The mathematician Girolamo Cardano is credited with the first use (in 1545) of negative square roots in solving the now-famous problem, “Find two numbers whose sum is 10 and whose product is 40.” Show that the complex numbers 5 + i 15 and 5 − i 15 satisfy the conditions of the problem. (Cardano did not use the symbolism i 15 or even − 15 . He wrote R.m 15 for − 15 , meaning “radix minus 15.” He regarded the numbers 5 + R.m 15 and 5 - R.m 15 as “fictitious” or "ghost numbers ,” and considered the problem “manifestly impossible.” But in a mathematically adventurous spirit, he exclaimed, “Nevertheless, we will operate.”)
The mathematician Girolamo Cardano is credited with the first use (in 1545) of negative square roots in solving the now-famous problem, “Find two numbers whose sum is 10 and whose product is 40.” Show that the complex numbers 5 + i 15 and 5 − i 15 satisfy the conditions of the problem. (Cardano did not use the symbolism i 15 or even − 15 . He wrote R.m 15 for − 15 , meaning “radix minus 15.” He regarded the numbers 5 + R.m 15 and 5 - R.m 15 as “fictitious” or "ghost numbers ,” and considered the problem “manifestly impossible.” But in a mathematically adventurous spirit, he exclaimed, “Nevertheless, we will operate.”)
Solution Summary: The author calculates the sum and product of two numbers, 10 and 40, using the FOIL method.
The mathematician Girolamo Cardano is credited with the first use (in 1545) of negative square roots in solving the now-famous problem, “Find two numbers whose sum is 10 and whose product is 40.” Show that the complex numbers
5
+
i
15
and
5
−
i
15
satisfy the conditions of the problem. (Cardano did not use the symbolism
i
15
or even
−
15
. He wrote R.m 15 for
−
15
, meaning “radix minus 15.” He regarded the numbers 5 + R.m 15 and 5 - R.m 15 as “fictitious” or "ghost numbers ,” and considered the problem “manifestly impossible.” But in a mathematically adventurous spirit, he exclaimed, “Nevertheless, we will operate.”)
Combination of a real number and an imaginary number. They are numbers of the form a + b , where a and b are real numbers and i is an imaginary unit. Complex numbers are an extended idea of one-dimensional number line to two-dimensional complex plane.
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