The mathematician is entirely free, within the limits of his imagination, to construct what worlds he pleases. What he is to imagine is a matter for his own caprice; he is not thereby discovering the fundamental principles of the universe nor becoming acquainted with the ideas of God. If he can find, in experience, sets of entities which obey the same logical scheme as his mathematical entities, then he has applied his mathematics to the external world; he has created a branch of science.

John William Navin SULLIVAN

Welcome to the blog Math1089 – Mathematics for All.

2021 is about to pass and 2022 is coming. This is the moment of leaving the old to welcome the new. Let’s hope that 2022

add the joys;

subtract the sorrows;

multiply the happiness; and

divide the love;

among your loved ones. Happy New Year

2022

In this blog, let us consider the number 2022 as the point of discussion. The prime factors of 2022 are 2, 3, and 337. Hence

2022 = 2 × 3 × 337.

Using Roman numerals, we can write

2022 = MMXXII.

The new year 2022 can be written using the digits 1, 2, . . . , 8, 9 once, in ascending order and using basic mathematical operations like +, −, ×, and ÷.

• 2022 = 1234 + 5 – 6 + 789
• 2022 = −1 × 2 + (3 + 4 × 5) × (6 – 7 + 89)

We can also represent 2022 using the digits 1, 2, . . . , 8, 9 once, in descending order and using basic mathematical operations like +, −, ×, and ÷.

• 2022 = 9 – 876 + (5 + 4) × 321
• 2022 = {9 + 8 × (7 × 6 – 5 + 4)} × (3 + 2 + 1)

Likewise, 2022 can be represented by using the digits 1, 2, . . . , 9, 10 once in descending order and using basic mathematical operations like +, −, ×, and ÷.

2022 = 10 + (9 × 8 × 7 – 6 + 5) × (4 + 3 – 2 − 1)

If we welcome power, then 2022 can be represented by using the digits 1, 2, . . . , 9, 10 once in ascending order and using basic mathematical operations like +, −, ×, and ÷.

2022 = 1 × 2 × (−3 + 4⁵) − 6 – 7 – 8 – 9 + 10

If we allow factorial to come in, then using it

• 2022 = −1 – 2 + 3 × (−45 + 6!)
• 2022 = 7! − (6 + 5!)  × 4! + 3 + 2 + 1
• 2022 = 1 + 2 + 3 − 4! × (5! + 6) + 7!

Few other representations of 2022 using various mathematical operations and using the digits 1, 2, . . . , 9 are

•  2022 = −(1 + 2) + 3⁴ × (−5 + 6 + 7 + 8 + 9)
• 2022 = 1− 2^3! × 4! + 5 × (6! − 7) − 8
• 2022 = −9 + 87 + (6⁵ ÷ 4) × {3 ÷ (2 + 1)}
• 2022 = 9 – 8 – 7 + 6 × {− 5 + (4 + 3)^{2 + 1}}

In binary and octal, 2022 can be written as

• 2022 = (11111100110)₂
• 2022 = (3746)₈

Allowing square root to come in, 2022 can be written as

2022 = √9 × (– 8 + 7 + 6!) + 5 × (4 − 32 + 1)

We can also represent 2022 as equality expressions written in such a way that both sides have the same digits.

2022 = 1⁹ + 44² + 72⁰ + 84¹ = 19 + 442 + 720 + 841.

Some strange multiplications (swapping like) with 2022 are given below

• 2022 × 11 = 1011 × 22
• 2022 × 111 = 1011 × 222.

As a power of 2, we can represent 2022 as

2022 = 2¹¹ – 26.

In conclusion, 2022 has a strange property: when divided by 337, the result is same as the sum of digits of 2022.

Once again, “Happy New Year 2022!”. Stay healthy and safe.

References:

Inder J. Taneja: Mathematical Beauty of 2022; https://inderjtaneja.com