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 Mathematics is a huge and fascinating subject, with many startling and little-known facts. Here are a few that demonstrate its richness and diversity: 

1. The Origin of "Hundred": The word "hundred" is derived from the old Norse phrase "hundrath," which indicates 120 rather than 100. 

2. Birthday Paradox: Two people in a group of 23 have a 50% probability of sharing the same birthday. The birthday problem refers to this seemingly contradictory possibility. 

3. Late Introduction of Mathematical Symbols: The majority of mathematical symbols, including the plus and minus signs, were not introduced until the 16th century. Previously, equations were written out in words.

4. "Forty" in Alphabetical Order: The only number with letters in alphabetical order is "Forty." On the other hand, "one" is the only number with letters arranged on the opposite side of the alphabet. 

5. Lack of Zero in Roman Numerals: There is no symbol for zero in the Roman numeral system. Later on in history, particularly by Indian mathematicians, the idea of zero as a number was established. 

6. Special Characteristics of 1000: "One thousand" is the first number in the English language to contain the letter "a."

 7. Spherical Bread Slicing: Assuming that the slices of a spherical loaf of bread have the same thickness, each slice will have an equal amount of crust.

8. Indian Mathematical Contributions: In 628 AD, Brahmagupta, an Indian mathematician, established the definition of zero and created guidelines for arithmetic operations involving negative and zero numbers. 

9. Fast Convergence to Pi: Srinivasa Ramanujan, a mathematician, found series that could be computed efficiently because they converged quickly to π. One example of a series is: The formula \frac{1}{\pi} = \frac{2\sqrt{2}}{9801} \sum_{k=0}^{\infty} \frac{(4k)!(1103 + 26390k)}{(k!)^4 396^{4k}} Only the first term gives an accurate π to eight decimal places because this series converges so fast.

10. The Kaprekar's Constant, or 6174: 6174 is referred to as Kaprekar's constant and was discovered by the Indian mathematician D.R. Kaprekar. In no more than seven iterations, you will reach 6174 if you take any four-digit number with at least two distinct digits, arrange the digits in descending and then ascending order to create two four-digit numbers, subtract the smaller from the larger, then repeat the process.

11. The Identity of Euler: Euler's identity is an equation that combines five of the most significant numbers in mathematics and is frequently praised for its beauty: This is the imaginary unit, the base of natural logarithms, and the ratio of the diameter to the circumference of a circle.

12. Fibonacci Sequence in Nature The Fibonacci sequence is commonly found in nature, with each number (1, 1, 2, 3, 5, 8,...) being the sum of the two numbers before it. Examples include the way leaves are arranged on stems, how trees branch, how pineapple fruit sprouts, and how artichokes bloom. 

13. Ideal Numbers: A positive integer is considered perfect if it equals the sum of its proper divisors, minus itself. Since 1 + 2 + 3 = 6, the lowest perfect number is 6 with the divisors 1, 2, and 3. Divisors 1, 2, 4, 7, and 14 make up the following number, which is 28 (1 + 2 + 4 + 7 + 14 = 28).

14. Benford's Law: The first digit is more likely to be tiny in many naturally occurring sets of numbers. Benford's Law states that the number 1 is the leading digit approximately 30% of the time, whereas larger numbers are less likely to be in this position. To identify fraud, this idea is applied in domains such as forensic accounting. 

15. Magic Squares: A grid of numbers with the identical sums in each row, column, and diagonal is called a magic square. The Lo Shu Square is the oldest known 3x3 magic square, and the idea originated in ancient China.

16.Imaginary Numbers: Because these numbers were seen as meaningless, the word "imaginary" was first used in the 17th century as a pejorative. Nonetheless, imaginary numbers are increasingly essential to complex number theory and have real-world uses in signal processing, physics, and engineering. 

17. The Golden Ratio: Often represented by the Greek symbol φ (phi), the golden ratio is roughly equal to 1.618 and may be found in many facets of nature, art, and architecture. It is defined so that the ratio of the larger quantity to the smaller one is equal to the ratio of the sum of two quantities to the larger quantity.

18. Möbius Strip: A surface with only one side and one edge is called a Möbius strip. One can be made by twisting a piece of paper half-way and then tying the ends together. Because of its special qualities, this object has captivated both mathematicians and artists.

 19. Prime Number Infinitude: The number of prime numbers is limitless. Around 300 BC, the Greek mathematician Euclid made the first demonstration of this. He demonstrates that there is always another prime that is absent from any finite collection of prime numbers. 

These facts highlight mathematics' richness and beauty while demonstrating its deep ties to the outside world.The breadth, history, and unexpected connections that emerge within the field of mathematics are all highlighted in these facts, which provide an insight into the intriguing field.