--- title: "Meaning & Key Takeaway of: The trouble with integers is that we have…" description: "\"The trouble with integers is that we have examined only the very small ones. Maybe all the exciting stuff happens at really big numbers, ones we can't ..." image: "https://sms4smile.com/quotes/i/895/ronald-graham-34796_berries_the-trouble-with-integers-is-that-we-hav_2929745.jpg" --- # The trouble with integers is that we have examined only… ![“The trouble with integers is that we have examined only the very small ones. Maybe all the exciting stuff happens at really big numbers, ones we can't even begin to think about in any very definite way. Our brains have evolved to get us out of the rain, find where the berries are, and keep us from…” quote by Ronald Graham](https://sms4smile.com/quotes/i/895/ronald-graham-34796_berries_the-trouble-with-integers-is-that-we-hav_2929745.jpg) [Download](https://sms4smile.com/quotes/i/895/ronald-graham-34796_berries_the-trouble-with-integers-is-that-we-hav_2929745.jpg) [Open image](https://sms4smile.com/quotes/i/895/ronald-graham-34796_berries_the-trouble-with-integers-is-that-we-hav_2929745.jpg) > “The trouble with integers is that we have examined only the very small ones. Maybe all the exciting stuff happens at really big numbers, ones we can't even begin to think about in any very definite way. Our brains have evolved to get us out of the rain, find where the berries are, and keep us from getting killed. Our brains did not evolve to help us grasp really large numbers or to look at things in a hundred thousand dimensions.” [Ronald Graham](https://sms4smile.com/quotes/author/ronald-graham/) ## About This Quote This interpretation was drafted with AI assistance. It is one reading of the quote, not the author's own explanation. Human cognition is tuned for small numbers and practical survival, making large or high‑dimensional numbers difficult to grasp. **In simple terms:** Our brains handle small numbers well, not huge ones. Key Takeaway Develop tools to comprehend large scales. Themes cognition number theory brain evolution scale perception mathematics Mood thoughtful inquisitive Type educational scientific When to use this quote - data analysis - scientific research - education - risk assessment - technology design Key Concepts large numbers dimensionality cognitive limits visualization techniques Questions to Reflect On - How can we improve intuition for large numbers? - What tools aid understanding high‑dimensional data? A Different Perspective Relying solely on intuition can mislead in complex calculations. ★ ★ ★ ★ ★ No ratings yet [Brain](https://sms4smile.com/quotes/about/brain/) [Dimensions](https://sms4smile.com/quotes/about/dimensions/) [Integers](https://sms4smile.com/quotes/about/integers/) [Large numbers](https://sms4smile.com/quotes/about/large-numbers/) [Math](https://sms4smile.com/quotes/about/math/) [Numbers and math](https://sms4smile.com/quotes/about/numbers-and-math/) [Rain](https://sms4smile.com/quotes/about/rain/) [Thinking](https://sms4smile.com/quotes/about/thinking/) ## Related quotes > [It is astonishing that human brains, which evolved to cope with the everyday world, have been able to grasp the counterintuitive mysteries of the…](https://sms4smile.com/quotes/martin-rees/it-is-astonishing-that-human-brains-which-evolved-to-cope-with-the/) [Martin Rees](https://sms4smile.com/quotes/author/martin-rees/) > [“Indeed, there is an emerging body of evidence to suggest that we are all modern cavemen, approaching the problems of our complex contemporary world…](https://sms4smile.com/quotes/anonymous/indeed-there-is-an-emerging-body-of-evidence-to-suggest-that-we-are/) [Anonymous](https://sms4smile.com/quotes/author/anonymous/) > [We frequently look into the future of mankind and see dangers. We see if we carry on doing what we are doing in 20…](https://sms4smile.com/quotes/richard-dawkins/we-frequently-look-into-the-future-of-mankind-and-see-dangers-we-see/) [Richard Dawkins](https://sms4smile.com/quotes/author/richard-dawkins/) > [“Development never truly ends, as our brains evolved to evolve...we are adapted to adapt, to continually redefine normality, transforming ones space of possibility with…](https://sms4smile.com/quotes/beau-lotto/development-never-truly-ends-as-our-brains-evolved-to-evolve-we-are/) [Beau Lotto](https://sms4smile.com/quotes/author/beau-lotto/) > [Our brains have evolved to help us survive within the orders of magnitude of size and speed which our bodies operate at. We never…](https://sms4smile.com/quotes/richard-dawkins/our-brains-have-evolved-to-help-us-survive-within-the-orders-of/) [Richard Dawkins](https://sms4smile.com/quotes/author/richard-dawkins/) > [“... the reason why we find some things intuitively easy to grasp and others hard, is that our brains are themselves evolved organs: on-board…](https://sms4smile.com/quotes/richard-dawkins/the-reason-why-we-find-some-things-intuitively-easy-to-grasp-and/) [Richard Dawkins](https://sms4smile.com/quotes/author/richard-dawkins/) > [That's the case with these exponential technologies; our brains, they struggle with it. We live in a world that is global and exponential, and…](https://sms4smile.com/quotes/jason-silva/that-s-the-case-with-these-exponential-technologies-our-brains-they/) [Jason Silva](https://sms4smile.com/quotes/author/jason-silva/) > [I figured the people who liked the sort of thing I was doing would come see it. 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He focused on making the best.](https://sms4smile.com/quotes/tim-cook/everybody-in-technology-seems-to-want-big-numbers-steve-never-got/) [Tim Cook](https://sms4smile.com/quotes/author/tim-cook/) > [You have to track every single thing you eat if you want to keep posting big numbers on the scale each week.](https://sms4smile.com/quotes/ali-vincent/you-have-to-track-every-single-thing-you-eat-if-you-want-to-keep/) [Ali Vincent](https://sms4smile.com/quotes/author/ali-vincent/) > [A lot of science doesn't require big "ns" but if you're trying to understand something about human health and you're looking at interventions that…](https://sms4smile.com/quotes/francis-collins/a-lot-of-science-doesn-t-require-big-ns-but-if-you-re-trying-to/) [Francis Collins](https://sms4smile.com/quotes/author/francis-collins/) > [When you lose and have big numbers, it's like, 'OK, sure, I played well, but how much does it really mean if you didn't…](https://sms4smile.com/quotes/nikola-vucevic/when-you-lose-and-have-big-numbers-it-s-like-ok-sure-i-played-well/) [Nikola Vucevic](https://sms4smile.com/quotes/author/nikola-vucevic/) > [You still just got to play your game. No matter where you're at, you've got to play your game, especially in majors. When you…](https://sms4smile.com/quotes/dustin-johnson/you-still-just-got-to-play-your-game-no-matter-where-you-re-at-you-ve/) [Dustin Johnson](https://sms4smile.com/quotes/author/dustin-johnson/) > [Maybe it will be a great thing when the Baby Boomers finally die out. In real life, it's not a matter of the good…](https://sms4smile.com/quotes/jack-bowman/maybe-it-will-be-a-great-thing-when-the-baby-boomers-finally-die-out/) [Jack Bowman](https://sms4smile.com/quotes/author/jack-bowman/) ## More by Ronald Graham > [Math is sometimes called the science of patterns.](https://sms4smile.com/quotes/ronald-graham/math-is-sometimes-called-the-science-of-patterns/) [Ronald Graham](https://sms4smile.com/quotes/author/ronald-graham/) > [The ultimate goal of mathematics is to eliminate all need for intelligent thought.](https://sms4smile.com/quotes/ronald-graham/the-ultimate-goal-of-mathematics-is-to-eliminate-all-need-for/) [Ronald Graham](https://sms4smile.com/quotes/author/ronald-graham/) > [Well, as you know, there are 24 hours in every day. And if that's not enough, you've always got the nights!](https://sms4smile.com/quotes/ronald-graham/well-as-you-know-there-are-24-hours-in-every-day-and-if-that-s-not/) [Ronald Graham](https://sms4smile.com/quotes/author/ronald-graham/) > [It wouild be very discouraging if somewhere down the line you could ask a computer if the Riemann hypothesis is correct and it said,…](https://sms4smile.com/quotes/ronald-graham/it-wouild-be-very-discouraging-if-somewhere-down-the-line-you-could/) [Ronald Graham](https://sms4smile.com/quotes/author/ronald-graham/) > [Someone has said that all the great jugglers are dead.](https://sms4smile.com/quotes/ronald-graham/someone-has-said-that-all-the-great-jugglers-are-dead/) [Ronald Graham](https://sms4smile.com/quotes/author/ronald-graham/) > [I was reminded of the Sydney Harris cartoon that said 'adding two numbers that have not been added before does not constitute a mathematical…](https://sms4smile.com/quotes/ronald-graham/i-was-reminded-of-the-sydney-harris-cartoon-that-said-adding-two/) [Ronald Graham](https://sms4smile.com/quotes/author/ronald-graham/) > [A lot of the high-level sports are really in your mind.](https://sms4smile.com/quotes/ronald-graham/a-lot-of-the-high-level-sports-are-really-in-your-mind/) [Ronald Graham](https://sms4smile.com/quotes/author/ronald-graham/) > [Someone has remarked that 'An ideal math talk should have one proof and one joke and they should not be the same'.](https://sms4smile.com/quotes/ronald-graham/someone-has-remarked-that-an-ideal-math-talk-should-have-one-proof/) [Ronald Graham](https://sms4smile.com/quotes/author/ronald-graham/) > [Some people think that mathematics is a serious business that must always be cold and dry; but we think mathematics is fun, and we…](https://sms4smile.com/quotes/ronald-graham/some-people-think-that-mathematics-is-a-serious-business-that-must/) [Ronald Graham](https://sms4smile.com/quotes/author/ronald-graham/) > [Juggling is sometimes called the art of controlling patterns, controlling patterns in time and space.](https://sms4smile.com/quotes/ronald-graham/juggling-is-sometimes-called-the-art-of-controlling-patterns/) [Ronald Graham](https://sms4smile.com/quotes/author/ronald-graham/) > [Incidentally, when we're faced with a "prove or disprove," we're usually better off trying first to disprove with a counterexample, for two reasons: A…](https://sms4smile.com/quotes/ronald-graham/incidentally-when-we-re-faced-with-a-prove-or-disprove-we-re-usually/) [Ronald Graham](https://sms4smile.com/quotes/author/ronald-graham/) > [AB=1/4((A+B)^2-(A-B)^2) is an amazing identity, and unfortunately, I have to remind my current students how to prove it.](https://sms4smile.com/quotes/ronald-graham/ab-1-4-a-b-2-a-b-2-is-an-amazing-identity-and-unfortunately-i-have-to/) [Ronald Graham](https://sms4smile.com/quotes/author/ronald-graham/) > [It would be very discouraging if somewhere down the line you could ask a computer if the Riemann hypothesis is correct and it said,…](https://sms4smile.com/quotes/ronald-graham/it-would-be-very-discouraging-if-somewhere-down-the-line-you-could/) [Ronald Graham](https://sms4smile.com/quotes/author/ronald-graham/) [Explore all 14 Ronald Graham quotes ](https://sms4smile.com/quotes/author/ronald-graham/) ## More Berries quotes > [“Yes, when I get big and have my own home, no plush chairs and lace curtains for me. And no rubber plants. I'll have…](https://sms4smile.com/quotes/betty-smith/yes-when-i-get-big-and-have-my-own-home-no-plush-chairs-and-lace/) [Betty Smith](https://sms4smile.com/quotes/author/betty-smith/) > [I’ve blamed her for all of this, for leaving, for ruining me. And maybe that was the seed of it, but from that one…](https://sms4smile.com/quotes/gayle-forman/i-ve-blamed-her-for-all-of-this-for-leaving-for-ruining-me-and-maybe/) [Gayle Forman](https://sms4smile.com/quotes/author/gayle-forman/) > [Even as a kid, I was a businessman. 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Wrapped tightly in our…](https://sms4smile.com/quotes/janine-benyus/a-solitary-american-monk-named-thomas-berry-writes-that-in-our/) [Janine Benyus](https://sms4smile.com/quotes/author/janine-benyus/) > [If you're going to do Chuck Berry, you got to, you know, go all out, and the duck walk is just kind of you…](https://sms4smile.com/quotes/mos-def/if-you-re-going-to-do-chuck-berry-you-got-to-you-know-go-all-out-and/) [Mos Def](https://sms4smile.com/quotes/author/mos-def/) > [After Halle Berry does her films and Queen Latifah does her films, it's left to all the black, Latino and Asian actresses to fight…](https://sms4smile.com/quotes/gabrielle-union/after-halle-berry-does-her-films-and-queen-latifah-does-her-films-it/) [Gabrielle Union](https://sms4smile.com/quotes/author/gabrielle-union/) > [I see what I want of Love... I see horses making the meadow dance, fifty guitars sighing, and a swarm of bees suckling the…](https://sms4smile.com/quotes/mahmoud-darwish/i-see-what-i-want-of-love-i-see-horses-making-the-meadow-dance-fifty/) [Mahmoud Darwish](https://sms4smile.com/quotes/author/mahmoud-darwish/) > [For a long time, no village girl would dress her hair or bosom with the sweetest flower from that field of death: and after…](https://sms4smile.com/quotes/charles-dickens/for-a-long-time-no-village-girl-would-dress-her-hair-or-bosom-with/) [Charles Dickens](https://sms4smile.com/quotes/author/charles-dickens/) > [I love dried mangoes, walnuts, and goji berries. The mix provides iron, antioxidants, and omega-3s and really kicks up my energy.](https://sms4smile.com/quotes/gretchen-bleiler/i-love-dried-mangoes-walnuts-and-goji-berries-the-mix-provides-iron/) [Gretchen Bleiler](https://sms4smile.com/quotes/author/gretchen-bleiler/) > [I've always had a thing for Catwoman. 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Wild seemed more enticing then a bowl of…](https://sms4smile.com/quotes/shannon-hale/no-one-had-ever-called-her-wild-before-she-wanted-to-be-wild-now-for/) [Shannon Hale](https://sms4smile.com/quotes/author/shannon-hale/) Browse all 222 Berries quotes ```json { "@context": "https://schema.org", "@graph": [ { "@type": "Quotation", "text": "The trouble with integers is that we have examined only the very small ones. Maybe all the exciting stuff happens at really big numbers, ones we can't even begin to think about in any very definite way. Our brains have evolved to get us out of the rain, find where the berries are, and keep us from getting killed. 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