MA 624 Test 1

The nursery rhyme: As I was going to St. Ives, I met a man with seven wives; every wife had seven sacks, every sack had seven cats, every cat had seven kits, Kits, cats, sack, and wives, How many were going to St. Ives?" has its roots in ________

Ahmes Papyrus

The Sand Reckoner is a work by ___ in which he set out to determine an upper bound for the number of grains of sand that fit into the universe. In order to do this, he had to estimate the size of the universe, building on the work of ___. In this work he also came up with a way to talk about ___.

Archimedes, Aristarchus, extremely large numbers

According to Boyer, 800 BECAUSE is roughly when the learning centers began to shift from the river valleys of ___________ to the banks of ___________.

Egypt and Mesopotamia, the Mediterranean Sea

____ discovered the theory of proportion used in book v of Euclid's Elements. The word ratio denoted essentially an undefined concept in Greek mathematics. Euclid's "definition" of ratio as a kind of relation in size between two magnitudes of the same type is quite inadequate. More significant is ______ statement that magnitudes are said to have a ratio to one another if a multiple of either can be found to exceed the other. This is essentially a statement of the so-called "Axiom of Archimedes" - a property that Archimedes himself attributed to Eudoxus. The Eudoxian concept of ratio consequently excludes ________ and clarifies what is meant by magnitudes of the same kind. A line segment, for example, is not to be compared, in terms of ratio, with an area; nor is an area to be compared with a volume.

Eudoxus, Euclid's, zero

The fundamental operation in Egyptian math was multiplication.

False

Euclid's Elements was NOT

contains the study of conics and higher planes a compendium of all geometric knowledge an advanced textbook covering all mathematics of his day contains the art of calculations

Aristotle made a sharp distinction between axioms (or common notions) and postulates, axioms were ___ and postulates were _____.

convincing in themselves, less obvious and do not presuppose the assent of the learner

In his Elements, Euclid lists five postulates and five common notions. Aristotle made a sharp distinction between axioms (or common notions) and postulates, axioms were ___ and postulates were ___.

convincing in themselves, less obvious and may not be intuitive

Aristarchus found the relative distance of the sun and moon from the earth, then used this ratio to determine the sizes of the earth.

false

Zeno's paradoxes are a set of philosophical problems generally thought to have been devised by Greek philosopher Zeno of Elea (c. 490-430 BC). In the paradox of Achilles and the tortoise, Zeno shows ____.

The faster runner could never overtake the slower runner if the slower runner had a head start.

Among non literate tribes there seems to have been virtually no need for fractions.

True

The Sand Reckoner is a work by Archimedes in which he set out to determine an upper bound for the number of grains of sand that fit into the universe. In order to do this, he had to estimate the size of the universe, building on the work of Aristarchus. In this work he also came up with a way to talk about extremely large numbers.

True

Archimedes saw his proof of the volume of a sphere as his greatest mathematical achievement, and gave instructions that his tombstone should contain ______

a sphere inscribed in a cylinder

The pythagoreans had discovered that the diagonal of a square does not have a common unit of measurement with the sides of the square; this is the famous discovery that the square root of 2 cannot be expressed as the ratio of two integers. This discovery had heralded the existence of commensurable quantities beyond the integers and rational fractions, but at the same time it threw into question the idea of measurement and calculations in geometry as a whole.

false - incommensurable

The method that Archimedes used to calculate and prove the formulas for the volume and surface area of a sphere is called the ____. It had been developed rigorously about a centry earlier by one of Archimedes' heroes, ___,

method of exhaustion, Eudoxus

The evidence of Egyptian mathematics contained _____

no formal proofs or theorems accurate approximations recognition of interrelationships between geometric figures

For people in Euclid's time, geometry was basis of everything. Shapes, points, lines, and triangles were fundamental objects. ____ arose from _____.

numbers, geometry

For people in Euclid's time, geometry was the basis of everything. Shapes, points, lines, and triangles were fundamental objects. ____ arose from _____

numbers, geometry

Angle trisection is a classical problem of ancient Greek mathematics. It concerns construction of an angle equal to ___ of a given arbitrary angle, using only two tools: an unmarked ruler and a compass.

one third

Archimedes imagined dividing the sphere up into rings inside of a cylinder. He found that the volumes of the rings added up to the volume of a cone whose base radius' and height were the same as the cylinder's. This meant the volume of the hemisphere must be equal to the volume of the cylinder minus the volume of the cone. The formula for the volume of the cylinder was known to be pir2h and the formula for the volume of a cone was known to be 1/3pir2h. In this example, r and h are identical. so the volumes are ____. Subtracting one from the other meant that the volume of a hemisphere must be ___ and since a sphere's volume is twice the volume of a hemisphere, the volume of a sphere is: V=4/3pir3

pir3 and 1/3pir3, 2/3pir3 pir2r and 1/3pir2r, (1-1/3)pir3

At the university of Alexandria, Euclid was known for his

teaching ability

According to Proclus Diadochus, Ptolemy once asked Euclid whether there was any shorter way to a knowledge of geometry than by a study of the Elements, whereupon Euclid answered ___________

that there is no royal road to geometry

The illustration from Euclids Elements represented the proof of the distributive law since the sum of

the area of the largest rectangle is equal to the sum of the 3 smaller areas (APQD, PRSQ, and RBCS)

Euclid's Elements

was an introductory textbook covering all elementary mathematics