Zero: The Biography of a Dangerous Idea: Chapters 5–6

by Charles Seife
Science journalist Charles Seife discusses the history of the number zero, from its origin as an Eastern philosophical concept to its rise as an important tool in mathematics to its current threat to modern physics.

Here are links to our lists for the book: Chapters 0–1, Chapter 2, Chapters 3–4, Chapters 5–6, Chapter 7–∞
35 words 42 learners

Learn words with Flashcards and other activities

Definition First
Print Flashcards

Other learning activities

Answer a few questions about each word. Use this to prep for your next quiz! Compete with other teams in real time to see who answers the most questions correctly! Test your spelling acumen. Read the definition, listen to the word and try spelling it!

Teaching tools

Create and assign quizzes to your students to test their vocabulary. Assign learning activities including Practice, Vocabulary Jams and Spelling Bees to your students, and monitor their progress in real-time.

Full list of words from this list:

words only definitions & notes
  1. stupor
    a state of being half-awake
    When, after a thousand-year stupor, European thought shook off the effect of the sleeping powders so skilfully administered by the Christian Fathers, the problem of infinity was one of the first to be revived.
  2. naive
    lacking information or instruction
    After all, the numbers in the sequence get closer and closer to zero; naively, one would guess that this would ensure that the sum remains finite.
  3. crude
    belonging to an early stage of technical development
    Johannes Kepler—the man who figured out that planets move in ellipses—spent that year gazing into wine barrels, since he realized that the methods that vintners and coopers used to estimate the size of barrels were extremely crude.
  4. tangent
    a line that touches a curve at only one point
    A tangent is a line that just kisses a curve. For any point along a smooth curve that flows through space, there is a line that just grazes the curve, touching at exactly one point. This is the tangent, and mathematicians realized that it is extremely important in studying motion.
  5. infuse
    fill, as with a certain quality
    Calculus was the very language of nature, yet its very fabric was infused with zeros and infinities that threatened to destroy the new tool.
  6. dubious
    open to doubt or suspicion
    Newton’s method of fluxions was very dubious. It relied upon an illegal mathematical operation, but it had one huge advantage. It worked.
  7. constancy
    the quality of being enduring and free from change
    These early equation-laws were extremely good at expressing simple relationships, but equations have limitations—their constancy, which prevented them from being universal laws.
  8. underpin
    confirm or support with evidence or authority
    It would be many years before mathematicians began to free calculus from its mystical underpinnings, for the mathematical world was busy fighting over who invented calculus.
  9. enigma
    something that baffles understanding and cannot be explained
    Zero was no longer an enemy to be avoided; it was an enigma to be studied.
  10. emanate
    proceed or issue forth, as from a source
    And as physicists and mathematicians all over the world were beginning to use calculus to explain nature, cries of protest emanated from the church.
  11. discourse
    an extended communication dealing with some particular topic
    In 1734, seven years after Newton’s death, an Irish bishop, George Berkeley, wrote a book entitled The Analyst, Or a Discourse Addressed to an Infidel Mathematician.
  12. impunity
    exemption from punishment or loss
    Calling infinitesimals “ghosts of departed quantities,” Berkeley showed how making these infinitesimals disappear with impunity can lead to a contradiction.
  13. rigorous
    demanding strict attention to rules and procedures
    Every theorem in geometry had been rigorously proved; by taking a few rules from Euclid and proceeding, very carefully, step by step, a mathematician could show how a triangle’s angles sum to 180 degrees, or any other geometric fact.
  14. chimera
    a grotesque product of the imagination
    A quantity is something or nothing; if it is something, it has not yet vanished; if it is nothing, it has literally vanished. The supposition that there is an intermediate state between these two is a chimera.
  15. caveat
    a warning against certain acts
    Though the two expressions look dramatically different, they are (with some caveats) exactly the same.
  16. prolific
    intellectually productive
    Euler was an excellent mathematician—in fact, he was one of the most prolific and influential in history—but in this case the careless manipulation of zero and infinity led him astray.
  17. foundling
    a child who has been abandoned and whose parents are unknown
    It was a foundling who finally tamed the zeros and infinities in calculus and rid mathematics of its mysticism. In 1717 an infant was found on the steps of the church of Saint Jean Baptiste le Rond in Paris.
  18. extrapolate
    estimate the value of
    Then, after all your manipulations are complete, you take the limit: you extrapolate and figure out where the expression is headed.
  19. adversary
    someone who offers opposition
    Zero and infinity are two sides of the same coin—equal and opposite, yin and yang, equally powerful adversaries at either end of the realm of numbers.
  20. ethereal
    characterized by lightness and insubstantiality
    Leibniz thought that i was a bizarre mix between existence and nonexistence, something like a cross between 1 (God) and 0 (Void) in his binary scheme. Leibniz likened i to the Holy Spirit: both have an ethereal and barely substantial existence.
  21. capitulate
    surrender under agreed conditions
    In 1763 the French capitulated and the Seven Years’ War was over.
  22. consummate
    having or revealing supreme mastery or skill
    Monge was a consummate geometer, specializing in three-dimensional geometry.
  23. molder
    decay or break down
    Moldering in a Russian prison, Poncelet founded a new discipline: projective geometry.
  24. culmination
    a concluding action
    Poncelet’s mathematics was the culmination of the work begun by the artists and architects of the fifteenth century, like Filippo Brunelleschi and Leonardo da Vinci, who discovered how to draw realistically—in perspective.
  25. projection
    the representation of a figure or solid on a plane
    This was the beginning of the discipline of projective geometry, where mathematicians look at the shadows and projections of geometric figures to uncover hidden truths even more powerful than the equivalence of parabolas and ellipses.
  26. archaic
    so extremely old as seeming to belong to an earlier period
    Upon his return from Russia (carrying a Russian abacus, by then an archaic oddity) he raised the discipline to a high art.
  27. relativity
    the theory that space and time are not absolute concepts
    Throughout his life Gauss worked on an incredible variety of topics—his work on curvature would become a key component of Einstein’s general theory of relativity—but it was Gauss’s way of graphing complex numbers that revealed a whole new structure in mathematics.
  28. reciprocal
    one of a pair of numbers whose product is 1
    Most interesting of all, if you take a number x and replace it with its reciprocal 1/x, that is equivalent to flipping the sphere upside down and reflecting it in a mirror.
  29. engulf
    flow over or cover completely
    Zero and infinity are eternally locked in a struggle to engulf all the numbers. Like a Manichaean nightmare, the two sit on opposite poles of the number sphere, sucking numbers in like tiny black holes.
  30. infallible
    incapable of failure or error
    My theory stands as firm as a rock; every arrow directed against it will return quickly to its archer. How do I know this? I have studied it....I have followed its roots, so to speak, to the first infallible cause of all created things.
  31. oscillate
    move or swing from side to side regularly
    Essential singularities are weird beasts; near a singularity of this sort, a curve goes absolutely berserk. It oscillates up and down faster and faster as it approaches the singularity, whipping from positive to negative and back again.
  32. abomination
    an action that arouses disgust or abhorrence
    The integers represented the purity of God, while the irrationals and other bizarre sets of numbers were abominations—figments of the imperfect human mind.
  33. vitriolic
    harsh, bitter, or malicious in tone
    Disgusted with Cantor, Kronecker launched vitriolic attacks against Cantor’s work and made it extremely difficult for him to publish papers.
  34. cohort
    a company of companions or supporters
    Adding up the size of the carpets, we see 1 + 1/2 + 1/4 + 1/8 + ...+ 1/2n goes to 2 as n goes to infinity. So we can cover the infinite cohorts of rational numbers in the number line with a set of carpets, and the total size of the carpets is 2. This means that the rational numbers take up less than two units of space.
  35. quark
    fundamental subatomic particle that has a fractional charge
    We can cover the rationals with carpets that, summed together, fit in the size of half an atom—or a neutron—or a quark—or as small as we can possibly imagine.
Created on Sun Feb 06 12:51:38 EST 2022 (updated Tue Aug 23 09:22:50 EDT 2022)

Sign up now (it’s free!)

Whether you’re a teacher or a learner, Vocabulary.com can put you or your class on the path to systematic vocabulary improvement.