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## Numbers and Algebra

### Basic Number Theory - Rational and Irrational Numbers

#### Q.01

'Exercise question 13 The leading digit\n505\nFor a non-negative real number a, where 0 ≤ r < 1, and a-r is an integer, the real number r is denoted by {a}. In other words, {a} represents the decimal part of a. (1) Find one positive integer n that makes the decimal part of {n log_10 2} less than 0.02. (2) Find one positive integer n where the leading digit of 2^n in decimal representation is 7. It is given that 0.3010 < log_10 2 < 0.3011, and 0.8450 < log_10 7 < 0.8451. [Kyoto University]'

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'Translate the given text into multiple languages.'

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'When x=π, y=π/12, the maximum value is 25/12 π; when x=0, y=5/12 π, the minimum value is 5/12 π'

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'Find the square root of a negative number. Let a be a positive real number.'

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"(1) Let's assume that there exists a rational number x that satisfies 3^{x}=5. Since 3^{x}=5>1, it implies that x>0. Therefore, x can be expressed as x=\x0crac{m}{n} where m and n are positive integers. When both sides are raised by n, we get 3^{m}=5^{n} (1). The left side is a multiple of 3, but the right side is not a multiple of 3, leading to a contradiction. Hence, x that satisfies 3^{x}=5 is not a rational number."

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'(1) If a > 0 and x > 0, then a^{1/2x} > 0, a^{-1/2x} > 0'

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'Translate the given text into multiple languages.'

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'Prove that the solution to the equation $3^{x}=5$ is not a rational number.'

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'Calculate the following sum: \\( \\frac{1}{(2n+1)(2n+3)}+\\frac{1}{(2n+3)(2n+5)}+\\frac{1}{(2n+5)(2n+7)} \\)。'

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'102 (1) x = 2 log_{2} 5 (2) 1/2 <= x <= log_{2} 5 (3) x = 9.81 (4) 0 < x <= sqrt(3)/9, 1 < x <= 81'

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'Express the following sets of numbers in terms of inequality.\n(1) \ \\log_{3} 5,2,2 \\log_{3} 2 \'

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'Find the square root of (1) to (3). Also, calculate (4) to (6).'

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'Express the following sets of numbers in terms of inequality.'

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'Solve the following inequalities: (1) $\\left(\\frac{1}{3}\\right)^{x}<\\frac{1}{81}$ (2) $5^{x+3}>\\frac{1}{25}$ (3) $2\\left(\\frac{1}{2}\\right)^{x^{2}}\\geqq\\left(\\frac{1}{128}\\right)^{x-1}$'

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'Find the value corresponding to the number 5.67 in the table below.'

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'Find the square root of (1) to (3). Perform calculations for (4) to (6).'

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"Exponents of rational numbers\nHere, let's define the power of a positive number a so that the exponent law on p.250 holds even when the exponent is a rational number (fraction)."

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'Express the relative sizes of the following sets of numbers using inequality symbols.'

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"(8) In Figure 6, there is a difference of 0.05 mm between the length of the main scale's 2mm increment and the minimum 1mm increment of the vernier scale, which is 2 - 1.95 = 0.05 (mm). Therefore, when the scale lines of the main scale and vernier scale are aligned and there is a misalignment of one increment on the vernier scale, a difference of 0.05mm occurs in the measured length (measurement value). As a result, the length that can be read with the caliper in Figure 6 is in increments of 0.05mm."

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'(1) Maximum value \\\sqrt{2}\, minimum value \-\\sqrt{2}\\\n(2) Maximum value 5, minimum value -5'

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'When the complex number z satisfies |z-1|≤|z-4|≤2|z-1|, illustrate the range in which the point z moves on the complex plane.'

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'Let α and z be complex numbers, with |α|>1. Compare the magnitudes of |z-α| and |αz-1|.'

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"About π: The fact that π is irrational can be proved within the scope of high school mathematics using methods such as proof by contradiction and integration by parts. Here we introduce Neve's proof published in 1947. Assuming that π is a rational number, π = {b} / {a} (a, b are natural numbers). Let f(x) = {1} / {n!} x ^ {n} (b - a x) ^ {n} = {a} ^ {n} / {n!} x ^ {n} (π - x) ^ {n}, consider the definite integral I = ∫_ {0} ^ {π} f(x) sinx dx. First, we prove that I is an integer. For I, by repeatedly using integration by parts, since f(x) is a 2n degree polynomial"

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'Find the argument \ \\theta \ of the complex number \ \\frac{5-2 i}{7+3 i} \. Ensure \ 0 \\leqq \\theta<2 \\pi \.'

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'Find the values of constants a and b such that the range of the function y=√(2x+4) is 1≤y≤3.'

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"The proposition 'Let n be an integer. If n^2 is a multiple of 7, then n is a multiple of 7' is true. Use this proposition to prove that √7 is irrational."

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'Basic Example 23 Rationalize the denominator Simplify the following expressions by rationalizing the denominator.'

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'When dividing 3 people into 3 groups of 3 each, if we eliminate the distinction between A, B, and C, then each set can be arranged in 3! ways, so what is the total number of ways to divide?'

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'Given x=(√2+√3)/(√2-√3), y=(√2-√3)/(√2+√3), find the values of the following expressions.'

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'Let the integer part of 1+√10 be a, and the decimal part be b. Find the following values: (1) a, b; (2) b + 1/b, b² + 1/b²'

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'Prove that when the fraction a = m / n (where m, n are integers and n>0) becomes an infinite decimal, a is a recurring decimal.'

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'√3 is an irrational number. Find the values of rational numbers a, b that satisfy 7+a√3/2+√3=b+9√3.'

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'(1) In order \ \\frac{\\sqrt{15}}{4}, -\\frac{1}{4}, -\\sqrt{15} \'

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"In a city with 6 north-south roads and 4 east-west roads, let's consider the shortest path from point P to point Q. During this journey, toss a coin: if it lands heads, move east by 1 block, if it lands tails, move north by 1 block. The probability of getting heads or tails is equal, both being 1/2. Furthermore, before reaching point Q, if the coin lands heads at the easternmost intersection or tails at the northernmost intersection, you cannot continue and must stay at that intersection."

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'The sum, difference, product, and quotient of two real numbers a and b are always real numbers. For example, even when adding rational numbers, the result is always a rational number. Explain that arithmetic operations are always possible within the range of rational and real numbers. However, division does not consider dividing by 0.'

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'Prove that the sum of a rational number and an irrational number is irrational.'

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'(1) Simplify \ \\frac{1}{1+\\sqrt{2}+\\sqrt{3}}+\\frac{1}{1+\\sqrt{2}-\\sqrt{3}}-\\frac{1}{1-\\sqrt{2}+\\sqrt{3}}-\\frac{1}{1-\\sqrt{2}-\\sqrt{3}} \.'

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'Volume \ \\frac{4}{3} \, Distance \ \\frac{2 \\sqrt{14}}{7} \'

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'Determine the range of the constant a and find the coordinates of the intersection points.'

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'86 x= \\sqrt{5}, \\quad -\\frac{1}{\\sqrt{2}}<x<\\frac{1}{\\sqrt{2}}, \\quad \\sqrt{5}<x'

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'Prove that the sum of a rational number and an irrational number is irrational.'

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'In a game where a coin is repeatedly tossed, and a prize is won when it lands heads up 3 times, with a maximum of 5 tosses allowed and no more tosses after the third heads up, how many possible sequences are there to win the prize if the first toss results in tails?'

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'Investigate the truth of the following propositions. However, use sets to investigate (2) and (3).\n(1) For real numbers a, b, if the square of a is equal to the square of b, then a equals b\n(2) For real numbers x, if |x|<3, then x<3\n(3) For real numbers x, if x<1, then |x|<1'

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'Example 98: Determine the range of existence of solutions for a quadratic equation with solutions in the ranges 0 < x < 1 and 1 < x < 2.'

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'A square root of a positive number a has two roots, which are equal in absolute value but differ in sign. The square root of 0 is 0. For example, the square root of 5 is sqrt{5} and -sqrt{5}. Calculation examples with square roots include: sqrt{3} × sqrt{7} = sqrt{21}. sqrt{5} / sqrt{2} = sqrt{5/2}.'

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'Explain the proof by contradiction method and prove the following proposition U using proof by contradiction. Proposition U: "√2 is an irrational number."'

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'(1) \\frac{4(\\sqrt{7}-1)}{3} (2) -4 (3) \\frac{110-32 \\sqrt{7}}{9}'

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'Rationalize the denominators of the following expressions.'

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'When x=(2+√3)/(2-√3) and y=(2-√3)/(2+√3), find the values of the following equations.'

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'Prove that sqrt(3) is an irrational number. Assume sqrt(3) is rational, i.e., there exist two natural numbers m and n without any common divisors other than 1 such that sqrt(3) = m/n. Therefore, m = sqrt(3)n. Squaring both sides gives m^2 = 3n^2, so m is a multiple of 3. Thus, there exists a natural number k such that m = 3k. Substituting this in, we get 9k^2 = 3n^2, which simplifies to n^2 = 3k^2, implying n is a multiple of 3, leading to a contradiction. Hence, sqrt(3) is irrational.'

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'Use proof by contradiction to prove the following proposition: At least one of x squared and x cubed is irrational.'

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'(1) Let a, b, c, and d be rational numbers, and √l be an irrational number. Prove that b=d when a+b√l=c+d√l. Also, prove that a=c in this case. (2) Find the values of rational numbers x and y that satisfy (1+3√2)x + (3+2√2)y = -5-√2.'

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'Rationalize the denominators of the following expressions.'

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'(1) (P) √5, (化) 10/3\n(2) (ウ) 3/5\n(3) (I) 2√5, (J) 5√5/4'

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'(1) For a certain natural number n, √n is a rational number, true. (2) For all real numbers x, x^2 ≠ x + 2, false.'

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'Remove the double square roots in the following expressions:'

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'Express the recurring decimals 0.2, 1.21, 0.13 as fractions.'

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'Answer the following question. 127 (1) Find the lengths of the other sides of a triangle with length b=2√7.'

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'Using the fact that √3 is irrational, prove that 1+2√3 is also irrational.'

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'(1) Let a, b be rational numbers. When a+b \\sqrt{3}=0, prove that \\sqrt{3} is irrational to show that a=b=0. \n(2) Find the values of rational numbers x, y that satisfy (2+3 \\sqrt{3}) x+(1-5 \\sqrt{3}) y=13.'

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'What are the solutions of the EXERCISES 14 7 15: (1) x = \\frac{2+\\sqrt{14}}{5}, -\\frac{6+3\\sqrt{14}}{5} (2) -\\frac{1}{3} \\leqq x < \\frac{7}{3} (3) x = \\frac{5}{4}, -\\frac{1}{2}?\u200b'

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'Explain the definition of irrational numbers and list two of their characteristics.'

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'What is a square root? Please explain with specific examples.'

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'Prove that √3 is irrational. Assume that √3 is rational and can be represented as √3 = p/q, where p and q are coprime integers. Squaring both sides gives us 3 = p^2/q^2, rearranging gives us 3q^2 = p^2. Therefore, p^2 is a multiple of 3, according to the assumption, p is a multiple of 3, so it can be written as p = 3m. Substituting back gives us 3q^2 = 9m^2, dividing by 3 gives q^2 = 3m^2, which means q is also a multiple of 3. This contradicts the fact that p and q are coprime, so the assumption is incorrect, and √3 is irrational.'

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'Remove the double square roots in the following expressions.'

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'Find the general term of the sequence 1/6, 1/9, 1/14, 1/21, 1/30.'

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'117\nFind the values of a and b, where a = (6 ± √14)/2 and b = (6 ∓ √14)/2\n(the signs are the same in both cases)'

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'Let a and b be non-zero real numbers. The following equalities hold true when a>0 and b>0, but what about in other cases? Please investigate the following scenarios: [1] a>0, b<0 [2] a<0, b>0 [3] a = √(a/b) (2) √(a) / √(b) = √(a/b) (3) √(a) √(b) = √(ab)'

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'For the sequence 1/1, 1/2, 3/2, 1/3, 3/3, 5/3, 1/4, 3/4, 5/4, 7/4, 1/5, ...'

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'(1) \ a_{2}=\\frac{4}{3}, a_{3}=\\frac{6}{5}, a_{4}=\\frac{8}{7} \,\\n\ a_{n}=\\frac{2 n}{2 n-1} \\\n(2) Summary'

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'Find the following values:\n14. (1) \x0crac{x}{(x+1)(x-1)}\n(2) 1'

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'Let a be a positive constant different from 1. If a^x=8 and a^y=25, express log_{10} 500 in terms of x and y.'

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"In problems where the coefficients of a high-degree equation are to be determined, the following [1], [2] are the basics of problem solving. First, let's grasp this most important point. x=α is a solution to the equation f(x)=0 if and only if f(α)=0 (holds when substituted) ⇐[1]⇔ f(x) has x−α as a factor ⇐[2] The most basic solution method is the strategy of [1] which is to 'substitute the solution'. In example problems 61, 62, we first show the answers using this strategy. However, when the solution is imaginary as in example 62, the calculations after substitution may become somewhat cumbersome."

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'Calculate the following expressions. (5) (sqrt{3}+sqrt{-1})(1-sqrt{-3})'

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'Calculate the following expressions. Assume a>0, b>0.'

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'Find the sum of the sequence from the first term to the nth term.'

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"(2) State the converse, contrapositive, and inverse of 'If xy is irrational, then at least one of x, y is irrational', and determine their truth values."

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'Using the fact that √3 is irrational, prove that 1/√2 + 1/√6 is irrational.'

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'124 (1) BE= (1+√5)/2, R= 2/√(10-2√5) (2) BG= √(10+2√5)/2 (3) In turn (3+√5)/48, (15+5√5)/12'

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'Using the proof by contrapositive from Proposition 61, prove that \ \\sqrt{7} \ is irrational, and then prove that \ \\sqrt{5}+\\sqrt{7} \ is irrational. Basic principle 2. It is difficult to directly show that a number is irrational (i.e., not rational). Therefore, we assume that the proposition to be proven is false, derive a contradiction, and prove that the proposition holds. '

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'In △ABC, where a=1+√3, b=2, and C=60°. Find the following:\n(1) Length of side AB\n(2) Measure of ∠B\n(3) Area of △ABC\n(4) Radius of circumcircle\n(5) Radius of incircle\n[Similar to Nara Education University]\np. 285 EX118,119'

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'For the following propositions, provide the contrapositive and inverse contrapositive, and determine if they are true or false.'

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"Focus on (5) \\( (\\sqrt{2})^{2}+(\\sqrt{3})^{2}=(\\sqrt{5})^{2} \\). Let's take a look at the given expressions:\n(1) \\( \\quad( \\text{Given expression} )=\\frac{3 \\sqrt{2} \\sqrt{3}}{2(\\sqrt{3})^{2}}-\\frac{\\sqrt{3} \\sqrt{2}}{3(\\sqrt{2})^{2}}=\\frac{3 \\sqrt{6}}{6}-\\frac{\\sqrt{6}}{6}=\\frac{2 \\sqrt{6}}{6}=\\frac{\\sqrt{6}}{3} \\)\n(2) \\( \\quad( \\text{Given expression} )=\\frac{6(3+\\sqrt{7})}{(3-\\sqrt{7})(3+\\sqrt{7})}=\\frac{6(3+\\sqrt{7})}{9-7} \\)\n=3(3+\\sqrt{7})=9+3 \\sqrt{7}\n(3) \\(\egin{aligned}\\text{Given expression}=\\frac{(\\sqrt{3}-\\sqrt{2})^{2}}{(\\sqrt{3}+\\sqrt{2})(\\sqrt{3}-\\sqrt{2})}-\\frac{(\\sqrt{5}+\\sqrt{3})^{2}}{(\\sqrt{5}-\\sqrt{3})(\\sqrt{5}+\\sqrt{3})} \\)\n& =\\frac{5-2 \\sqrt{6}}{3-2}-\\frac{8+2 \\sqrt{15}}{5-3}=5-2 \\sqrt{6}-(4+\\sqrt{15}) \\)\n& =1-2 \\sqrt{6}-\\sqrt{15}\n\\end{aligned}\\)\n(4)\n\egin{array}{l}\\(\\text {Given expression}=\\frac{1+\\sqrt{6}-\\sqrt{7}}{\\{(1+\\sqrt{6})+\\sqrt{7}\\}\\{(1+\\sqrt{6})-\\sqrt{7}\\}} \\)\n+\\frac{5-2 \\sqrt{6}}{(5+2 \\sqrt{6})(5-2 \\sqrt{6})} \\)\n=\\frac{1+\\sqrt{6}-\\sqrt{7}}{(1+\\sqrt{6})^{2}-(\\sqrt{7})^{2}}+\\frac{5-2 \\sqrt{6}}{25-24}=\\frac{1+\\sqrt{6}-\\sqrt{7}}{2 \\sqrt{6}}+5-2 \\sqrt{6} \\)\n=\\frac{(1+\\sqrt{6}-\\sqrt{7}) \\sqrt{6}}{2(\\sqrt{6})^{2}}+5-2 \\sqrt{6} \\)\n=\\frac{\\sqrt{6}+6-\\sqrt{42}}{12}+\\frac{12(5-2 \\sqrt{6})}{12}=\\frac{66-23 \\sqrt{6}-\\sqrt{42}}{12} \\)\n\\end{array}\\)\n(5)\n\\(\egin{aligned}\\text {Given expression}=\\frac{(\\sqrt{2}-\\sqrt{3}+\\sqrt{5})\\{(\\sqrt{2}+\\sqrt{3})+\\sqrt{5}\\}}{\\{(\\sqrt{2}+\\sqrt{3})-\\sqrt{5}\\}\\{(\\sqrt{2}+\\sqrt{3})+\\sqrt{5}\\}} \\)\n& =\\frac{\\{(\\sqrt{2}+\\sqrt{5})-\\sqrt{3}\\}\\{(\\sqrt{2}+\\sqrt{5})+\\sqrt{3}\\}}{(\\sqrt{2}+\\sqrt{3})^{2}-(\\sqrt{5})^{2}} \\)\n& =\\frac{(\\sqrt{2}+\\sqrt{5})^{2}-(\\sqrt{3})^{2}}{2 \\sqrt{6}}=\\frac{2+\\sqrt{10}}{\\sqrt{6}} \\)\n& =\\frac{(2+\\sqrt{10}) \\sqrt{6}}{(\\sqrt{6})^{2}}=\\frac{2 \\sqrt{6}+2 \\sqrt{15}}{6}=\\frac{\\sqrt{6}+\\sqrt{15}}{3} \\)\n\\end{aligned}\\)\n\\\leftarrow \\text{If the denominator is} \\sqrt{a},\\text{ then multiply the denominator and numerator by} \\sqrt{a}.\\n\\\leftarrow \\text{If the denominator is} \\a-\\sqrt{b},\\text{ then multiply the denominator and numerator by} \\a+\\sqrt{b}.\\n\\\leftarrow \\text{If the denominator is} \\sqrt{a}+\\sqrt{b},\\text{ then multiply the denominator and numerator by} \\sqrt{a}-\\sqrt{b};\\text{ if the denominator is} \\sqrt{a}-\\sqrt{b},\\text{ then multiply the denominator and numerator by} \\sqrt{a}+\\sqrt{b}.\"

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'In example 28, rationalizing the denominator of x yields x=5-2√6, and rationalizing the denominator of y yields y=5+2√6.'

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'Prove each part of the following problem. (2) Assuming √n and √(n+1) are both rational numbers. Prove that √n and √(n+1) are both positive integers. (3) Assuming √(n+1) - √n is a rational number. Show the properties of √n and √(n+1). Solve the following problem. Derive a x + y from (a x + y)/(1 - a) = a and solve the equation.'

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'Simplify the following expressions by rationalizing the denominators.'

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'Provide a counterexample to the following propositions.'

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'Practice simplifying the following expressions by rationalizing the denominators.'

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'Explain the properties of real numbers and square roots.'

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'Let A be the set of rational numbers and B be the set of irrational numbers in 33®. Let ∅ represent the empty set. Choose the appropriate symbol ∈, ∋, ⊆, ⊇, ∪, ∩ to fill in the blanks below.'

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'Prove that \ \\sqrt{2}+\\sqrt{3} \ is irrational. It is assumed that \ \\sqrt{2}, \\sqrt{3} \ are known to be irrational.'

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'Prove that for rational numbers a, b, c, d and an irrational number x, if a+bx=c+dx, then a=c and b=d.'

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'\ \\frac{\\sqrt{3}+\\sqrt{2}}{2 \\sqrt{3}-\\sqrt{2}} \'

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"From the given conditions, AC=BC=\\frac{6}{\\sqrt{2}}=3 \\sqrt{2}. Taking points D, E, F, G as shown in the figure, let the length of the rectangle's vertical side be x, then DE=AE=AC-CE=3 \\sqrt{2}-2 x, FG=AG=AC-GC=3 \\sqrt{2}-x. Also, since 0<CE<AC, we have 0<2 x<3 \\sqrt{2}, which means 0<x<\\frac{3 \\sqrt{2}}{2}. Let y be the sum of the areas of the two rectangles, then y =x(3 \\sqrt{2}-2 x)+x(3 \\sqrt{2}-x) = -3 x^{2}+6 \\sqrt{2} x = -3(x-\\sqrt{2})^{2}+6. The maximum value of y is 6 when x=\\sqrt{2}. Therefore, the maximum value of the sum of the areas of the two rectangles is 6."

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'Rationalize the denominators and simplify the following expressions:'

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'Prove that the sum of a rational number and an irrational number is an irrational number.'

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'Prove that PR√2+√3 is irrational. Assume that √2 and √3 are both irrational.'

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'Prove that the sum of a rational number and an irrational number is irrational.'

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'(5) \\( \egin{aligned} (\\sqrt{10}-2 \\sqrt{5})(\\sqrt{5}+\\sqrt{10}) &= (\\sqrt{2} \\cdot \\sqrt{5}-\\sqrt{2} \\sqrt{10})(\\sqrt{5}+\\sqrt{10}) \\\\ &= \\sqrt{2}(\\sqrt{5}-\\sqrt{10})(\\sqrt{5}+\\sqrt{10}) \\\\ &= \\sqrt{2}(5-10)=-5 \\sqrt{2} \\end{aligned} \\)'

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'From \\\sqrt{3} \\tan \\theta+1=0\, it follows that \\\tan \\theta=-\\frac{1}{\\sqrt{3}}\. Let \\\mathrm{T}\ be the point on the line \x=1\ where the \y\ coordinate is \-\\frac{1}{\\sqrt{3}}\. The intersection of the line \OT\ and the semicircle with radius 1 is point \\\mathrm{P}\ in the figure. The required \\\theta\ is \\\angle \\mathrm{AOP}\.'

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'Given x = \\frac{\\sqrt{2}+\\sqrt{3}}{\\sqrt{2}-\\sqrt{3}}, y = \\frac{\\sqrt{2}-\\sqrt{3}}{\\sqrt{2}+\\sqrt{3}}, find the values of the following expressions.'

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'Number of solutions when 84 a > -1 / 8, number of solutions when a = -1 / 8, number of solutions when a < -1 / 8'

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'When x=\\frac{1-\\sqrt{2}}{1+\\sqrt{2}}, y=\\frac{1+\\sqrt{2}}{1-\\sqrt{2}}, find the values of the following expressions.\\n(1) x+y, x y\\n(2) 3 x^{2}-5 x y+3 y^{2}'

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'Prove using proof by contradiction that √3 is irrational.'

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'79 (1) Given z=√3+i, -√3-i (2) Given z=2i, -√3-i, √3-i'

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'84 (1) (√3 - 1/2) + (√3/2 + 2)i (2) √2/2 + (3√2/2 + 1)i (3) -1 + 3i (4) 1 - i'

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'(2) The point z satisfies the equation |z-(1-√3 i)|=1 w=(2+2 √3 i) z, i.e., w=2(1+√3 i) z From z =w/(2(1+√3 i))=w(1-√3 i)/(2(1+√3 i)(1-√3 i)) =w(1-√3 i)/8, substitute into (1) to get |w(1-√3 i)/8-(1-√3 i)|=1, i.e., |(1-√3 i)/8||w-8|=1|(1-√3 i)/8|=2/8=1/4, so |w-8|=4 Therefore, the point w draws a circle centered at point 8 with a radius of 4. Reference 2+2 √3 i=4(cos(π/3)+i sin(π/3)) So, the point (2+2 √3 i) z is the point obtained by rotating point z around the origin by π/3 and multiplying by 4. Therefore, the center point 1-√3 i of the circle |z-(1-√3 i)|=1 moves to point 8, and the radius of the circle is 4. Therefore, the point w draws a circle centered at point 8 with a radius of 4.'

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"Please explain the following terms: bounded, finite definite value, directed line segment, focal conic, rational function, positive function, sum of squares, eccentric angle, eccentricity, volume of a solid, limacon, zero divisor, zero matrix, zero vector, lemniscate, continuous, Leibniz series, L'Hôpital's rule, Rolle's theorem"

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'For a complex number \ \\alpha=a+b i \, where \ \\overline{\\alpha}=a-b i \ is the conjugate of \ \\alpha \, prove the following:\n\n(1) If \ \\alpha \ is a real number, then \ \\overline{\\alpha}=\\alpha \. If \ \\alpha \ is a purely imaginary number with \ \\alpha \\neq 0 \, then \ \\overline{\\alpha}=-\\alpha \.\n(2) Prove that \ \\alpha+\\overline{\\alpha} \ is a real number.\n(3) Prove that \ \\overline{\\alpha+\eta}=\\overline{\\alpha}+\\overline{\eta} \.\n(4) Prove that \ \\overline{\\alpha\eta}=\\overline{\\alpha}\\overline{\eta} \.'

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'(1) s^2 - t^2/a^2 = 1\nFrom (1), s^2/b^2 + t^2 = 1\nSubstitute (2) into t^2 = 1 - s^2/b^2\nObtain s^2 - (1/a^2)(1 - s^2/b^2) = 1\nSimplify to s^2 = b^2(a^2 + 1)/(a^2 b^2 + 1)\ns > 0, b > 0, thus s = b sqrt((a^2 + 1)/(a^2 b^2 + 1))\nFrom (2), (3) we get t^2 = 1 - (1/b^2) * b^2(a^2 + 1)/(a^2 b^2 + 1) = a^2(b^2 - 1)/(a^2 b^2 + 1)\nt > 0, a > 0, b > 1, thus t = a sqrt((b^2 - 1)/(a^2 b^2 + 1))'

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'The answer to the exercise problem is t=\\frac{\\pi}{6}+\\frac{1}{2}, V(t)=\\frac{\\pi}{24}(2 \\pi-3 \\sqrt{3}+1)'

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'Practice 44\\n(1) (Solution 1) x=1/(y^2-2y) From this, we have y^2-2y-1/x=0 Let the discriminant of this quadratic equation be D\\nD/4=(-1)^2-1(-1/x)=1/x+1\\nD/4 >= 0 implies 1/x+1 >= 0 Therefore, x<=-1,0<x Thus, when x<=-1,0<x we have y=1±√(1+1/x)'

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'Exercise question answer 62 (1) a= \\ frac{9}{8}'

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'A container has a volume of π/4(h²+h) cm³ and a water surface area of π/2(h+1/2) cm² when the height is h cm. Assuming water is poured into the container at a rate of π cm³ per second. Determine the following after 5 seconds of pouring water: (A) the height h from the bottom of the water surface, (B) the speed v at which the water surface rises, (C) the rate of increase w in the water surface area.'

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'Exercise problem solution 58 (3) \\frac{\\sqrt{3}}{12}'

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'Find the n-th root of 1 and explain which position on the unit circle each value corresponds to.'

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'Find the maximum and minimum values of the following problems.'

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'(1) \ z \ is all real numbers except 0, 1, and -1'

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'Read the following proof to show that e is irrational.'

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'(1)\\\ \\frac{1+i}{2} \\alpha + \\frac{1-i}{2} \eta \'

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'Prove the inequality e^{x}>1+\\sum_{k=1}^{n} \\frac{x^{k}}{k!} (x>0)'

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'Exercise 41 III: Prove the inequality \ t \\geqq \\tan t - \\frac{\\tan^{3} t}{3} \.'

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'(1) \ z_2 = \\frac{3+\\sqrt{3} i}{2}, \\quad z_3 = 1+\\sqrt{3} i \'

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"Please explain the proof that 'e is irrational'. Show the procedure of proving that e is irrational using the method of proof by contradiction and infinite series."

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'When α=√3+i and β=2-2i, express αβ and α/β in polar form, where the argument θ is in the range 0≤θ<2π.'

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'Let a, b be non-negative real numbers. The following equations hold when a > 0, b > 0, but what about other cases? Investigate in the following cases.'

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'Calculate the following mathematical expressions.'

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'(1) Let a = 1/4, b = 3/4, and 2ab = 3/8, a^2 + b^2 = 5/8,\n\nIt is expected that a<2ab<1/2<a^2 + b^2 < b.'

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'Answer the following questions. Assume that $\\sqrt{3}$ is irrational.\n(1) Prove that $\\log_{3} 4$ is irrational.\n(2) Find one pair of real numbers $(a, b)$, where $a$ and $b$ are irrational, and $a^{b}$ is rational.'

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'Find the values of θ such that 21θ is between 210° and π/2. Although cos θ is not a rational number, find the values of θ where both cos 2θ and cos 3θ are rational numbers.'

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'Determining the number of digits and the first decimal place using common logarithms'

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'(Given that a > 0 and 9^a + 9^-a = 14, find the value of the following expressions: 1) 3^a + 3^-a 2) 3^a - 3^-a 3) 27^a + 27^-a 4) 27^a - 27^-a)'

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'Find the rational numbers x and y that satisfy the equation 20^x = 10^(y+1).'

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'Translate the given text into multiple languages.'

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'22 \\\frac{a+2}{a+1}\, \\\sqrt{2}\, \\\frac{a}{2}+\\frac{1}{a}\'

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'Find the complex number z that satisfies the equation z^2=2+2sqrt(3)i.'

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'32 Common Logarithms\n33 Related Advanced Problems EXERCISES'

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'Determining the number of digits using common logarithms, and the position of the first non-zero digit in decimal'

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'The maximum value is (9 + 4√3) / 9, and the minimum value is (9 - 4√3) / 9'

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'Answer the following question: When the logarithm of n with base 5832, log_{5832} n, is a rational number and satisfies 1/2 < log_{5832} n < 1, find the value of n.'

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'Express the following sets of numbers in order using inequality signs.'

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'When 147a <- √{3}, - \\frac{\\sqrt{3}}{2}a+\\frac{1}{4}, - \\sqrt{3} \\leqq a < \\sqrt{2}, \\frac{a^{2}}{4} + 1, \\sqrt{2} \\leqq a, \\frac{\\sqrt{2}}{2}a+\\frac{1}{2}'

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'Let a be a positive real number on the complex plane, w=a(cosπ/36+isinπ/36). Define the sequence of complex numbers {zn} as z1=w, zn+1=znw^(2n+1) (n=1,2,…). (1) Find the argument of zn. (2) In the complex plane, with the origin as O and representing zn as point Pn. Find the values of n and a such that △OPnPn+1 is a right isosceles triangle for 1≤n≤17.'

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'(3) x = √5/10 has a maximum value of √5/2 at x = -1/2 and a minimum value of -1/2'

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'Please explain the method of using the concept of infinite geometric series to express a repeating decimal as a fraction.'

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'(2) Represent the integer part of a real number a (k ≤ a < k+1 and k is an integer) as [a]. Find the number of distinct items among [f(1)], [f(2)], [f(3)], ..., [f(1000)]. Calculate as needed using log 10 = 2.3026.'

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'Given text to be translated into multiple languages.'

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'When a ball is dropped on the floor, it bounces back up to 3/5 of the falling height.'

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'Prove that the equation α^{2}+β^{2}+γ^{2}-αβ-βγ-γα=0 holds true when triangle ABC with vertices A(-1), B(1), C(√3i) is an equilateral triangle and triangle PQR with vertices P(α), Q(β), R(γ) is also an equilateral triangle.'

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'Graph and range of irrational functions\nGraph and intersection points of irrational numbers, irrational inequalities'

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'(1) When |x| is sufficiently small, find the first-order and second-order approximations of the following functions.'

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'Let α and z be complex numbers, with |α|>1. Compare the magnitudes of |z-α| and |α z-1|.'

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'Express the following complex numbers in polar form. The argument 𝜃 should satisfy 0 ≤ 𝜃 < 2𝜋.'

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'(1) 1/(x+3) ≥ 1/(3-x) (2) 3/(1+2/x) ≥ x^2. Assume (1) as y=1/(x+3) and (2) as y=1/(3-x). Solving this gives x=0. The required solution for the inequality is the range of x values where the graph of (1) is above the graph of (2) or where they have common points. Therefore, from the figure, the range of x values we are looking for is -3 < x ≤ 0, 3 < x.'

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'When a=-\\frac{24}{\\pi^{2}} and b=\\frac{12}{\\pi^{2}}, the minimum value is -\\frac{48}{\\pi^{4}}+\\frac{1}{2}'

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'Find the maximum and minimum values of |z+√3| when a complex number z satisfies |z-i|=1, and determine the corresponding values of z.'

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'When a= \\frac{2}{e+1}, the minimum value is \\( (e+1) \\log \\frac{2}{e+1}+e \\)'

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'There is a type of growth that cannot be expressed in numbers.'

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'Let a complex number z satisfy |z| ≤ 1. Consider the complex number w = z-√2. Answer the following questions about w: (1) What kind of shape does the point w trace on the complex plane? Illustrate it. (2) If we represent the absolute value of w^2 as r and the argument as θ, find the range of r and θ. Note that 0 ≤ θ < 2π.'

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'432 Basic Example 85 Product and Quotient of Complex Numbers\nLet α=1-i, β=√3+i. Where, the argument is 0 ≤ θ < 2π.\n(1) Express αβ and α/β in polar form respectively.\n(2) Find arg(β^4), |α/β^4|.\n(3) Refer to p.429 Basics 1 1, 2'

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'Arrange the values of the following function in descending order: (11^1/10, 13^1/12, 15^1/14)'

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'Prove the inequality \ \\frac{1}{n}+\\log n \\leqq \\sum_{k=1}^{n} \\frac{1}{k} \\leqq 1+\\log n \.'

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'Let \ a \ be a positive constant. Find the range of values of \ a \ such that the inequality \ a^{x} \\geqq x \ holds for all positive real numbers \ x \.'

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'If a complex number z satisfies |z|=1, then the maximum value of |z^3-1/z^3| is a.'

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'Expressing 1+√3i and 1+i in polar form, find the values of cos(π/12) and sin(π/12) respectively.'

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'When z = \ \\frac{\\sqrt{3}}{2} + \\frac{3}{2} i \, the maximum value is 3. When z = -\\frac{\\sqrt{3}}{2} + \\frac{1}{2} i \\), the minimum value is 1.'

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'34 (1) \ \\frac{\\sqrt{2}}{12} \\\n(2) \ \\frac{\\sqrt{2}}{324} \\\n(3) \ \\frac{9 \\sqrt{2}}{104} \'

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'Express the following recurring decimal as a fraction.'

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'Prove that there does not exist a natural number n such that both √n and √(n+1) are rational numbers.'

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'Exercise 4 | II --> Book p.59\n(2)\n4/(1+√2+√3)\n= 4(1+√2-√3)/{((1+√2)+(√3))((1+√2)-(√3))}\n= 4(1+√2-√3)/(3+2√2-3)= 4(1+√2-√3)/(2√2)\n= 2(1+√2-√3)/√2= 2(1+√2-√3)√2/(√2)^2= 2(1+√2-√3)√2/2\n= √2+2-√6'

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'In Example 31, the stone has a probability of 1/2 to move from point A to B and C respectively. Similarly, it has a probability of 1/2 to move from point B to C and D. By observing the stones arriving at each point from A and B to C, from B and C to D, from C and D to E, and so on. Therefore, it can be concluded that when the stone moves from point P, Q to point R, the probabilities of reaching points P, Q, R are p, q, r respectively, then r=1/2 p+1/2 q. Using this, the probabilities of reaching each point can be calculated successively.'

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'Please read the explanation about real numbers and their properties and answer the following questions:\n1. How do you classify real numbers?\n2. How do you represent the point P corresponding to coordinate a on the number line?\n3. Please provide the definition of absolute value.\n4. Define square root and explain the difference between positive and negative square roots.'

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'Please provide an example of how irrational numbers appear in specific mathematical problems.'

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'From number 128 in Mathematics I (3), considering the inequality \ \\sqrt{3} \\tan \\theta-1 \\geqq 0 \\], we get \\[ \\tan \\theta \\geqq \\frac{1}{\\sqrt{3}} \. Solving the equation \ \\tan \\theta=\\frac{1}{\\sqrt{3}} \ yields \ \\theta=30^{\\circ} \. Since the solution lies on the line \ x=1 \ with a \ y \ coordinate greater than or equal to \ \\frac{1}{\\sqrt{3}} \, the range of solutions is \ 30^{\\circ} \\leqq \\theta<90^{\\circ} \'

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'Practice proving the following using the uniqueness of prime factorization.'

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'Let x be a positive number. A rectangle with both sides being rational numbers can be tiled with one type of square. In other words, for a rectangle with side lengths of 1 and x, if x is a rational number, it can be tiled with one type of square. If it cannot be tiled with one type of square, then the other side length of the rectangle is irrational. Using this fact, prove that √10 is irrational.'

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'Find the range of the function y = \\frac{8x+4}{x^{2}-2x+5}.'

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Rationalize the denominator of the following expressions.
(1) rac{10}{\sqrt{5}}
(2) rac{\sqrt{9}}{\sqrt{8}}
(3) rac{1}{\sqrt{2}+1}
(4) rac{2+\sqrt{3}}{2-\sqrt{3}}

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27^3 x is an irrational number. Prove the following proposition using reductio ad absurdum. At least one of x^2 or x^3 is irrational.

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Find the values of the expressions given the values of x and y defined as: x=rac{\sqrt{2}+1}{\sqrt{2}-1}, y=rac{\sqrt{2}-1}{\sqrt{2}+1}. (1) x+y, xy (2) x^{2}+y^{2} (3) x^{4} y^{2}+x^{2} y^{4} (4) x^{3}+y^{3}

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TRAINING 27 (1) From the following (1) to (4), choose all the correct statements. (1) $\sqrt{0.25}= \pm 0.5$. (2) $\sqrt{0.25}=0.5$. (3) The square root of rac{49}{64} is \pm rac{7}{8} . (4) The square root of rac{49}{64} is only rac{7}{8} . (2) Find the values of \( (\sqrt{3})^{2},\left(-\sqrt{rac{3}{2}}
ight)^{2}, \sqrt{(-7)^{2}},-\sqrt{(-9)^{2}} \) respectively.

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Consider the polynomial P = 3x^3 - 3xy^2 + x^2 - y^2 + ax + by in terms of x and y, where a and b are rational constants. (1) When x = 1/(2-√3) and y = 1/(2+√3), find the values of x + y and x - y. (2) For the x and y values in (1), if P = 4, find the values of a and b.

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Given 5 $a=\frac{1-\sqrt{5}}{2}$, find the values of the following expressions.
(1) $a^{2}-a-1$
(2) $a^{6}$

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Given x=rac{2+\sqrt{3}}{2-\sqrt{3}}, y=rac{2-\sqrt{3}}{2+\sqrt{3}}, find the values of the following expressions: (1) x+y, xy (2) x^{2}+y^{2} (3) x^{4} y^{3}+x^{3} y^{4} (4) x^{3}+y^{3}

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Regarding the product x y, for x and y, since the denominator of x and the numerator of y are the same, and the numerator of x and the denominator of y are the same, we can calculate x y=1 without rationalizing the denominator. Reciprocal relationship: \frac{A}{B}, \frac{B}{A}

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Proof by Contradiction (2)
(1) Prove that $\sqrt{2}$ is an irrational number using proof by contradiction. Assume, for contradiction, that $\sqrt{2}$ is a rational number. Then there exist two integers $p$ and $q$ with no common factors such that \sqrt{2} = rac{p}{q}. Squaring both sides yields 2 = rac{p^2}{q^2}, i.e., 2q^2 = p^2. Since p^2 is even, p must also be even. Thus, let $p=2k$ for some integer $k$. Substituting gives 2q^2 = (2k)^2, i.e., 2q^2 = 4k^2. Simplifying results in q^2 = 2k^2, so q must also be even. This means p and q share a common factor of 2, contradicting the assumption that p and q have no common factors. Therefore, $\sqrt{2}$ is irrational.

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Prove that TRAINING 59 (3) $\sqrt{3}$ is an irrational number. You may use the fact that if the square of an integer $n$ is a multiple of 3, then $n$ is a multiple of 3.

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What do you call a number that can be expressed as a fraction rac{m}{n} using an integer $m$ and a non-zero integer $n$?

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When rac{30}{7} is expressed as a decimal, find the digit in the 100th decimal place.

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Rationalize the denominators of the following expressions.
(1) rac{\sqrt{2}}{\sqrt{3}}
(2) rac{2}{\sqrt{12}}
(3) rac{1}{\sqrt{5}+\sqrt{3}}
(4) rac{\sqrt{5}}{2-\sqrt{5}}

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(1) Assume $a, b, c, d$ are rational numbers and $\sqrt{l}$ is an irrational number. Prove that $b = d$ when $a + b \sqrt{l} = c + d \sqrt{l}$. Also, prove that $a = c$ in this case. (2) Find the rational values of $x$ and $y$ that satisfy \( (1 + 3 \sqrt{2}) x + (3 + 2 \sqrt{2}) y = -5 - \sqrt{2} \).

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What is a decimal that ends at a certain decimal place called?

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Using the fact that √6 is an irrational number, prove that the following numbers are irrational: (1) 1-√24 (2) √2+√3

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What is the decimal called where the same sequence of digits repeats below a certain place?

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Remove the double radicals from the following expressions.
(1) $\sqrt{4+2 \sqrt{3}}$
(2) $\sqrt{9-2 \sqrt{20}}$
(3) $\sqrt{11+4 \sqrt{6}}$
(4) $\sqrt{4-\sqrt{15}}$

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(1) Choose all that are correct from the following 1〜(4). (1) The square root of 7 is $\pm \sqrt{7}$ (3) \sqrt{rac{9}{16}}= \pm rac{3}{4} (2) The square root of 7 is only $\sqrt{7}$ (4) \sqrt{rac{9}{16}}=rac{3}{4} (2) Find the values of \( (\sqrt{13})^{2},(-\sqrt{13})^{2}, \sqrt{5^{2}}, \sqrt{(-5)^{2}} \)

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Express the repeating decimals $1 . \dot{5}, 0 . \dot{6} \dot{3}$ as fractions.

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TRAINING 42
If the integral part of \sqrt{6}+3 is a and the fractional part is b, the value of a^{2}+b^{2} is \square.

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Using the fact that √3 is an irrational number, prove that 1+2√3 is an irrational number.

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What is a decimal called where the digits after the decimal point go on indefinitely?

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The point \( (-\sqrt{6}-\sqrt{2} i) z \) is how the point $z$ is moved. The range of the rotation angle $heta$ is $-\pi< heta \leqq \pi$.

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Since the point rac{z}{z-2} lies on the imaginary axis, the real part of rac{z}{z-2} is 0.

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Absolute Value of a Complex Number
For a complex number $z=a+bi$, the distance between the point $z$ and the origin $O$ given by $\sqrt{a^{2}+b^{2}}$ is called the absolute value of the complex number $z=a+bi$, and it is denoted by $|z|$.
In other words, the absolute value of a complex number is a real number.
Find the absolute value of the following complex numbers $z$:
1. $z = 3 + 4i$
2. $z = -1 + i$
3. $z = 2 - 2i$

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Properties of Conjugate Complex Numbers: Regarding complex numbers α and β, the following holds.

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5 (1) $\frac{1}{\sqrt{5}}$
(2) 5
(3) Minimum value $2 \sqrt{5}$ at $t=-1$

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Calculate the following complex numbers:
(1) $i$
(2) rac{1}{256}-rac{1}{256} i
(3) -rac{1}{512}
(4) -64
(5) 1024

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Calculate the argument of the following complex numbers.
(1) rac{1}{\sqrt{3}} i
(2) In order rac{\pi}{2}, rac{\pi}{6}, rac{\pi}{3}

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The imaginary part of the complex number $z$ is positive, and the three points \( A(z), B(z^2), C(z^3) \) are vertices of a right isosceles triangle. Find $z$.

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For a complex number $z$, show that |z|=|-\overline{z}| .

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78 1, 1/√2 + 1/√2i, i, -1/√2 + 1/√2i, -1, -1/√2 - 1/√2i, -i, 1/√2 - 1/√2i

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Let the real part and imaginary part of the complex number lpha both be positive. Also, let |lpha|=|eta|=1 . If the complex numbers i lpha, rac{i}{lpha}, eta represent three points on the complex plane that form the vertices of an equilateral triangle, find lpha and eta .