# How many times does the digit 1 appear in the numbers 1 to 1,000?

Number Systems is a value used for counting and measuring objects, and for performing arithmetic calculations. It is a method of writing for expressing numbers. It provides a unique representation to every number and presents the arithmetic and algebraic form of the number. It enables us to operate arithmetic operations like addition, subtraction, multiplication, and division.

An equation is a declaration that links two algebraic expressions of equal 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 linked with ‘=’ sign.

**What is a Number?**

A word or symbol that indicates 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 formed by the combination of digits. These numbers are used to express algebraic quantities. An integer is an indication from a set of 10 characters ranging from 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. Any amalgamation of integers represents a number. The size of a Number depends on the count of digits that are used for its development. For Example: 126, 128, 0.356, -12, 78, 94 etc.

**Whole Numbers**

Whole Numbers are the same as Natural Numbers, but they also include ‘zero’. We can donate whole numbers by the symbol W. Whole numbers include all the natural numbers and 0 (zero).

We know that the numbers 1, 2, 3, 4, 5 are natural numbers. And the numbers 0, 1, 2, 3, 4, 5, 6 .. etc., are the whole numbers as they include 0 as well. Whole numbers are represented as W and Natural numbers are represented as N. Therefore, it is correct to say,

W= {N} + 0

On the number line, everything which is on the right side of 0 (including 0) comes under whole numbers.

### How many times does the digit 1 appear in the numbers 1 to 1,000?

**Solution:**

Hint: The easy method is to write the numbers in the range of 10, 100 and then count the number of 1’s in them

Steps to find out the solution:First we will take the numbers in range of 10:

From the range of numbers 0 to 10, the digit 1 appears 2 time.

Now we add the numbers in range of 100

From the range of numbers 0 to 99, the digit 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)

Observe that in number 11, 1 appears two times.

Now from 100 to 199 the digit 1 appears 120 times.

In this case the digit at hundreds place is 1. Therefore there are 120 1’s from 100 to 199.

Explanation- we have to find 1’s from 100, 101, 102……….. 199. At hundreds place of these 3 digit numbers 1 will come 100 times. At tens place 1 will come 10 times

That is in cases of 110, 111, 112, 113, 114…… 119. At units place 1 will come 10 times(101, 111, 121, 131……. 191)

So add these:

100 + 10 + 10 = 120 so 1 appears 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 digit 1 appear 20 times.

Similarly,

From 300 to 399 we have 301,…311,….391. Again 20 times

From 400 to 499 we have 401,…411,…491. Again 20 times

From 500 to 599 we have 501,…511,…591. Again 20 times.

From 600 to 699 we have 601,…611,…691. Again 20 times

From 700 to 799 we have 701,…711,…791. Again 20 times

From 800 to 899 we have 801,…811,…891. Again 20 times

From 900 to 999 we have 901,…911,…991. Again 20 times

and then we have 1000 in which digit 1 appears 1 time

By adding all of the above, we get:

Total number of times = (20).(9) + 120 + 1

= 180 + 120 + 1

= 301

Therefore when we list numbers from 1 to 1000 the digit 1 is written 301 times.

Note:Don’t forget to count the digit 1 twice in numbers like 11, and also do not forget to count the digit 1 in hundredth’s place in the numbers from 100 to 199 and also in the thousand place number like 1000.

**Similar Questions**

**Question 1: While listing the numbers from 1 to 1000, how many times the digit 5 appears.**

**Answer:**

Steps to find the solution:

First, we will take the numbers in the range of 10:

From the range of numbers 0 to 10, the digit 5 appears 1 time.

Now we add the numbers in the range of 100

From the range of numbers 0 to 99, the digit 5 appears 20 times.

(They are: 5, 15, 25, 35, 45, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 65, 75, 85, 95)

Observe that in number 55, 5 appears two times.

Now from 100 to 199 we have: 105, 115, 125, 135, 145, 150, 151, 152, 153, 154, 155, 156, 157, 158, 159, 165, 175, 185, 195. The digit 5 appears 20 times.

Similarly,

From 200 to 299 we have 205,215…….245, 250,…255,…..,295. Again 20 times.

From 300 to 399 we have 305,…355,…,395. Again 20 times

From 400 to 499 we have 405,…455,…,495. Again 20 times

From 600 to 699 we have 605,…655,…,695. Again 20 times

From 700 to 799 we have 705,…755,…,795. Again 20 times

From 800 to 899 we have 805,…855,…,895. Again 20 times

From 900 to 999 we have 905,…955,…,995. Again 20 times

Now the remaining range is 500 to 599:

Now from 500 to 599, the digit 5 appears 120 times.

In this case, the digit at hundred’s places is 5. Therefore there are 120 5’s from 500 to 599.

Explanation:We have to find 5’s from 500, 501, 502……….. 599. At hundred’s place of these 3 digit numbers, 5 will come 100 times. At tens place, 5 will come 10 timesThat is in cases of 550, 551, 552, 553, 554…… 559. At units place 5 will come 10 times(505, 555, 525, 535……. 595)

So add these:

100 + 10 + 10 = 120 so 5 appear 120 time’s.

By adding all above, we get:

Total number of times = (20).(9) + 120

= 180 + 120

= 300

Therefore when we list numbers from 1 to 1000 the digit 5 is written 300 times.

Note:Don’t forget to count the digit 5 twice in numbers like 55, and also do not forget to count the digit 5 in hundredth’s place in the numbers from 500 to 599.

**Question 2: How many times does the digit 2 appear in numbers from 1 to 100?**

**Answer:**

Complete step-by-step solution:

We have to find the number of times the digit 2 appears in numbers from 1 to 100.

From 1 to 10, the digit 2 appears only once for 2.

From 11 to 20, the digit 2 appears two times for 12 to 20.

From 21 to 30, the digit 2 appears in 21, 22, 23, 24, 25, 26, 27, 28, 29 and in 22 it

appears twice. So, it appears ten times.

From 31 to 40, the digit 2 appears only once for 32.

From 41 to 50, the digit 2 appears only once for 42.

From 51 to 60, the digit 2 appears only once for 52.

From 61 to 70, the digit 2 appears only once for 62.

From 71 to 80, the digit 2 appears only once for 72.

From 81 to 90, the digit 2 appears only once for 82.

From 91 to 100, the digit 2 appears only once for 92.

So, the total number of times of repetition of 2 is 1+2+10+1+1+1+1+1+1+1 = 20.

Hence, the digit 2 appears in numbers from 1 to 100 for 20 times.