In base x system , 1/5 and (bar on 17)0.17 are numerals for the same number . Find x.
Urgent
just as in base 10 .17 = 17/99 = 17/(10^2-1)
in base n,
.17 = (x+7)/(x^2-1) = 1/5
x = 9
check: .179 = 16/80 = 1/5
To find the value of x, we need to equate the two numerals and solve for x.
The numeral 1/5 in base x system is represented as 0.2 (read as zero point two).
The numeral (bar on 17)0.17 represents a repeating decimal. The bar on top of the 17 indicates that 17 repeats infinitely in the decimal representation.
Let's write the decimal representation of (bar on 17)0.17:
(Bar on 17)0.17 = 17.171717...
Now, equating the two numerals, we have:
0.2 = 17.171717... (equation 1)
To solve this equation, we need to convert the repeating decimal 17.171717... to a fraction.
Let's say the fraction equivalent to 17.171717... is y:
y = 17.171717...
Multiply both sides of equation 1 by 100 to eliminate the decimal places:
100(0.2) = 100(17.171717...)
20 = 1717.171717...
Now, subtract the equation 1 from the manipulated equation:
1717.171717... - 0.171717... = 1717
1717 = 1716
We have reached a contradiction. The equation is false.
Therefore, there is no value of x that satisfies the given condition.
To find the value of x in the base-x system where 1/5 is equivalent to (bar on 17)0.17, we can use the fact that both these numbers represent the same value.
Let's break down the representation of each number:
1/5 in base-x system: This can be written as 0.2 in base-x, where the digit '2' represents 1 divided by x.
(bar on 17)0.17 in base-x system: The bar on the 17 indicates a repeating pattern. So, (bar on 17) can be represented as the fraction 17/x^2 + 7/x + 1, and adding 0.17 gives us: 17/x^2 + 7/x + 1 + 0.17.
Now, since both these numbers represent the same value, we can equate them and solve for x.
0.2 = 17/x^2 + 7/x + 1 + 0.17
Simplifying the equation, we get:
0.2 = (17 + x^2 + 7x + x)/x^2
0.2x^2 = 24x + 17
0.2x^2 - 24x - 17 = 0
This equation is in quadratic form, and we can solve it using the quadratic formula:
x = [-(-24) ± √((-24)^2 - 4(0.2)(-17))] / (2(0.2))
Simplifying further, we get:
x = [24 ± √(576 + 13.6)] / 0.4
x = [24 ± √(589.6)] / 0.4
Calculating the square root of 589.6 gives us approximately 24.264.
x = [24 ± 24.264] / 0.4
Considering both the positive and negative roots:
x₁ = (24 + 24.264) / 0.4 ≈ 120.66
x₂ = (24 - 24.264) / 0.4 ≈ -0.664
Since x cannot be negative (as base-x system only allows positive bases), the value of x is approximately 120.66.
Therefore, in the base-120.66 system, 1/5 is equal to (bar on 17)0.17.