The World of Numbers Lecture Notes

Flashcards covering the history and classification of number systems, from ancient counting bones and Indian philosophical roots to real and imaginary numbers.

Natural Numbers ($\mathbb{N}$)

The basic set of counting numbers, denoted as $\{1, 2, 3, 4, \dots\}$, originating from the human necessity to keep count.

One-to-one correspondence

A concept used by early humans to ensure a count was accurate by matching one object (like a pebble) to another (like a cow) without written symbols.

Lebombo Bone

An artifact discovered between South Africa and Swaziland dating back approximately $35,000$ years, featuring $29$ notches believed to be a lunar or menstrual calendar.

Ishango bone

A 20,000 BCE mathematical marvel found in the Democratic Republic of Congo featuring notches grouped into prime numbers and demonstrations of doubling.

Parārdha

The name given to the power of $10^{12}$ in the ancient Indian Vedas.

Tallakṣhaṇa

The name described by Buddha in the 4th century BCE for the power of $10^{53}$.

Śhūnya

The Sanskrit word for zero, which was formally transformed into a functional number by Brahmagupta in $628\,CE$.

Śhūnyatā

The philosophical concept of emptiness or nothingness in Indian traditions that provided the conceptual framework for the mathematical zero.

Bakhśhālī Manuscript

An early centuries CE document that features a bold dot, known as a bindu, to physically represent zero.

Brāhmasphuṭasiddhānta

The seminal work written by Brahmagupta in $628\,CE$ where he defined zero as the result of subtracting a number from itself ($a - a = 0$).

Dhana (Fortunes)

Brahmagupta's term for positive numbers, representing wealth or assets in a commercial context.

Ṛiṇa (Debts)

Brahmagupta's term for negative numbers, representing financial obligations or deficits.

Integers ($\mathbb{Z}$)

The set containing positive natural numbers, their negative counterparts, and zero, derived from the German word 'Zahlen'.

Rational Numbers ($\mathbb{Q}$)

Any number that can be expressed in the form $\frac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$. The 'Q' stands for quotient.

Equivalent rational numbers

Different fractions that represent the same value on a number line, such as $\frac{1}{2}$ and $\frac{2}{4}$, allowed by dividing out common factors.

Absolute value

The distance of a rational number $x$ from $0$ on the number line, written as $|x|$, and is always non-negative ($|x| \geq 0$).

Density of Rational Numbers

The property that between any two rational numbers, there are infinitely many more rational numbers that can be found by taking averages.

Irrational Numbers

Numbers on the number line that cannot be expressed as a ratio of integers ($\frac{p}{q}$), such as $\sqrt{2}$ or $\pi$.

Proof by Contradiction

A logical technique used by Hippasus to prove the irrationality of $\sqrt{2}$ by assuming the opposite and showing it leads to a logical disaster.

Mādhava of Sangamagrama

Founder of the Kerala School of Mathematics who discovered an infinite series formula for exactly expressing $\pi$ in the 14th century.

Real Numbers ($\mathbb{R}$)

The unbroken, continuous line formed by the union of all rational and irrational numbers.

Terminating decimal

A rational number's decimal expansion that stops because the division eventually leaves a remainder of $0$.

Repeating decimal

A decimal expansion where the division never reaches a remainder of $0$, but a sequence of digits loops infinitely.

Pure repeating decimal

A decimal in which the repetition of digits begins immediately after the decimal point.

General repeating decimal

A decimal containing some non-repeating digits immediately after the decimal point followed by a repeating block.

Cyclic number

A repeating block of digits, such as $142857$ from $\frac{1}{7}$, where the digits shift in a circle when multiplied by certain integers.

Imaginary Numbers ($i$)

A dimension of numbers invented to represent the square root of $-1$, which cannot exist on the Real Number line.