Answer to Question #114301 in Algebra for Felicia Ngubandze

Question

1. Express 54069,269 in words; place value; and in terms of the powers of 10.
2. Differentiate between real and non-real numbers (give an examples).

Answer 1

1. Fifty-four million sixty-nine thousand two hundred sixty-nine.

Place value: 54069,269 = (5 x 10000000) + (4 x 1000000) + (6 x 10000) + (9 x 1000) + (2 x 100) + (6 x 10) + (9 x 1) = 50000000 + 4000000 + 60000 + 9000 + 200 + 60 + 9

5 - ten million's place

4 - million's place

6 - ten thousand's place

9 - thousand's place

2 - hunderd's place

6 - ten's place

9 - one's place

54069,269 = 0.54069269 x 10⁸ = 54.069269 x 10⁶ = 54069.269 x 10³ = 54069,269 x 10⁰ =

= 5406926,900 x 10^-2 = 54069269000,000 x 10^-6

2. Main differences between complex and real numbers:

• complex numbers cannot be compared (< or >)
• every polynomial with real or complex coefficients has so many complex roots what is its degree
• in a system of real numbers, a root of even degree cannot be extracted from a negative number. For complex numbers, it is possible to extract the root from any number of any degree, however, the result is ambiguous - the complex root of the nth degree from a nonzero number has n different complex values

Examples: you can compare real numbers, for instance, -120 < 5.67 or $\dfrac{1}{3}$ > $\dfrac{2}{27}$ , but you can't do this to complex numbers. Polynomial x⁴ + 2x³ + x² + 5x +12 would have 4 roots(with complex roots), because n = 4. Also for equation like x² = -1 there will be two solutions: i and -i.