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Numeral System is a mathematical notation used for counting and calculating objects, and for executing arithmetic calculations. It is a writing system for representing numbers. It gives an exceptional depiction of every number and constitutes the arithmetic and algebraic form of the number. It allows us to operate arithmetic operations like addition, subtraction, multiplication, and division.
An equation is a declaration that joins two algebraic expressions of the same values with the '=' sign. For example: In equation 8x + 4 = 7, 8x + 4 is the left-hand side expression and 7 is the right-hand side expression connected with the '=' sign.
A word or sign that designates an amount is known as a number. The numbers 4,6,8 etc. are even numbers and 3,5,7 etc. are odd numbers. A number is a value generated by a mixture of digits. These numbers are used to represent an algebraic number. A digit is an indication from a group of 10 numbers ranging from 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. Any combination of integers represents a number. The size of a Number depends on the count of digits that are used for its growth. For Example: 136, 198, 0.245, -16, 98, 96 etc.
Types of Numbers
Numbers are of various types depending upon the pattern of digits that are used for their development. Various characters and rules are also put in the Numbers which classify them into a diversity of different types,
Integers are a group of Whole Numbers plus the negative values of the Natural Numbers. Integers do not cover fraction numbers i.e. they can't be written in a/b form. The scope of Integers is from the Infinity at the Negative end and Infinity at the Positive end, including zero. Integers are shown by the symbol Z. Integers are those digits whose fractional part is 0 like -5, -4, 1, 0, 20, 200.
Natural Numbers are numbers that scope from 1 to infinity. These numbers are also described as Positive Numbers or Counting Numbers. We can also show Natural numbers by the symbol N. All the integers which are greater than 0 are natural numbers, Counting numbers like 5,6,7,8,9,10.
Whole Numbers are similar to Natural Numbers, but they also include 'zero'. Whole numbers can also be shown by the symbol W. Whole numbers include all natural numbers and 0 (zero).
Prime Numbers and Composite Numbers
All those numbers which are having only two definite components, the number itself and 1, are called prime numbers. All the numbers which are not Prime Numbers are known as Composite Numbers except 0. Zero is neither a prime nor a composite number. Some prime numbers are 3, 5, 7, 57, 51, 67, and 391. All numbers which are greater than 1 are composite numbers. Some composite numbers are 7, 5, 3, 17, 15, and 200.
Fractions are the integers that correspond in the shape of a/b, where, a represent Whole numbers and b represent Natural Numbers, i.e., b can never be 0. The upper part of the fraction i.e. a is described as a Numerator whereas the lower part i.e. b is termed as Denominator. Example: -1/5, 0.25, 2/5, 18/4, …
Rational numbers are the numbers that can be shown in the fraction form i.e. a/b. Here, a and b both are numbers and b is not equal to 0. All the fractions are rational numbers but not all the rational numbers are fractions. Example: -2/5, 0.54, 1/5, 13/4, …
Irrational Numbers
Irrational numbers are the numbers that can not be shown in the form of fractions i.e. they can not be written as a/b. Example: √2, √3, √.434343, π…
Real and Imaginary Numbers
Real numbers are numbers that can be shown in decimal form. These numbers involve whole numbers, integers, fractions, etc. All the numbers belong to Real numbers but all the real numbers do not belong to the integers. Imaginary Numbers are all those numbers that are not real numbers. These numbers when squared will show a negative number. The √-1 is represented as i. These numbers are also called complex numbers. Example: √-2, √-5,…
First we will take the digit in range of 10:
From the range of 0 to 10, the number 1 appears 2 time.
Now we calculate the integers in range of 100
From the range of digits 0 to 99, the number 1 appears 20 times.
(They are: 1, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 31, 41, 51, 61, 71, 81, 91)
Check out that in digit 11, 1 seem two times.
Now from numbers 100 to 199 the digit 1 seem 120 times.
In this case the numbers at hundreds place is 1. Therefore there are 120 1's from 100 to 199.
Explanation- we have to calculate 1's from 100, 101, 102……….. 199.
At tens place 1 will come 10 times. At hundreds place of these 3 digit numbers 1 will come 100 times.
That is 110, 111, 112, 113, 114…… 119.
At units place 1 will appear 10 times(101, 111, 121, 131……. 191)
So add these:
100 + 10 + 10 = 120 so 1 comes 120 times.
Now from 200 to 299 we have 201, 210, 211, 212, 213, 214, 215, 216, 217, 218, 219, 221, 231, 241,
251, 261, 271, 281, 291 the number 1 appear 20 times.
From 300 to 399 we have 301,…311,….391. Again 20 times
From 400 to 499 we have 401,…411,…491. Again 20 times
By adding all of these we get:
Total number of times = (20).(4) + 120
= 80 + 120
Therefore when we calculate digits from 1 to 500 the digit 1 appear 200 times.
Note: Don't forget to check the number 1 twice in digits like 11, and also do not forget to check the number 1 in hundredth's place in the digits from 100 to 199.
Question 1: How many times will the digit two appear between 1 and 500?
(They are: 2, 12, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29 32, 42, 52, 62, 72, 82, 92)
Now from 100 to 199 we have 102, 112, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 132, 142, 152, 162, 172,
182, 192, the number 2 appear 20 times.
Note: Don't forget to check the number 2 twice in digits like 22, and also do not forget to check the
number 2 in hundredth's place in the digits from 200 to 299.
Question 2: How many times will the digit three appear between 1 and 500?
From 100 to 199 we have 103, 113, 123, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 143, 153, 163, 173, 183,
193 Again 20 times
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ManasChhabra2
@ManasChhabra2
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