The World of Numbers Lecture Notes Flashcards

A Comprehensive Vocabulary List on Number Systems and their History.

One-to-one correspondence

The simple act of matching one object to another used by early humans to ensure no livestock was lost, marking the birth of Natural Numbers.

Natural Numbers ($\mathbb{N}$)

The set of basic counting numbers, defined as $\mathbb{N} = {1, 2, 3, 4, \dots }$.

Lebombo Bone

An artifact discovered in the Lebombo Mountains containing 29 distinct, deliberately-carved notches dating back approximately $35,000$ years.

Age of Lebombo Bone

Approximately $35,000$ years old.

Notch count of Lebombo Bone

It features $29$ distinct, uniformly-sized notches, believed to track time such as lunar phases or a menstrual calendar.

Ishango bone

A mathematical artifact found near the headwaters of the Nile River in the Democratic Republic of Congo, dating to around $20,000$ BCE.

Age of Ishango bone

Dating to approximately $20,000$ BCE.

Prime number grouping (Ishango)

Specific tallies found on the Ishango bone grouped into $11$, $13$, $17$, and $19$.

Doubling (Ishango)

A concept of multiplication by $2$ found in one of the columns of the Ishango bone.

Urban centers (Indus Valley)

Ancient cities like Lothal and Harappa where standardized weights and measures were crucial for trade.

Ancient trade materials

Items such as terracotta pottery, lapis lazuli, or cotton traded with Mesopotamia.

Parārdha

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

Lalitavistara

A text from the $4^{th}$ century BCE where Buddha describes names for powers of $10$.

Tallakṣhaṇa

The specific name given to the power of $10^{53}$ as described in the Lalitavistara.

Concept of zero

Considered perhaps the most important mathematical invention in human history, born from the Indian place-value system.

Joint counting

A practice using the $3$ joints of each finger and the thumb to count, relating to ancient base-12 systems.

Placeholders (Babylonian/Mayan)

Symbols used by early civilizations to indicate an empty column in a number, though not treated as an operational number.

Brahmagupta

The Indian mathematician who formally transformed the void into an operational number in $628$ CE.

Date of Brahmagupta

He lived around $628$ CE.

Śhūnyatā

A philosophical concept of emptiness or nothingness from the Upanishads and Buddhist literature.

Emptiness

The translation of the word śhūnyatā, a state aimed for in yoga and meditation.

Vṛttis

Fluctuations of the mind that one seeks to empty to reach stillness, as described in meditation contexts.

Patanjali

The author who described how śhūnyatā leads to control over the mind and body in the Yoga Sutras around the $3^{rd}$ century BCE.

Yoga Sutras period

Compiled around the $3^{rd}$ century BCE.

Fields using zeroness

Fields such as architecture, literature, linguistics, and eventually mathematics.

Āryabhaṭa

An early Indian mathematician whose work preceded Brahmagupta's mathematical formalization of zero.

Bakhśhālī Manuscript

An early document dated to the early centuries CE featuring the physical symbol for zero.

Bindu

A bold dot used in the Bakhśhālī Manuscript to represent zero.

Brāhmasphuṭasiddhānta

Brahmagupta's seminal work from $628$ CE defining the fundamental laws of arithmetic.

Mathematical definition of zero

Defined by Brahmagupta as the result of subtracting a number from itself ($a - a = 0$).

Brahmagupta’s Rule (Addition)

When zero is added to a number, the number remains unchanged ($a + 0 = a$).

Brahmagupta’s Rule (Subtraction)

When zero is subtracted from a number, the number remains unchanged ($a - 0 = a$).

Brahmagupta’s Rule (Multiplication)

When any number is multiplied by zero, the result is zero ($a \times 0 = 0$).

Fortunes (Dhana)

Positive numbers representing wealth or assets in commercial mathematics.

Debts (Ṛiṇa)

Negative numbers representing financial liabilities or debts.

Negative Numbers

Numbers formally introduced by moving to the left of zero on the number line.

Integers ($\mathbb{Z}$)

The set created by positive natural numbers, their negative counterparts, and zero.

Origin of symbol $\mathbb{Z}$

Derived from the German word Zahlen, meaning numbers.

Arithmetic of Integers: Rule 1

A fortune plus a fortune is a fortune ($5 + 4 = 9$).

Arithmetic of Integers: Rule 2

A debt plus a debt is a debt ($(-5) + (-4) = -9$).

Arithmetic of Integers: Rule 3

A fortune minus zero is a fortune, and a debt minus zero is a debt.

Arithmetic of Integers: Rule 4

The product of a debt and a fortune is a debt ($(-3) \times 4 = -12$).

Arithmetic of Integers: Rule 5

The product of two debts is a fortune ($(-3) \times (-4) = 12$).

Removal of debt

A conceptual explanation for why a negative times a negative equals a positive ($- \times - = +$).

Fractions

Numbers that represent parts of a whole, necessitated by measuring and complex societal growth.

Negative fractions

The additive inverses of positive fractions, where the sign can be placed on the numerator, denominator, or the front.

Rational Numbers ($\mathbb{Q}$)

The set combining all integers and all fractions, both positive and negative.

Definition of a rational number

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

$q \neq 0$ constraint

The essential requirement in the definition of a rational number because division by zero is not defined.

Equivalent rational numbers

Different representations of the same value in $\frac{p}{q}$ form, such as $\frac{-1}{3} = \frac{-2}{6}$.

Representing 12/30 as 2/5

The process of dividing both the numerator and denominator by a common factor ($6$) to find an equivalent fraction.

Co-prime

The state of two integers having no common factors other than $1$.

Representative value on number line

The choosing of a single fraction in simplest form (where $p$ and $q$ are co-prime) to represent all equivalent fractions.

Rational Equality Law

Two rational numbers $\frac{a}{b}$ and $\frac{c}{d}$ are equal if $ad = bc$.

Rational Addition Law

$\frac{a}{b} + \frac{c}{b} = \frac{a + c}{b}$, where both fractions share a common denominator.

Rational Multiplication Law

$\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}$ (where $b \neq 0$ and $d \neq 0$).

Rational Division Law

$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{ad}{bc}$ (where $b, d, c \neq 0$).

Commutative property

The mathematical law stating that the order of addition or multiplication for rational numbers does not change the result.

Law of distributivity

The property followed by rational numbers expressed as $p(q + r) = pq + pr$.

Rational closure

The property where adding, subtracting, or multiplying two rational numbers always results in another rational number.

Origin

The point marked as $0$ on a number line from which all other points are measured.

Unit interval

The distance between two consecutive integers on a number line.

Representation of $p/q$

To locate on a number line, divide the unit interval into $q$ equal parts and move $p$ parts from $0$.

Absolute value ($|x|$)

The distance of a rational number from $0$ on the number line, always resulting in a non-negative value.

Sign result of absolute value

The absolute value of any rational number is always non-negative ($|x| \geq 0$).

Distance between numbers $a$ and $b$

Calculated as $|a - b|$ on the number line.

Density of rational numbers

The property that there are infinitely many rational numbers between any two points on the number line.

Finding a rational between $a$ and $b$

A method using the average formula $\frac{a + b}{2}$ to find a number between any two rational numbers.

Baudhāyana

The author of the Śhulbasūtra around $800$ BCE who encountered lengths that could not be expressed as fractions.

Śhulbasūtra

An ancient manual used for constructing complex geometric fire altars.

Length of diagonal (unit square)

An irrational length expressed as $\sqrt{2}$ based on the Baudhāyana-Pythagoras Theorem.

Irrational Numbers

Numbers on the number line that cannot be expressed as a ratio of integers.

Hippasus

The mathematician from the Pythagorean school (c. $400$ BCE) credited with the first proof of the irrationality of $\sqrt{2}$.

Proof by Contradiction

A logic technique where one assumes the opposite of what is to be proven to show it leads to a logical failure.

Step 1 (Irrationality of $\sqrt{2}$)

Assume $\sqrt{2}$ is a rational number that can be written in its simplest form $\frac{p}{q}$.

Step 4 (Irrationality of $\sqrt{2}$)

Deduce that $p^2$ is even, meaning $p$ must also be an even integer.

Step 7 (Irrationality of $\sqrt{2}$)

Deduce that $q^2$ is even, meaning $q$ must also be an even integer.

Step 8 (Irrationality of $\sqrt{2}$)

Conclude the assumption is false because $p$ and $q$ were assumed to share no common factors, but both are even.

Rule of ruler and compass (distance)

To construct $\sqrt{2}$, draw a unit perpendicular at $1$, find the hypotenuse, and arc it to the number line.

$\pi$ (Pi)

The ratio of a circle's circumference to its diameter, which is an irrational number.

Āryabhaṭa's approximation of $\pi$

The value $\frac{3927}{1250} = 3.1416$ given in $499$ CE.

Asanna

A term used by Āryabhaṭa to indicate that his value for $\pi$ was only an approximation.

Johann Lambert

The mathematician who formally proved the irrationality of $\pi$ in $1761$.

Mādhava of Sangamagrama

Founder of the Kerala School of Mathematics credited with the first infinite series formula for $\pi$.

Kerala School of Mathematics

An influential mathematical school launched by Mādhava in the $14^{th}$ century.

Mādhava’s infinite series for $\pi$

$\pi = 4 \times (1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \dots)$, representing $\pi$ as an infinite sum.

Real Numbers ($\mathbb{R}$)

The set formed by uniting all Rational and Irrational numbers to create an unbroken, continuous line.

Signature of rational numbers

In decimal form, they either terminate or repeat in a looped sequence.

Terminating decimal expansion

A decimal that eventually leaves a remainder of $0$ and stops, such as $\frac{3}{8} = 0.375$.

Repeating decimal expansion

A decimal where the division never reaches $0$ and a sequence of digits begins to loop infinitely.

Cause of repeating decimals

The limited number of possible remainders when dividing by an integer, which eventually forces repetition.

Predicting terminating expansions

A rational number $\frac{p}{q}$ terminates if the prime factors of $q$ are only $2$, only $5$, or both.

Pure repeating decimal

A decimal in which the repeating sequence begins immediately after the decimal point.

General repeating decimal

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

Cyclic Number

A sequence of digits like $142857$ that shifts in a cyclic circle when multiplied by certain digits.

Cyclic property of $1/7$

The repeating block $142857$ shifts its order ($285714$, $428571$, etc.) when multiplying by $1$ through $6$.

Irrational decimal expansion

An expansion that possesses an infinite number of digits without any terminating or repeating pattern.

Non-uniqueness ($0.999 \dots$)

The mathematical fact that $\dots$ repeating nines is exactly equal to the next whole number ($0.999 \dots = 1$).

Imaginary Numbers ($i$)

A new dimension of numbers invented to solve equations like the square root of $-1$.

Square root of $-1$

Represented by the letter $i$, which does not exist on the Real Number line.