When R is positive, [R]*{R} will always be less than R, and never equal. When R is negative, {R} = [R]/([R] - 1).
A table follows for the resulting R (i.e., [R] + {R}), from the values of [R] at decreasing integer values -1, -2, -3, ..., together with 5*{R} - [R]/4:
-.5 2.75
-1.333333333333333 3.833333333333333
-2.25 4.5
-3.2 5
-4.166666666666667 5.416666666666667
-5.142857142857143 5.785714285714286
-6.125 6.125
-7.111111111111111 6.444444444444445
-8.1 6.75
-9.090909090909092 7.045454545454545
-10.08333333333333 7.333333333333333
-11.07692307692308 7.615384615384616
-12.07142857142857 7.892857142857143
-13.06666666666667 8.166666666666666
-14.0625 8.4375
-15.05882352941176 8.705882352941176
-16.05555555555556 8.972222222222221
-17.05263157894737 9.236842105263158
-18.05 9.5
-19.04761904761905 9.761904761904761
-20.04545454545455 10.02272727272727
-21.04347826086957 10.28260869565217
-22.04166666666667 10.54166666666667
-23.04 10.8
-24.03846153846154 11.05769230769231
-25.03703703703704 11.31481481481481
-26.03571428571428 11.57142857142857
-27.03448275862069 11.82758620689655
-28.03333333333333 12.08333333333333
-29.03225806451613 12.33870967741936
-30.03125 12.59375
-31.03030303030303 12.84848484848485
-32.02941176470588 13.10294117647059
-33.02857142857143 13.35714285714286
-34.02777777777778 13.61111111111111
-35.02702702702702 13.86486486486486
-36.02631578947368 14.11842105263158
-37.02564102564103 14.37179487179487
-38.025 14.625
-39.02439024390244 14.87804878048781
-40.02380952380953 15.13095238095238
-41.02325581395349 15.38372093023256
-42.02272727272727 15.63636363636364
-43.02222222222223 15.88888888888889
-44.02173913043478 16.14130434782609
-45.02127659574468 16.3936170212766
-46.02083333333334 16.64583333333333
-47.02040816326531 16.89795918367347
-48.02 17.15
-49.01960784313726 17.40196078431373
-50.01923076923077 17.65384615384615
-51.0188679245283 17.90566037735849
-52.01851851851852 18.15740740740741
-53.01818181818182 18.40909090909091
-54.01785714285715 18.66071428571428
-55.01754385964912 18.91228070175439
-56.01724137931034 19.16379310344827
-57.01694915254237 19.41525423728813
-58.01666666666667 19.66666666666667
-59.01639344262295 19.91803278688525
-60.01612903225806 20.16935483870968
-61.01587301587302 20.42063492063492
-62.015625 20.671875
-63.01538461538462 20.92307692307692
-64.01515151515152 21.17424242424243
-65.01492537313433 21.42537313432836
-66.01470588235294 21.67647058823529
-67.01449275362319 21.92753623188406
-68.01428571428572 22.17857142857143
-69.01408450704226 22.42957746478873
-70.01388888888889 22.68055555555556
-71.01369863013699 22.93150684931507
-72.01351351351352 23.18243243243243
-73.01333333333334 23.43333333333333
-74.01315789473684 23.68421052631579
-75.01298701298701 23.93506493506494
-76.01282051282051 24.18589743589744
-77.01265822784811 24.4367088607595
-78.0125 24.6875
-79.01234567901234 24.93827160493827
-80.01219512195122 25.1890243902439
-81.01204819277109 25.43975903614458
-82.01190476190476 25.69047619047619
-83.01176470588236 25.94117647058824
-84.01162790697674 26.19186046511628
-85.01149425287356 26.44252873563218
-86.01136363636364 26.69318181818182
-87.01123595505618 26.9438202247191
-88.01111111111111 27.19444444444444
-89.01098901098901 27.44505494505495
It seems that only -3.2 works, producing the integer 5 for 5*{R} - [R]/4.
DEFDBL A-Z
CLS
FOR n = -1 TO -90 STEP -1
x = n / (n - 1)
R = n + x
PRINT R; TAB(30); 5 * x - n / 4
NEXT
using n to represent [R] and x to represent {R}.
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Posted by Charlie
on 2009-09-12 15:19:33 |
computer exploration
