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Chapter 1: Problem 7
Which is bigger: \(17 \%\) of nineteen million, or \(19 \%\) of seventeen million?
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(a) Compute by mental arithmetic (using pencil only to record results), then learn by heart: (i) the squares of positive integers: first up to \(12^{2}\); then to \(31^{2}\) (ii) the cubes of positive integers up to \(11^{3}\) (iii) the powers of 2 up to \(2^{10}\). (b) How many squares are there: (i) \(<1000 ?\) (ii) \(<10000 ?\) (iii) \(<100000 ?\) (c) How many cubes are there: (i) (ii) (iii) \(<1000000 ?\) (d) (i) Which powers of 2 are squares? (ii) Which powers of 2 are cubes? (e) Find the smallest square greater than 1 that is also a cube. Find the next smallest. Evaluating powers, and the associated index laws, constitute an example of a direct operation. For each direct operation, we need to think carefully about the corresponding inverse operation - here "extracting roots". In particular, we need to be clear about the distinction between the fact that the equation \(x^{2}=4\) has two different solutions, while \(\sqrt{4}\) has just one value (namely 2).
(a) Which of the prime numbers \(<100\) can be written as the sum of two squares? (b) Find an easy way to immediately write \(\left(a^{2}+b^{2}\right)\left(c^{2}+d^{2}\right)\) in the form \(\left(x^{2}+y^{2}\right)\). (This shows that the set of integers which can be written as the sum of two squares is "closed" under multiplication.) (c) Prove that no integer (and hence no prime number) of the form \(4 k+3\) can be written as the sum of two squares. (d) The only even prime number can clearly be written as a sum of two squares: \(2=1^{2}+1^{2}\). Euler \((1707-1783)\) proved that every odd prime number of the form \(4 k+1\) can be written as the sum of two squares in exactly one way. Find all integers \(<100\) that can be written as a sum of two squares. (e) For which integers \(N<100\) is it possible to construct a square of area \(N\), with vertices having integer coordinates? In Problem 25 parts (a) and (d) you had to decide which integers \(<100\) can be written as a sum of two squares as an exercise in mental arithmetic. In part (b) the fact that this set of integers is closed under multiplication turned out to be an application of the arithmetic of norms for complex numbers. Part (e) then interpreted sums of two squares geometrically by using Pythagoras' Theorem on the square lattice. These exercises are worth engaging in for their own sake. But it may also be of interest to know that writing an integer as a sum of two squares is a serious mathematical question \- and in more than one sense. Gauss \((1777-1855),\) in his book Disquisitiones arithmeticae (1801) gave a complete analysis of when an integer can be represented by a 'quadratic form', such as \(x^{2}+y^{2}\) (as in Problem 25) or \(x^{2}-2 y^{2}\) (as in Problem \(\mathbf{5 4}(\mathrm{c})\) in Chapter 2 ). A completely separate question (often attributed to Edward Waring \((1736-1798))\) concerns which integers can be expressed as a \(k^{\text {th }}\) power, or as a sum of \(n\) such powers. If we restrict to the case \(k=2\) (i.e. squares), then: \- When \(n=2,\) Euler \((1707-1783)\) proved that the integers that can be written as a sum of two squares are precisely those of the form $$ m^{2} \times p_{0} \times p_{1} \times p_{2} \times \cdots \times p_{s} $$ where \(p_{0}=1\) or \(2,\) and \(p_{1}
Let \(\Delta=\operatorname{area}(\triangle A B C)\) (a) Prove that $$ \Delta=\frac{1}{2} \cdot a b \cdot \sin C . $$ (b) Prove that \(4 R \Delta=a b c\).
(a) Evaluate $$ \left(1+\frac{1}{2}\right)\left(1+\frac{1}{3}\right)\left(1+\frac{1}{4}\right)\left(1+\frac{1}{5}\right) . $$ (b) Evaluate $$ \sqrt{1+\frac{1}{2}} \times \sqrt{1+\frac{1}{3}} \times \sqrt{1+\frac{1}{4}} \times \sqrt{1+\frac{1}{5}} \times \sqrt{1+\frac{1}{6}} \times \sqrt{1+\frac{1}{7}} $$ (c) We write the product " \(4 \times 3 \times 2 \times 1 "\) as "4!" (and we read this as "4 factorial"). Using only pencil and paper, how quickly can you work out the number of weeks in \(10 !\) seconds? \(\Delta\)
Problem 39 (Shadows) Can one use the Sun's rays to produce a plane shadow of a cube: (i) in the form of an equilateral triangle? (ii) in the form of a square? (iii) in the form of a pentagon? (iv) in the form of a regular hexagon? (v) in the form of a polygon with more than six sides?
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