Express 0.1111111...+0.01010101010101....+ 0.000100010001 as a common fraction.
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First, illustrate the problem by drawing a square inside a circle as shown in the first picture. Connect each corner of the square to the center of the circle. Since the square is inscribed in the circle, they have the same center points. Each segment drawn to the corners is a radius of the circle measuring 1 unit. Also, a square has equal sides. So, the angle made between those segments are equal. You can determine each angle by dividing the whole revolution into 4. Thus, each point is 360°/4 = 90°.
Next, cut a portion of one triangle from the circle as shown in the second picture. Since one of the angles is 90°, this is a right triangle with s as the hypotenuse. Applying the pythagorean theorem,
s = √(1²+1²) = √2
So each side of the square is √2 units. The area of the square is, therefore,
A = s² = (√2)² = 2
The area of the square is 2 square units.
Next, cut a portion of one triangle from the circle as shown in the second picture. Since one of the angles is 90°, this is a right triangle with s as the hypotenuse. Applying the pythagorean theorem,
s = √(1²+1²) = √2
So each side of the square is √2 units. The area of the square is, therefore,
A = s² = (√2)² = 2
The area of the square is 2 square units.