he author, a mathematician, responds when a young woman named Meg asks if she must give up a sense of beauty to study math.,end italics,
from ,begin bold,Why Do Math?,end bold,
paragraph 1,What math does for me is this: It makes me aware of the world I inhabit in an entirely new way. It opens my eyes to nature's laws and patterns. It offers an entirely new experience of beauty.
paragraph 2,When I see a rainbow, for instance, I don't just see a bright, multicolored arc across the sky. I don't just see the effect of raindrops on sunlight, splitting the white light from the sun into its constituent colors. I still find rainbows beautiful and inspiring, but I appreciate that there's more to a rainbow than mere refraction of light. The colors are, so to speak, a red (and blue and green) herring. What require explanation are the shape and the brightness. Why is a rainbow a circular arc? Why is the light from the rainbow so bright?
paragraph 3,You may not have thought about those questions. You know that a rainbow appears when sunlight is refracted by tiny droplets of water, with each color of light being diverted through a slightly different angle and bouncing back from the raindrops to meet the observing eye. But if that's all there is to a rainbow, why don't the billions of differently colored light rays from billions of raindrops just overlap and smear out?
paragraph 4,The answer lies in the geometry of the rainbow. When the light bounces around inside a raindrop, the spherical shape of the drop causes the light to emerge with a very strong focus along a particular direction. Each drop in effect emits a bright cone of light, or, rather, each color of light forms its own cone, and the angle of the cone is slightly different for each color. When we look at a rainbow, our eyes detect only the cones that come from raindrops lying in particular directions, and for each color, those directions form a circle in the sky. So we see lots of concentric circles, one for each color.
paragraph 5,The rainbow that you see and the rainbow that I see are created by different raindrops. Our eyes are in different places, so we detect different cones, produced by different drops.
paragraph 6,Rainbows are personal.
paragraph 7,Some people think that this kind of understanding "spoils" the emotional experience. I think this is rubbish. It demonstrates a depressing sort of aesthetic complacency. People who make such statements often like to pretend they are poetic types, wide open to the world's wonders, but in fact they suffer from a serious lack of curiosity: they refuse to believe the world is more wonderful than their own limited imaginations. Nature is always deeper, richer, and more interesting than you thought, and mathematics gives you a very powerful way to appreciate this. The ability to ,begin italics,understand,end italics, is one of the most important differences between human beings and other animals, and we should value it. Lots of animals emote, but as far as we know, only humans think rationally. I'd say that my understanding of the geometry of the rainbow adds a new dimension to its beauty. It doesn't take anything away from the emotional experience.
paragraph 8,The rainbow is just one example. I also look at animals differently, because I'm aware of the mathematical patterns that underlie their movements. When I look at a crystal, I am aware of the beauties of its atomic lattice as well as the charm of its colors. I see mathematics in waves and sand dunes, in the rising and the setting of the sun, in raindrops splashing in a puddle, even in birds sitting on telephone cables. And I'm aware—dimly, as if looking out over a foggy ocean—of the infinity of things we ,begin italics,don't,end italics, know about these everyday wonders.
paragraph 9,Then there's the inner beauty of mathematics, which should not be underrated. Math done "for its own sake" can be exquisitely beautiful and elegant. Not the "sums" we all do at school; as individuals those are mostly ugly and formless, although the general principles that govern them have their own kind of beauty. It's the ideas, the generalities, the sudden flashes of insight, the realization that trying to trisect an angle with straightedge and compass is like trying to prove that 3 is an even number, that it makes perfect sense that you can't construct a regular seven-sided polygon but you can construct one with seventeen sides, that there is no way to untie an overhand knot, and why some infinities are bigger than others whereas some that ought to be bigger are actually equal, that the ,begin italics,only,end italics, square number (other than 1, if you want to be picky) that is the sum of consecutive squares, 1 + 4 + 9 + . . ., is the number 4900.
paragraph 10,You, Meg, have the potential to become an accomplished mathematician. You have a logical mind and also an inquiring one. You're not convinced by vague arguments; you want to see the details and check them out for yourself. You don't just want to know how to make things work, you want to know ,begin italics,why ,end italics,they work. And your letter made me hope that you'll come to see mathematics as I see it, as something fascinating and beautiful, a way of seeing the world that is like no other.
paragraph 11,I hope this sets the scene for you.
Yours,
Ian
(From ,begin underline,Letters to a Young Mathematician,end underline, by Ian Stewart, copyright © 2006. Reprinted by permission of Basic Books, an imprint of Hachette Book Group, Inc.)
Question
Select the ,begin emphasis,two,end emphasis, sentences that ,begin emphasis,best,end emphasis, describe how paragraph 2 contributes to the development of ideas in the passage.
Answer options with 5 options
1.
Paragraph 2 provides an example supporting the general statements in paragraph 1.
2.
Paragraph 2 introduces the mathematical ideas that are explained in paragraphs 3 and 4.
3.
Paragraph 2 defines the mathematical terms used in paragraphs 3–5.
4.
Paragraph 2 creates a contrast to the important topics discussed in paragraph 7.
5.
Paragraph 2 gives supporting evidence for the description of Meg in paragraph 10.
1. Paragraph 2 provides an example supporting the general statements in paragraph 1.
2. Paragraph 2 introduces the mathematical ideas that are explained in paragraphs 3 and 4.