Mathematics is on the artistic side a creation of new rhythms, orders, designs, harmonies, and on the knowledge side, is a systematic study of various rhythms, orders, designs and harmonies.William L. Schaaf
Welcome to the blog Math1089 – Mathematics for All.
2023 is about to pass, and 2024 is coming. This is the moment to leave the old and welcome the new. Let’s hope that 2024 will
add the joys;
subtract the sorrows;
multiply the happiness; and
divide the love
among your loved ones.
1704047400 days hours minutes secondsuntil
In this blog post, consider the number 2024 and let us discuss the mathematical beauties of this number.
The factors of 2024 are 1, 2, 4, 8, 11, 22, 23, 44, 46, 88, 92, 184, 253, 506, 1012, and 2024. The prime factorisation of 2024 is 2 × 2 × 2 × 11 × 23.
We find that in 2024
(2 + 0 + 2) ÷ 4 = 1 as its factor;
(20 + 2)! – 4 = 2 as a factor;
2 + 0 + 2 + 4 = 8 as its factor;
20 × 2 × 4 = 8 as a factor;
(2 + 0! + 2 × 4) = 11 as its factor;
(20 – 2 + 4) = (–2 + 024) = (2 + 0!)! + 24 = 22 as a factor;
(–20 + 24) = – (2 ÷ 02) + 4! = 23 as its factor;
20 + 24 = 20 × 2 + 4 = 44 as a factor
2 × (–0! + 24) = –2 + 0 + 2 × (4!) = 46 as its factor;
(20 + 2) × 4 = 88 as its factor.
It is an even number, and the year is a leap year. The corresponding representation of 2024 in Roman numeral is MMXXIV and in binary is (11111101000)2.
Adding to 2024 its reverse 4202, we get a palindrome 6226.
Various Representations of 2024
(a) Using the digits 1 to 9 and basic mathematical operations
Below are representations of 2024 using the digits 1, 2, . . . , 9 (in the same order) and basic mathematical operations like addition, subtraction, multiplication etc.
2024 = 123 + 4 × (5 + 6 × 78) + 9
2024 = 1 − 2 + 3 × (4 + 5) × (6 + 78 − 9)
(b) Using the digits 9 to 1 and basic mathematical operations
Below are representations of 2024 using the digits 9, 8, . . . , 0 (in the same order) and basic mathematical operations like addition, subtraction, multiplication etc.
2024 = 9 × 8 + 7 + 6 × 54 × 3 × 2 + 1
2024 = (98 − 76) × (5 + 43 × 2 + 1)
(c) Using the digits 1 to 9 and various powers
Allowing for exponentiation, here are a few representations of 2024 using the digits 1, 2, …, 9 (in the same order) and basic mathematical operations such as addition, subtraction, multiplication, etc.
2024 = (−1 + 2 − 3 + 4)5+6 − 7 − 8 − 9
2024 = 1 × 2 × (3 + 45) − 6 − 7 − 8 − 9
(d) Using the digits 1 to 9 and factorial operation
Alternatively, if we allow the factorial operation to come into the picture, a few different expressions are given below.
2024 = [1 + (2 − 3) × 4!] × (5 − 6 − 78 − 9)
2024 = 9 − 8 + 7 + (−6 + 5!) × 4! − (3 + 2 + 1)!
2024 = (−9 − 8) × (7 − 6 − 5!) + 4 + 3 − (2 + 1)!
(e) Using the digits 0 and 1 only
Expression of 2024 using the digits 0, 1 and various mathematical operations
2024 = (1 + 1) × 1001 + (1 + 1) × 11
2024 = 11 + 1001 + 1001 + 11
(f) As the sum of digits
If we divide 2024 by 253, we get 8 as the answer which is same as 2 + 0 + 2 + 4, the sum of the digits of the number.
(g) Using all the digits of the number
The digits of the number 2024 are 0, 2 and 4. Using these, we can write
2024 = (20 + 24) + (20 + 24)(20 + 24) + (20 + 24)
(h) As a trigonometric equation
If sin x1 + sin x2 + ···+ sin x2024 = 0, and sin xi are non-negative, then
cos x1 + cos x2 + ···+ cos x2024 = 2024.
(i) As the sum of two primes
It’s easy to represent 2024 as the sum of two primes. A few examples are provided below.
2024 = 73 + 1951
2024 = 991 + 1033
(j) As the sum of three primes
Similarly, we can represent 2024 as the sum of three primes. A few examples are provided below.
2024 = 2 + 739 + 1283
(k) As the sum of consecutive integers
Below is an example where we represent 2024 as the sum of consecutive integers.
2024 = 77 + 78 + ‧‧‧ + 98 + 99
2024 = 119 + 120 + ‧‧‧ + 133 + 134
It’s easy to represent 2024 as the sum of consecutive even numbers:
2024 = 246 + 248 + 250 + 252 + 254 + 256 + 258 + 260
2024 = 174 + 176 + 178 + 180 + 182 + 184 + 186 + 188 + 190 +192 + 194
2024 = 66 + 68 + 70 + 72 + 74 + 76 + 78 + 80 + 82 + 84 + 86 + 88 + 90 + 92 + 94 + 96 + 98 + 100 + 102 + 104 + 106 + 108 + 110.
(l) As a product of sub-factorials
By employing subfactorials, it becomes straightforward to express 2024 as the product of two terms.
2024 = !5 × (!5 + 2)
(m) Expression for 2024 as a binomial coefficient
(n) As a power of 2, with powers in decreasing order
It’s easy to represent 2024 as the sum/ difference of various powers of 2. Here are few examples.
2024 = 210 + 29 + 28 + 27 + 26 + 25 + 24 − 23
(o) As a sum of three squares
We can represent 2024 as the sum of three squares. Few examples are
2024 = 22 + 162 + 422
2024 = 82 + 142 + 422
2024 = 102 + 182 + 402
2024 = 162 + 182 + 382
(p) As a sum of four squares
We can represent 2024 as the sum of four squares. Few examples are
2024 = 42 + 182 + 282 + 302
2024 = 82 + 222 + 242 + 302
(q) 2024 as a sum of five squares
As above, we can represent 2024 as the sum of four squares. Few examples are
2024 = 52 + 192 + 222 + 232 + 252
2024 = 132 + 172 + 192 + 232 + 262
(r) As a sum of six squares
Here is the desired expression for 2024.
2024 = 72 + 172 + 192 + 202 + 212 + 222
(s) As a sum of cubes
Similarly, we can represent 2024 as the sum of cubes. Few examples are
2024 = 113 + 73 + 63 + 53 + 23 + 13
2024 = 93 + 83 + 73 + 63 + 53 + 43 + 33 + 23
(t) As the sum/ difference of various powers of numbers
2024 = −17 − 25 + 36 + 43 − 52 + 64 − 71
2024 = 17 − 25 + 36 + 41 + 52 + 64 + 70
2024 = 18 − 24 + 36 + 45 + 51 + 63 + 70 + 82
2024 = 17 + 28 − 39 + 46 + 52 + 60 + 75 + 83 + 91
(u) As a limit
2024 is the result of the following limit.
(v) as a definite integral
The digits of the number 2024 are 0, 2 and 4. Using these, we can write
(w) In a magic square with a magic sum
In the following magic square, the magic sum is 2024.
(x) As a determinant
(y) As a continued fraction
We have the following identity:
Repeated application of this identity yields
As a result, we can write
(z) As an infinite series
We can also represent 2024 as the sum of an infinite series. For example
Since 2024 is the sum of the (sixteen) numbers from 119 to 134, and sixteen is even, we have the following relation:
2024 = 1342 – 1332 + 1322 – 1312 + 1302 – 1292 + 1282 – 1272 + 1262 – 1252 + 1242 – 1232 + 1222 – 1212 + 1202 – 1192.
Single Digit Representation
A single-digit representation of any number is always a challenge. The same goes when we represent 2024 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 2024 using the digit 1
A few representations using only 1’s are given below:
Representation of 2024 using the digit 2
A few representations using only 2’s are given below:
Representation of 2024 using the digit 3
A few representations using only 3’s are given below:
Representation of 2024 using the digit 4
A few representations using only 4’s are given below:
Representation of 2024 using the digit 5
A few representations using only 5’s are given below:
Representation of 2024 using the digit 6
A few representations using only 6’s are given below:
Representation of 2024 using the digit 7
A few representations using only 7’s are given below:
Representation of 2024 using the digit 8
A few representations using only 8’s are given below:
Representation of 2024 using the digit 9
A few representations using only 9’s are given below:
We can represent 2024 as the difference of two squares in the following ways:
2024 = 5072 – 5052;
2024 = 2552 – 2512;
2024 = 572 – 352; and
2024 = 452 – 12.
Number Patterns
Patterns starting with the number 2024
Same digit equality expressions
Look for the presence of the same digits on both sides and ensure that each of the expressions is equal to 2024. Here, we have used basic mathematical operations.
2024 = 11 + 442 + 710 + 861 = 11 + 442 + 710 + 861
2024 = 21 + 442 + 710 + 851 = 21 + 442 + 710 + 851
Pythagorean triples
Below are a few examples of Pythagorean triples with one member being 2024.
20242 + 24152 = 31512
20242 + 37952 = 43012
20242 + 39902 = 44742
20242 + 53822 = 57502
20242 + 56432 = 59952
20242 + 83432 = 85852
Representation of 2024 using the numbers 10, 9, . . . , 1 in that order using various mathematical operations:
2024 = (10 + 9 − 8) × (7 + 6 − 5) × (4 × 3 × 2 − 1)
2024 = (10 + 9 − 8) × (7 + 6 − 5) × (4 × 3 × 2 − 1)
2024 = {10 + (9 + 8 × 7) × 6} × 5 + 4 × 3 × 2 × 1
2024 = (10 + [9 × {8 − (7 – 6 × 5)}]) × (4 + 3) + (2 − 1)
2024 = 10 × 9 × (8 + 7 + 6) + 5! + 4 × 3 + 2 × 1
2024 = 10 × (9 + 8) × 7 + 6! + 5! − 4 − 3 + 2 −1
2024 = (10 + 9) × (8 + 7 + 6) × 5 + 4! + 3 + 2 × 1
2024 = 10 × (9 + 8 × 7 − 6 + 5! + 4!) − 3 × 2 × 1
During your explorations, if you find something not listed here, feel free to mail us or add yours using the comments.
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