Category: Numbers

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Roman Numerals vs Hindu-Arabic Numbers in Psychology

What Our Brains See vs What They Read

The way our brain interprets Roman numerals and Hindu-Arabic numbers reveals an interesting distinction in psychology—especially when viewed through the lens of communication theory.

According to the psychologist and communication theorist Paul Watzlawick, signs and symbols can be divided into two categories: analogical and digital.

· Analogical signs resemble what they represent. They are intuitive and visually descriptive.

· Digital signs are symbolic. They rely on learned codes and have no visual connection to what they signify.

In this sense, Roman numerals (like I, II, III) are analogical. When we see “II”, we can immediately see two units. The visual repetition reflects the quantity directly—our brain interprets the number almost as a drawing of its value.

On the other hand, Hindu-Arabic numbers (like 2, 3, 4) are digital. The symbol “2” doesn’t visually resemble two objects—it’s abstract. Understanding it depends on prior learning and decoding. The brain treats it more like language than image.

This distinction matters. Roman numerals engage perception in a way that mimics reality. Arabic numerals, by contrast, engage abstract reasoning. The first shows, the second tells.

In daily life, we may not notice the difference—but psychologically, the visual nature of Roman numerals connects us to meaning more directly, while the efficiency of Arabic numerals supports speed, calculation, and abstraction.

In short:
Roman numerals speak to the eye.
Arabic numbers speak to the mind.

⇨ More about numbers.

roman numbers vs arabic numbers
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The Origins of Our Numerals

The Kitāb al-Fuṣūl fī al-Ḥisāb al-Hindī (كتاب الفصول في الحساب الهندي), or The Book of Chapters on Hindu Arithmetic, authored by Abū al-Ḥasan Aḥmad ibn Ibrāhīm al-Uqlīdisī in 952 CE, is the earliest known Arabic treatise detailing Indian arithmetic and the use of Hindu-Arabic numerals. A unique manuscript of this work is preserved in the Yeni Cami Library in Istanbul. The treatise also offers the earliest documentation of numerals in use in Damascus and Baghdad.

Another significant reference is found in Talqīḥ al-Afkār bi-Rusūm Ḥurūf al-Ghubār (تلقيح الأفكار برُسوم حروف الغبار), or Fertilization of Thoughts with the Help of Dust Letters, by the Berber mathematician Ibn al-Yāsamīn (ابن الياسمين), who died in 1204. In the excerpt shown below, he presents the Indian numerals, stating:​

“Know that specific forms have been chosen to represent all numbers; they are called ‘ghubār’ (dust), and they are these (first row). They may also appear like this (second row). However, among us, people use the first type of forms.”​

An intriguing anecdote about Ibn al-Yāsamīn is that he composed mathematical poems, such as the Urjūza fī al-Jabr wa al-Muqābala, to make algebra more accessible. These poetic works were not only educational tools but also reflected the rich interplay between mathematics and literature in the Islamic Golden Age.

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Benford’s Law: Why 1 Comes First

Benford’s Law is a curious mathematical rule that describes how often different digits (1–9) appear as the first digit in many real-life datasets. Surprisingly, lower digits (like 1) show up much more frequently than higher ones (like 9).

Simple Formula

The probability of a digit d being the first digit is:

📌 P(d) = log₁₀(1 + 1/d)

For example, the number 1 appears as the first digit about 30% of the time, while 9 appears only about 5% of the time! This pattern shows up in finance, science, populations, and even street addresses.

A fascinating rule of nature—numbers aren’t as random as they seem!

benford's law

⇨ Further reading.

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The Metallic Ratios: Beyond the Golden Ratio

Many people are familiar with the Golden Ratio (φ), but it is just one member of a broader family known as the Metallic Ratios. These ratios describe a recursive relationship between the sides of a rectangle. Given a rectangle with side lengths A and B (B > A), the Metallic Ratios satisfy the equation:

A / B = (B – n × A) / A

where n is a fixed integer.

For the Golden Ratio (n = 1):
A / B = (B – A) / A
which leads to:
φ² – φ – 1 = 0
solving this gives:
φ = (1 + √5) / 2 ≈ 1.618…

For the Silver Ratio (n = 2):
A / B = (B – 2A) / A
which leads to:
ψ² – 2ψ – 1 = 0
solving this gives:
ψ = 1 + √2 ≈ 2.414…

Different values of n define other Metallic Ratios. The table below presents some of them.

Curious minds, don’t stop here—explore the fascinating properties of the other Metallic Ratios and see what patterns and surprises you can uncover!

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The Zaniest Word in Math: Zenzizenzizenzic

One of the most peculiar numeral words in English, zenzizenzizenzic (/’zɛnziːzɛnziːzɛnzik/), denotes the square of the square of a number’s square. It appeared only once in English, in Robert Recorde’s The Whetstone of Wit (1557). The term derives from the obsolete zenzic, meaning the square of a number. Zenzic was borrowed from German, where mathematicians of the 14th and 15th centuries adopted it from the medieval Italian censo, itself a descendant of Latin census. Italian algebraists used censo to translate the Arabic māl (literally “possessions” or “property”), the standard term for a squared number. This association arose because early mathematicians, including the Arabs, conceptualized squared numbers as representing areas, particularly land—hence, property.

Notably, zenzizenzizenzic is the only English word with six Zs.

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How a Schoolboy Outsmarted a Tedious Task

1780s, Germany. A schoolteacher, desperate for some peace, gives his 8-year-olds a tedious task: add up all the numbers from 1 to 100. That should keep them busy, right?

Enter young Carl Friedrich Gauss. While his classmates grind away, he takes a quick look, “folds” the numbers—1 pairs with 100, 2 with 99, and so on—realizing each pair sums to 101. With 50 such pairs, he multiplies: 50 × 101 = 5050.

Boom. Two minutes, problem solved. Teacher stunned. Classmates still counting. Gauss goes on to become one of history’s greatest mathematicians.

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Creating Perfect Squares from Odd Integers

It’s visually easy to see that the sum of consecutive odd numbers forms perfect squares—this brilliant animation is perfect for empirically understanding why. But how can we explain it in words?

1️⃣ The sum of consecutive odd numbers produces perfect squares
– The sequence of odd numbers:
1, 3, 5, 7, 9, …
– The sum of the first n odd numbers follows the formula:
1 + 3 + 5 + … + (2n-1) = n²
– This can be proven by induction.

2️⃣ The discrete derivative of n² is 2n + 1
– The discrete derivative (forward difference) of a function f(n) is:
Δ f(n) = f(n+1) – f(n)
– Applying it to f(n) = n²:
(n+1)² – n² = n² + 2n + 1 – n² = 2n + 1
– This shows that the difference between consecutive squares is always an odd number—specifically, the (n+1)th odd number!

A simple—well, for those who love math—yet beautiful mathematical insight!