Numbers
What is a Number?
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A number is a mathematical value used for counting or measuring or labelling objects. Numbers are used to performing arithmetic calculations. Examples of numbers are natural numbers, whole numbers, rational and irrational numbers, etc. 0 is also a number that represents a null value.
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A number has many other variations such as even and odd numbers, prime and composite numbers. Even and odd terms are used when a number is divisible by 2 or not, whereas prime and composite differentiate between the numbers that have only two factors and more than two factors, respectively.
Mathematical Definition:
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A member of the set of positive integers; one of a series of symbols of unique meaning in a fixed order that can be derived by counting.
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A member of any of the following sets of mathematical objects: integers, rational numbers, real numbers, and complex numbers. These sets can be derived from the positive integers through various algebraic and analytic constructions.
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What is Number System in Maths?
A number system is defined as a system of writing to express numbers. It is the mathematical notation for representing numbers of a given set by using digits or other symbols in a consistent manner. It provides a unique representation of every number and represents the arithmetic and algebraic structure of the figures. It also allows us to operate arithmetic operations like addition, subtraction, multiplication and division.
The value of any digit in a number can be determined by:
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The digit
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Its position in the number
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The base of the number system
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Digits and Numeral:
In Hindu Arabic system, we use ten symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 called digits to represent any number.
A group of digits, denoting a number is called a numeral.
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Example:
123, 5968, 1929384,
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Types of Number Systems
There are different types of number systems in which the four main types are as follows.
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Binary number system (Base - 2)
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Octal number system (Base - 8)
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Decimal number system (Base - 10)
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Hexadecimal number system (Base - 16)
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Place Value or Local Value of a Digit in a Numeral:
In the numeral: 689745132
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Place value of 2 is (2 x 1) = 2;
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Place value of 3 is (3 x 10) = 30;
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Place value of 1 is (1 x 100) = 100 and so on.
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Place value of 6 is 6 x 10 = 600000000
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Face Value:
The face value of a digit in a numeral is the value of the digit itself at whatever place it may be. In the above numeral, the face value of 2 is 2; the face value of value of 3 is 3 and so on
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TYPES OF NUMBERS
Natural Numbers:
Counting numbers 1, 2, 3, 4, 5...... are called natural numbers.
Whole Numbers:
All counting numbers together with zero form the set of whole numbers. Thus,
(i) 0 is the only whole number which is not a natural number.
(ii) Every natural number is a whole number.
Integers:
All natural numbers, 0 and negatives of counting numbers i.e., {…, - 3 , - 2 , - 1 , 0, 1, 2, 3,…..} together form the set of integers.
(i) Positive Integers: {1, 2, 3, 4, …...} is the set of all positive integers.
(ii) Negative Integers: {- 1, - 2, - 3.…..} is the set of all negative integers.
(iii) Non-Positive and Non-Negative Integers: 0 is neither positive nor negative. So, {0, 1,
2, 3,….} represents the set of non-negative integers, while {0, - 1 , - 2 , - 3 ,…..} represents the set of non-positive integers.
Even Numbers:
A number divisible by 2 is called an even number, e.g., 2, 4, 6, 8,10, etc.
Odd Numbers:
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A number not divisible by 2 is called an odd number. e.g., 1, 3, 5, 7,9, 11, etc.
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Prime Numbers:
A number greater than 1 is called a prime number, if it has exactly two factors, namely 1 and the number itself.
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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.
Prime numbers Greater than 100: 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499, 503, 509, 521, 523, 541, 547, 557, 563, 569, 571, 577, 587, 593, 599, 601, 607, 613, 617, 619, 631, 641, 643, 647, 653, 659, 661, 673, 677, 683, 691, 701, 709, 719, 727, 733, 739, 743, 751, 757, 761, 769, 773, 787, 797, 809, 811, 821, 823, 827, 829, 839, 853, 857, 859, 863, 877, 881, 883, 887, 907, 911, 919, 929, 937, 941, 947, 953, 967, 971, 977, 983, 991, 997.
Composite Numbers:
Numbers greater than 1 which are not prime, are known as composite numbers, e.g., 4, 6, 8, 9, 10, 12.
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Note:
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1 is neither prime nor composite.
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2 is the only even number which is prime.
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There are 25 prime numbers between 1 and 100.
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Co-primes:
Two numbers a and b are said to be co-primes, if their H.C.F. is 1. e.g., (2, 3), (4, 5), (7, 9), (8, 11), etc. are co-primes.
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Methods to Find Prime Numbers Easily
There are various methods to determine whether a number is prime or not. The best way for finding prime numbers is by factorization method. By factorization, the factors of a number are obtained and, thus, one can easily identify a prime number.
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Finding Prime Numbers Using Factorization
Factorization is the best way to find prime numbers. The steps involved in using the factorization method are:
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Step 1: First find the factors of the given number.
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Step 2: Check the number of factors of that number.
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Step 3: If the number of factors is more than two, it is not a prime number.
Example:
Take a number, say, 36.
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Now, 36 can be written as 2 × 3 × 2 × 3. So, the factors of 36 here are 1, 2, 3, 4, 6, 9, 12, 18, and 36.
Since the number of factors of 36 is more than 2, it is not a prime number but a composite number.
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Now, if we take the example of 19. The prime factorization of 19 is 1 x 19. You can see here, there are two factors of 19. Hence, it is a prime number.
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Shortcut to Find Prime Numbers
One of the shortcuts to finding the prime numbers are given below.
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Step 1: Write all the numbers from 1 to 100 with 6 numbers in a row (as shown in the figure).
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Step 2: As the square root of 100 is ±10, the multiples of numbers till 10 has to be crossed out.
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Step 3: Choose 2 and cross the entire column as all are multiple of 2. Also, cross out the entire columns of 4 and 6 as those are also 2’s multiples.
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Step 4: Now move to 3 and cross out the entire column.
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Step 5: Take 5 and cross out the diagonally towards left. Then, cross out diagonally from numbers 30, 60, and 90. Now, all the multiples of 5 are crossed out.
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Step 6: Choose 7 and cross out diagonally towards the right. Then, check the next number on that column which is divisible by 7 and cross diagonally right. The first number on that column that is divisible by 7 is 49 and then 91. Crossing diagonally right from 49 to 91 leaves no multiples of 7 on the list.
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Other Methods to Find Prime Numbers:
Prime numbers can also be found by the other two methods using the general formula. The methods to find prime numbers are:
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Method 1:
Two consecutive numbers which are natural numbers and prime numbers are 2 and 3. Apart from 2 and 3, every prime number can be written in the form of 6n + 1 or 6n – 1, where n is a natural number.
For example:
6(1) – 1 = 5
6(1) + 1 = 7
6(2) – 1 = 11
6(2) + 1 = 13
6(3) – 1 = 17
6(3) + 1 = 19…..so on
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Method 2:
To find the prime numbers greater than 40, the general formula that can be used is n2+ n + 41, where n are natural numbers 0, 1, 2, …., 39.
For example:
(0) + 0 + 0 = 41
(1) + 1 + 41 = 43
(2) + 2 + 41 = 47
(3) + 3 + 41 = 53
(4) + 2 + 41 = 59…..so on
Note: These both are the general formula to find the prime numbers. But values for some of them will not yield a prime number.
2
2
2
2
2
TESTS OF DIVISIBILITY
Divisibility By 2:
A number is divisible by 2, if its unit's digit is any of 0, 2, 4, 6, 8.
Ex. 84932 is divisible by 2, while 65935 is not.
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Divisibility By 3:
A number is divisible by 3, if the sum of its digits is divisible by 3.
Ex.592482 is divisible by 3, since sum of its digits = (5 + 9 + 2 + 4 + 8 + 2) = 30, which is divisible by 3.
But 864329 is not divisible by 3, since sum of its digits =(8 + 6 + 4 + 3 + 2 + 9) = 32, which is not divisible by 3.
Divisibility By 4:
A number is divisible by 4, if the number formed by the last two digits is divisible by 4.
Ex. 892648 is divisible by 4, since the number formed by the last two digits is 48, which is divisible by 4.
But 749282 is not divisible by 4, since the number formed by the last tv/o digits is 82,
which is not divisible by 4.
Divisibility By 5:
A number is divisible by 5, if its unit's digit is either 0 or 5.
Thus, 20820 and 50345 are divisible by 5, while 30934 and 40946 are not.
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Divisibility By 6:
A number is divisible by 6, if it is divisible by both 2 and 3.
Ex. The number 35256 is clearly divisible by 2. Sum of its digits = (3 + 5 + 2 + 5 + 6) = 21, which is divisible by 3. Thus, 35256 is divisible by 2 as well as 3. Hence, 35256 is divisible by 6.
Divisibility By 8:
A number is divisible by 8, if the number formed by the last three digits of the given number is divisible by 8.
Ex. 953360 is divisible by 8, since the number formed by last three digits is 360, which is
divisible by 8.
But, 529418 is not divisible by 8, since the number formed by last three digits is 418,
which is not divisible by 8.
Divisibility By 9:
A number is divisible by 9, if the sum of its digits is divisible by 9.
Ex. 60732 is divisible by 9, since sum of digits * (6 + 0 + 7 + 3 + 2) = 18, which is
divisible by 9.
But, 68956 is not divisible by 9, since sum of digits = (6 + 8 + 9 + 5 + 6) = 34, which is
not divisible by 9.
Divisibility By 10:
A number is divisible by 10, if it ends with 0.
Ex. 96410, 10480 are divisible by 10, while 96375 is not.
Divisibility By 11:
A number is divisible by 11, if the difference of the sum of its digits at odd places and the sum of its digits at even places, is either 0 or a number divisible by 11.
Ex. The number 4832718 is divisible by 11, since: (sum of digits at odd places) - (sum of digits at even places) = (8 + 7 + 3 + 4) - (1 + 2 + 8) = 11, which is divisible by 11.
Divisibility By 12:
A number is divisible by 12, if it is divisible by both 4 and 3.
Ex. Consider the number 34632.
(i) The number formed by last two digits is 32, which is divisible by 4,
(ii) Sum of digits = (3 + 4 + 6 + 3 + 2) = 18, which is divisible by 3. Thus, 34632 is
divisible by 4 as well as 3. Hence, 34632 is divisible by 12.
Divisibility By 14:
A number is divisible by 14, if it is divisible by 2 as well as 7.
Divisibility By 15:
A number is divisible by 15, if it is divisible by both 3 and 5.
Divisibility By 16:
A number is divisible by 16, if the number formed by the last 4 digits is divisible by 16.
Ex.7957536 is divisible by 16, since the number formed by the last four digits is 7536,
which is divisible by 16.
Divisibility By 24:
A given number is divisible by 24, if it is divisible by both 3 and 8.
Divisibility By 40:
A given number is divisible by 40, if it is divisible by both 5 and 8.
Divisibility By 80:
A given number is divisible by 80, if it is divisible by both 5 and 16.
Note:
If a number is divisible by p as well as q, where p and q are co-primes, then the given number is divisible by pq.
If p arid q are not co-primes, then the given number need not be divisible by pq, even when it is divisible by both p and q.
Ex. 36 is divisible by both 4 and 6, but it is not divisible by (4x6) = 24, since 4 and 6 are not co-primes.
General rules of divisibility for all numbers:
♦ If a number is divisible by another number, then it is also divisible by all the factors of the other number.
♦ If two numbers are divisible by another number, then their sum and difference is also divisible by the other number.
♦ If a number is divisible by two co-prime numbers, then it is also divisible by the product of the two co-prime numbers.
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