n this first article of our topic ‘Numbers’, we deal with ‘Numbers’
themselves. What are numbers ? Where do we see and meet them ? Are they of
different types ? Can we learn these distinct types ?
Well, numbers are all around us, floating all over the place. They demand our attention and careful evaluation. A number here or there can make a huge difference. In face a misplaced zero or decimal is the difference between one being poor or rich. Such being their importance, it is critical we begin with the most basic concepts in Mathematics: types of numbers.
In Hindu Arabic System, we use ten symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 called digits to represent any number. This is the decimal system where we use the numbers 0 to 9. '0' is called insignificant digit.
Types of Numbers
-Whole numbers : If we include 0 among the natural numbers, then the numbers 0, 1, 2, 3, 4, 5 ... are called whole numbers.The set of whole number can be represented by W= {0, 1, 2, 3, 4, 5...}. Every natural number is a whole number but 0 is a whole number which is not a natural number.
-Integer : All counting numbers and their negatives including zero are known as integers.
The set of integers can be represented by Z or I = {...-4,-3,-2,-1, 0, 1, 2, 3, 4 ...}
-Positive Integers :The set I = {1, 2, 3, 4...} is the set of all positive integers. Positive integers and natural numbers are synonyms.
-Negative Integers : The set I = {-1, -2, -3...} is the set of all negative integers. 0 is neither positive nor negative. Non-negative Integers : The set {0, 1, 2, 3...} is the set all non-negative integers.
-Rational Numbers : The numbers of the form p/q, where p and q are integers and q ≠ 0, are known as rational numbers, e.g. 4/7, 3/2, -5/8, 0/1, -2/3, etc. The set of all rational numbers is denoted by Q. i.e. Q ={x:x =p/q; p, q belong to I,q≠0}. Since every natural number ‘a’ can be written as a/1, every natural number is a rational number. Since 0 can be written as 0/1 and every non-zero integer ‘a’ can be written as a/1, every integer is a rational number. Every rational number has a peculiar characteristic that when expressed in decimal form is expressible rather in terminating decimals or in non-terminating repeating decimals. For example, 1/5 =0.2, 1/3 = 0.333...22/7 = 3.1428704287,
8/44 = 0.181818...., etc. The recurring decimals have been given a short notation as 0.333.... = 0.3
4.1555... = 4.05, 0.323232...= 0.32.
-Irrational Numbers : Those numbers which when expressed in decimal from are neither terminating nor repeating decimals are known as irrational number, e.g. √2, √3, √5, π, etc.
Note that the exact value of is not 22/7. 22/7 is rational while π irrational number. 22/7 is approximate value of π. Similarly, 3.14 is not an exact value of it.
-Real Numbers : The rational and irrational numbers combined together are called real numbers, e.g.13/21, 2/5, -3/7;√3, 4 +√2, etc. are real numbers.The set of all real numbers is denoted by R.
Note that the sum, difference or product of a rational and irrational number is irrational, e.g. 3+√2, 4 -√3, 2/3-√5, 4√3,-7√5 are all irrational.
-Even Numbers : All those numbers which are exactly divisible by 2 are called even numbers.
e.g.2, 6, 8, 10, etc., are even numbers.
-Odd Numbers : All those numbers which are not exactly divisible by 2 are called odd numbers, e.g. 1,3, 5, 7 etc., are odd numbers.
-Prime Numbers : A natural number other than 1 is a prime number if it is divisible by 1 and itself only. For example, each of the numbers 2, 3, 5, 7 etc., are prime numbers. 2 is the only even number which is prime number. Two numbers which have only 1 as the common factor are called co-relatively prime to each other, e.g. 3 and 5 are co-primes.
Prime numbers up to 100 are: 2, 3, 5, 7, 11, 13, 17, 19,23, 29, 31, 37, 41,43, 47, 53, 59,
61, 67, 71, 73, 79, 83, 89, 97, i.e. 25 prime numbers between 1 and 100.
-Composite Numbers : Natural numbers greater than 1 which are not prime, are known as composite
numbers. For example, each of the numbers 4, 6, 8, 9, 12, etc., are composite numbers. The number 1 is neither a prime number nor composite number.
Some more properties on natural numbers :
Well, numbers are all around us, floating all over the place. They demand our attention and careful evaluation. A number here or there can make a huge difference. In face a misplaced zero or decimal is the difference between one being poor or rich. Such being their importance, it is critical we begin with the most basic concepts in Mathematics: types of numbers.
In Hindu Arabic System, we use ten symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 called digits to represent any number. This is the decimal system where we use the numbers 0 to 9. '0' is called insignificant digit.
Types of Numbers
- Natural numbers
- Whole numbers
- Integers
- Positive Integers
- Negative Integers
- Non-negative Integers
- Rational Numbers
- Irrational Numbers
- Real Numbers
- Even Numbers
- Odd Numbers
- Prime Numbers
- Composite Numbers
-Whole numbers : If we include 0 among the natural numbers, then the numbers 0, 1, 2, 3, 4, 5 ... are called whole numbers.The set of whole number can be represented by W= {0, 1, 2, 3, 4, 5...}. Every natural number is a whole number but 0 is a whole number which is not a natural number.
-Integer : All counting numbers and their negatives including zero are known as integers.
The set of integers can be represented by Z or I = {...-4,-3,-2,-1, 0, 1, 2, 3, 4 ...}
-Positive Integers :The set I = {1, 2, 3, 4...} is the set of all positive integers. Positive integers and natural numbers are synonyms.
-Negative Integers : The set I = {-1, -2, -3...} is the set of all negative integers. 0 is neither positive nor negative. Non-negative Integers : The set {0, 1, 2, 3...} is the set all non-negative integers.
-Rational Numbers : The numbers of the form p/q, where p and q are integers and q ≠ 0, are known as rational numbers, e.g. 4/7, 3/2, -5/8, 0/1, -2/3, etc. The set of all rational numbers is denoted by Q. i.e. Q ={x:x =p/q; p, q belong to I,q≠0}. Since every natural number ‘a’ can be written as a/1, every natural number is a rational number. Since 0 can be written as 0/1 and every non-zero integer ‘a’ can be written as a/1, every integer is a rational number. Every rational number has a peculiar characteristic that when expressed in decimal form is expressible rather in terminating decimals or in non-terminating repeating decimals. For example, 1/5 =0.2, 1/3 = 0.333...22/7 = 3.1428704287,
8/44 = 0.181818...., etc. The recurring decimals have been given a short notation as 0.333.... = 0.3
4.1555... = 4.05, 0.323232...= 0.32.
-Irrational Numbers : Those numbers which when expressed in decimal from are neither terminating nor repeating decimals are known as irrational number, e.g. √2, √3, √5, π, etc.
Note that the exact value of is not 22/7. 22/7 is rational while π irrational number. 22/7 is approximate value of π. Similarly, 3.14 is not an exact value of it.
-Real Numbers : The rational and irrational numbers combined together are called real numbers, e.g.13/21, 2/5, -3/7;√3, 4 +√2, etc. are real numbers.The set of all real numbers is denoted by R.
Note that the sum, difference or product of a rational and irrational number is irrational, e.g. 3+√2, 4 -√3, 2/3-√5, 4√3,-7√5 are all irrational.
-Even Numbers : All those numbers which are exactly divisible by 2 are called even numbers.
e.g.2, 6, 8, 10, etc., are even numbers.
-Odd Numbers : All those numbers which are not exactly divisible by 2 are called odd numbers, e.g. 1,3, 5, 7 etc., are odd numbers.
-Prime Numbers : A natural number other than 1 is a prime number if it is divisible by 1 and itself only. For example, each of the numbers 2, 3, 5, 7 etc., are prime numbers. 2 is the only even number which is prime number. Two numbers which have only 1 as the common factor are called co-relatively prime to each other, e.g. 3 and 5 are co-primes.
Prime numbers up to 100 are: 2, 3, 5, 7, 11, 13, 17, 19,23, 29, 31, 37, 41,43, 47, 53, 59,
61, 67, 71, 73, 79, 83, 89, 97, i.e. 25 prime numbers between 1 and 100.
-Composite Numbers : Natural numbers greater than 1 which are not prime, are known as composite
numbers. For example, each of the numbers 4, 6, 8, 9, 12, etc., are composite numbers. The number 1 is neither a prime number nor composite number.
Some more properties on natural numbers :
- Sum of 1st n Natural number i.e 1 + 2 + 3 + …. + n = {n (n + 1)} /2
- Sum of squares of 1st n natural number i.e { n(n+l)(2n+l) }/6
- Sum of cubes of 1st n Natural number i.e [{n(n+1)}/2]^2
- Sum of squares of 1st n Odd natural number i.e {n (2n -1 )(2n + 1)}/3
- Sum of squares of 1st n Even natural number i.e {2n(n+1)(2n+1)}/3
- Sum of cubes of 1st n Odd natural number i.e n^2 (2n^2 -1)
- Sum of cubes of 1st n Even natural number i.e 2 [n (n + 1)]^2
- From 1 – 9, the digit 2 is used 1 time
- From 1 – 99, the digit 2 is used 20 times.
- From 1 – 999, the digit 2 is used 300 times
- From 1 – 9999, the digit 2 is used 4000 times and so on
Well, there are two series of three’s here: 3, 13,…33,…93 = ten 3s at the unit’s place. 30,31…39= ten 3s at the ten’s place. So a total of 20 threes make an appearance here.
6 when multiplied with 7 give 4 and 2
6 x 7 = 42
66 x67 = 4422
666 x 667 = 444222
6666 x 6667 = 44442222
For multiplication such as the above : The number of digits in both the numbers is same , the 2nd number has one 6 less than the number of 6’s the 1st number has . The number of 4’s and 2’s is equal to the number of 6’s in the 1st number
Cyclic numbers
142857 is called as the cyclic number, since its digits are rotated around when multiplied by any number from 1 to 6
142857 x 1 = 142857
142857 x 5= 714285
142857 x 4= 571428
142857 x 6 = 857142
142857 x 2= 285714
142867 x 3 = 428571
Magical 1
3 x 37 = 111
33 x 3367 = 111,111
333 x 333667 = 111,111,111
3333 x 33336667= 111,111,111,111
33333 x 3333366667= 111,111,111,111,111
333333 x 333333666667 = 111,111,111,111,111,111
Consider the L.H.S of the above numbers. The number of 3s in both the numbers are same and 6 is one less than the number of 3 and there is only one 7. Then in the final product, N would appear three times the number of threes in the first number on the LHS.
Some Other Properties of Numbers
Squares: There is a definite relationship between the unit digits when square of a number is considered, we will see that one by one.
If unit digit of a perfect square is I then ten’s digit has to be Even (e.g. 81, 121, 441 etc)
If unit digit of a number is 2 then it can’t be a perfect square.
If unit digit of a number is 3 then it can’t be a perfect square.
If unit digit of a perfect square is 4 then ten’s digit has to be Even (e.g. 64,144 etc)
If unit digit of a perfect square is 5 then ten’s digit has to be 2 (e.g. 25, 225, 625 etc)
If unit digit of a perfect square is 6 then ten’s digit has to be Odd and multiple of 3.
If unit digit of a number is 7 & 8 then it can’t be a perfect square
If unit digit of a perfect square is 9 then ten’s digit has to be Even (e.g. 49, 169 etc)
If unit digit of a perfect square is 0 then ten’s digit has to be 0 (e.g. 100,400,900 etc)
Hence if a number end with 2, 3, 7 or 8 then it can’t be perfect square.
Properties of Integers : We have studied about integers, the entities which can be shown on the number line is called integers there are some properties of Integers in Nature. Usually these properties are used in the GRE examination
- Among all the numbers positive and negative whole numbers are included in the integers
- Zero is neither rational nor irrational but it is an Integer
- There are some numbers which follow one another which are called as consecutive integers
- There is an Integer to which a number is multiplied the value will never change and that integer is 1
As like BODMAS in CAT exam, GRE believes in PEMDAS and this stands for
P = Parentheses
E = Exponents
M= Multiplication
D = Division
A = Addition
S= Subtraction
TOOLTIP 1: What is the approximate value of N= [{2(4 +5)3 x 4}/10 -23]
Step by step
[{2(9)3 x 4}/10 -23] => [{2(729) x 4}/10 -23] => [{1458 x 4}/10 -23] => [5832/10 -23] => 560.2 = 560 approximate Consecutive Integers: The integers which follow one another are called consecutive integers
For example 3,4,5,6 are consecutive integers. An individual random number never be a consecutive integer.
Some more points about Consecutive Numbers
TOOLTIP 1
If there is odd number of digits in the set of consecutive numbers like set of three consecutive numbers (4,5,6,) or 5 digits are there say (3,4,5,6,7). Then in this case the sum of all integers always divisible by the number of digits of set
For example 2+3+4 = 9 divisible by number of digits i.e. 3
For example 1+2+3+4+5 = 15 divisible by 5
TOOLTIP 2
On the other hand ,If there is even number of digits in set of consecutive numbers like set of four consecutive numbers (4,5,6,7) or 6 digits are there say (3,4,5,6,7,8). Then in this the sums of all integers is never be divisible by the number of digits of the set
For example 2+3+4+5 = 14 never divisible by number of digits i.e. 4
For example 1+2+3+4+5+6 = 21 never divisible by 6
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