Category : 7th Class
INTEGERS
FUNDAMENTALS
N= (1, 2, 3, 4,............)
Elementary Question - 1: Which is the smallest natural number?
Ans.: 1
Elementary Question - 2: Which is the smallest whole number?
Ans.: 0
W= (0, 1, 2, 3,............)
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:
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
Note:
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:
For \[a,b\in Z,a+b\in Z,a-b\in Z\] \[and\text{ }a\times b\in Z.\]
If \[a,b\in Z,\]the a + b = b + a and \[a\times b\text{ }=b\times a.\]
If \[a,\text{ }b,\text{ }c\text{ }\in \text{ }Z.\]then a + (b + c) = (a + b) + c and \[a\times \left( b\times c \right)=\left( a\times b \right)\times c.\]
For a, b and \[c\in Z,\text{ }a\text{ }\left( b+c \right)=ab+ac\]and \[a\left( b-c \right)=ab-ac.\]
For \[a\in Z,a+0=a=0+a\]and \[a\times 1=a=1\times a.\]
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.
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