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Number

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A number is a mathematical object used to count, measure, and label.

Ishango bone (cropped)
Ishango bone (cropped)

Numbers allow us to quantify the world around us. We use them to determine the size of a collection, known as a cardinal number. We also use them for ordering, such as with serial numbers, or for coding, such as with ISBNs. While people often use the terms interchangeably, a number is different from a numeral. A numeral is the specific symbol used to represent a number. For example, "eleven" is a number word, while "11" is the corresponding numeral.

To represent any number efficiently, we use a numeral system. The most common is the Hindu-Arabic system, a decimal system using ten digits. This system can display any non-negative integer through organized combinations of symbols.

Mathematics has expanded the concept of numbers over many centuries. The most basic are natural numbers, such as 1, 2, and 3. Mathematicians also use rational numbers, which are fractions representing parts of a whole. Real numbers include values like pi (π) or the square root of 2.

Archimedes pi
Archimedes pi

Beyond these are complex numbers, which extend real numbers by using the square root of -1, represented as *i*.

Venn Diagram of Numbers Expanded
Venn Diagram of Numbers Expanded

We perform calculations with these objects using arithmetic operations. These include addition, subtraction, multiplication, division, and exponentiation. The study of these properties is called arithmetic or number theory.

Humanity's relationship with numbers began with very simple tools.

YBC-7289-OBV-labeled
YBC-7289-OBV-labeled

Archaeologists have found bones with marks cut into them, which may be tally marks. The Lebombo bone dates to about 43,000 years ago. The Ishango bone is estimated to be between 22,000 and 30,000 years old. These marks might have recorded lunar cycles or animal quantities. Tallying systems are the first abstract numeral systems, though they lack place value. This limitation makes it difficult to represent very large numbers. Later, the Mesopotamians developed a base 60, or sexagesimal, system around 3400 BC. The Egyptians used a base 10 system starting around 3100 BC.

One of the most significant shifts in math was the treatment of zero.

Khmer Numerals - 605 from the Sambor inscriptions
Khmer Numerals - 605 from the Sambor inscriptions

Initially, zero served only as a placeholder in place-value systems. The Babylonians and Egyptians used zero in this way. In the New World, the Olmec people used a shell glyph as a placeholder by 38 BC. The Maya later developed zero as a cardinal number in their base 20 system.

Maya
Maya

In the Old World, the Indian mathematician Brahmagupta formulated the concept of zero as a number in AD 628. He treated zero as an integer and established rules for its use. He discussed how zero interacts with positive and negative numbers. He even addressed the operation of division by zero.

Negative numbers also represent a major conceptual development. As early as 100–50 BC, mathematicians in China recognized negative numbers. They used red and black rods to denote positive and negative coefficients. In India, negative numbers were used during the 600s to represent debts. European mathematicians resisted this idea for a long time. Even in the 18th century, many ignored negative results as meaningless. René Descartes referred to them as "false roots" when they appeared in algebraic polynomials. It was not until the 17th century that the concept gained broader acceptance in Europe.

Rational and irrational numbers provide even more precision. Rational numbers include fractions, which the Ancient Egyptians used in texts like the Rhind Mathematical Papyrus. These texts show how to derive areas using fractional notation. Irrational numbers, such as the square root of 2, cannot be expressed as simple fractions. Interestingly, the Babylonians showed they could approximate these values as early as 1800 BCE. Their clay tablet, YBC 7289, shows an approximation of the square root of 2 with great accuracy. This demonstrates that humans have been seeking precise numerical values for millennia.

Today, the concept of numbers continues to grow through algebraic structures. In the 19th century, mathematicians developed systems that share properties with numbers. Some of these, like p-adic or hypercomplex numbers, are explicitly called numbers. Others are categorized differently based on mathematical convention. These structures extend our ability to model complex systems. From ancient tally marks to modern algebraic theory, numbers remain the fundamental language of measurement and logic.

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🖼️ Images & Media (15)
File:Venn Diagram of Numbers Expanded.svg
Venn Diagram of Numbers Expanded.svg
File:Archimedes pi.svg
Archimedes pi.svg
File:SimilarGoldenRectangles.svg
SimilarGoldenRectangles.svg
File:Largest known prime number.svg
Largest known prime number.svg
File:NumberSetinR2.svg
NumberSetinR2.svg
File:Numeral Systems of the World.svg
Numeral Systems of the World.svg
File:Maya.svg
Maya.svg
File:Euler's formula caimi.svg
Euler's formula caimi.svg
File:Nat num.svg
Nat num.svg
File:The Ancient Quipu Plate XXI.jpg
The Ancient Quipu Plate XXI.jpg
File:Mandel zoom 00 mandelbrot set.jpg
Mandel zoom 00 mandelbrot set.jpg
File:YBC-7289-OBV-labeled.jpg
YBC-7289-OBV-labeled.jpg

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