placed in brackets to avoid confusion arising due to two signs in evaluations simultaneously,

e.g., \[3+\left( -\,5 \right)=-2\]

2.0 is not included in either \[{{Z}^{+}}\]c or\[{{Z}^{-}}\]. Hence, it is non-negative integer

 

Common use of numbers

(i) To represent quantities like profit, income, increase, rise, high, north, east, above, depositing, climbing and so on, positive numbers are used.

(ii) To represent quantities like loss, expenditure, decrease, fall, low, south, west, below, withdrawing, sliding and so on, negative numbers are used.

(iii) On a number line, when we

  1. Add a positive integer, we move to the right.
  2. Add a negative integer, we move to the left.
  3. Subtract a positive integer, we move to the left.
  4. Subtract a negative integer, we move to the right
  5. Notation e means belongs to; a, b \[\in \] means the numbers a and b belong to \[\] (set of integers)

 

Note

  1. 0 is neither positive nor negative.
  2. The + sign is not written before a positive number
  3. \[\frac{1}{2}\] and 0.3 are not integers as they are neither not whole numbers nor are they negative of natural numbers.

 

Mod of a number

Mod or modulus of a number denotes the positive value of that number.

Thus, |a| = +a if a > 0 and |a| = - a if a < 0

Elementary questions - 3, find |6| - | - 3|

Ans.\[\left| 6 \right|=+\,6\,\And \left| -3 \right|=+\,3\therefore \left| 6 \right|-|-3|=6-3=3.\]

 

A short note on notations

The brilliance of a mathematician or mathematical student lies in his/ her ability to tell more things in less words. To illustrate \[\in \] means "belongs to": means such that; \[\forall \] means for all. For example, to represent that x is a natural number. We write \[x\in \,N;\]

Further, \[\subset \] means is a "subset of'. A subset is a smaller set wholly contained in larger set.

For e.g., if A = {1, 2, 3} and B = {1, 2, 3, 4, 5} then \[A\subset B\]i.e., A is a subset of B.

Elementary question - 4: Is set of natural numbers, a subset of integers?

Answer: Yes, it is represented as\[N\subset Z\]

 

Properties of integers:

  • Closure property: Closure property is satisfied with respect to addition, subtraction and multiplication in the set of integers.

For \[a,b\in Z,a+b\in Z,a-b\in Z\] \[and\text{ }a\times b\in Z.\]

  • Commutative property: Commutative property is satisfied with respect to addition and multiplication in the set of integers.

If \[a,b\in Z,\]the a + b = b + a and \[a\times b\text{ }=b\times a.\]

  • Associative property: Associative property is satisfïed with respect to addition and multiplication m the set of integers.

If \[a,\text{ }b,\text{ }c\text{ }\in \text{ }Z.\]then a + (b + c) = (a + b) + c and \[a\times \left( b\times c \right)=\left( a\times b \right)\times c.\]

  • Distributive property: Multiplication is distributed over addition and subtraction in the set of integers.

For a, b and \[c\in Z,\text{ }a\text{ }\left( b+c \right)=ab+ac\]and \[a\left( b-c \right)=ab-ac.\]

  • Identity element: 0 is the identity dement under addition and 1 is the identity element under multiplication.

For \[a\in Z,a+0=a=0+a\]and \[a\times 1=a=1\times a.\]

  • Multiplication by zero: For any integer a, \[a\times 0=0\times a=0.\]
  • Additive inverse: For an integral number 'a', \[\left( -a \right)\]is the additive Inverse.
  • Multiplicative Inverse: For an integer 'a', \[\frac{1}{a}\] is the multiplicative inverse.

 

Elementary Question - 5:

Show by a practical example that closure property is not satisfied with respect to division in the set of integers.

Ans. Consider two numbers 2 and 3

Let us do \[2\div 3=\frac{2}{3}\]

Now. \[2\in Z,\,\,3\in Z\]

But \[\frac{2}{3}\cancel{\in }\,Z\] as \[\frac{2}{3}\] is not an integer.

Notes - Integers
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