*While asleep, I had an unusual experience. There was a red screen formed by flowing blood, as it were. I was observing it. Suddenly a hand began to write on the screen. I became all attention. That hand wrote a number of elliptic integrals. They stuck to my mind. As soon as I woke up, I committed them to writing.*

**S. Ramanujan**

Welcome to the blog **Math1089 – Mathematics for All**.

2022 is about to pass, and **2023** is coming. This is the moment to leave the old and welcome the new. Let’s hope that **2023** will

addthe joys;

subtractthe sorrows;

multiplythe happiness; and

dividethe love

among your loved ones. **Happy New Year**!

**2023**

In this blog post, consider the number **2023** and let us discuss the mathematical beauties of this number.

The **factors** of 2023 are 1, 7, 17, 119, 289 and 2023. The **prime factors** are 7 and 17, whereas the **prime factorisation** of 2023 is 7 × 17^{2}.

The corresponding representation of 2023 in **Roman numeral **is MMXXIII and in **binary** is (11111100111)_{2}.

**Various Representations of 2023**

**Using the digits 1 to 9 and basic mathematical operations**

Below are representations of 2023 using the digits 1, 2, . . . , 9 and basic mathematical operations like addition, subtraction, multiplication etc.

2023 = 12 × 3 × (4 + 5) × 6 + 7 + 8 × 9

2023 = 9 × 8 + 7 + 6 × 54 × 3 × 2 × 1

**Using the numbers 1 to 10 and basic mathematical operations**

Below are representations of 2023 using the numbers 1, 2, . . . , 10 and basic mathematical operations like addition, subtraction, multiplication, exponentiation etc.

2023 = 12 × (3 × 4 + 5) × 6 + 789 + 10

2023 = 10 + 98 + (7 + 6) × 5 + 43

^{2}+ 1

**Using the digits 1 to 9 and various powers**

We can represent 2023 using the digits 1 to 9, where the bases are in increasing order and powers in decreasing order.

2023 = 1

^{9}– 2^{8}+ 3^{7}+ 4^{6 }– 5^{5}– 6^{4}+ 7^{3}+ 8^{2}+ 9^{1}

**As the sum of digits**

If we divide 2023 by 289, we get 7 as the answer which is same as 2 + 0 + 2 + 3, the sum of the digits of the number.

**Using all the digits of the number**

The digits of the number are 0, 2 and 3. Using these, we can write

2023 = (2 + 0 + 2 + 3)(2

^{2}+ 0^{2}+ 2^{2}+ 3^{2})^{2}

**Using algebraic identities**

Applying the difference of squares identity, we have

**As a sum of squares**

We can write 2023 as the sum of squares in the following ways:

2023

^{2}= 952^{2}+ 1785^{2}2023

^{2}= 1127^{2}+ 1680^{2}

**As a sum of cubes**

We can write 2023 as the sum of cubes in the following way:

2023 = 2

^{3}+ 5^{3}+ 6^{3}+ 7^{3}+ 11^{3}

**As various powers of 2**

It’s easy to represent 2023 as the sum/ difference of various powers of 2. Here are few examples.

2023 = 2

^{11}‒ 2^{5}+ 2^{2}+ 2^{1}+ 2^{0}2023 = 2

^{10}+ 2^{9}+ 2^{8}+ 2^{7}+ 2^{6}+ 2^{5}+ 2^{2}+ 2^{1}+ 2^{0}2023 = 2

^{10}+ 2^{9}+ 2^{8}+ 2^{7}+ 2^{6}+ 2^{5}+ 2^{4}‒ 2^{3}‒ 2^{2}+ 2^{1}+ 2^{0}

**Using factorials**

Allowing for factorial representation, another way to represent is

**As a trigonometric equation**

If cos

x_{1}+ cosx_{2}+ ···+ cosx_{2023}= 0, thensin

x_{1}+ sinx_{2}+ ···+ sinx_{2023}= 2023.

**As a limit**

We can write 2023 as a limit in the following way:

**As a definite integration**

Allowing for definite integral representation, another way to represent is

**Using matrices and determinants**

If we consider a square matrix *M*, its determinant value is 2023.

**As a continued fraction**

We have the following identity:

Repeated application of this identity yields

As a result, we can write

**As an infinite series**

We can also represent 2023 as the sum of an infinite series. For example

In fact, a simple computation shows that

The cube of 2023 is 8279186167. We can write the same number by rearranging the digits in their order and using various mathematical operations. As a result, we have the following number relationship:

2023

^{3}= 8279186167 = (82 + 79 + 1861 − 6 + 7)^{3}

**Single Digit Representation**

A single-digit representation of any number is always a challenge. The same goes when we represent 2023 using only the digits 1 or 2 or 3 . . . or 9 etc. Here, we are presenting four types of representations, but the reader can explore more. During your explorations, if you find something not listed here, feel free to mail us or add yours using the comments.

**Representation of 2023 using the digit 1**

A few representations using only 1’s are given below:

**Representation of 2023 using the digit 2**

Similar representations using only 2’s are given below:

**Representation of 2023 using the digit 3**

Using only 3’s, a few representations are given below:

**Representation of 2023 using the digit 4**

A few representations using only 4’s are given below:

**Representation of 2023 using the digit 5**

Similar representations using only 5’s are given below:

**Representation of 2023 using the digit 6**

Using only 6’s, a few representations are given below:

**Representation of 2023 using the digit 7**

A few representations using only 7’s are given below:

**Representation of 2023 using the digit 8**

Similar representations using only 8’s are given below:

**Representation of 2023 using the digit 9**

Using only 9’s, a few representations are given below:

**During your explorations, if you find something not listed here, feel free to mail us or add yours using the comments.** Your suggestions are eagerly and respectfully welcome! See you soon with a new mathematics blog that you and I call **“****Math1089 – Mathematics for All!**“.

**References:**

*Inder J. Taneja*: 23 and 2023 in Numbers and Patterns; https://inderjtaneja.com

Great work and appreciate you as maths enthusiast. You inspired us to look at any number mathematically. Thank you so much.