566.5555555559: Round off the (terminating) decimal number to the nearest one (1 whole place)

Rounding a number means replacing it with a simpler, shorter approximation, while keeping its value close to the initial one. The rounded number will be less accurate, but easier to work with.

Counting by units (1 whole place at a time), our number is sitting on the axis of numbers between two consecutive neighboring numbers:
566 < 566.5555555559 < 567

Our number is to be rounded off to one of these neighbors, the closer one.

The middle of this interval, the number that is equally close to the either neighbor, is: (566 + 567) ÷ 2 = 566.5

Our number, 566.5555555559, is larger than 566.5, so it is closer to the larger neighbor: 567

Except for 'To Ceiling, To Floor', the rounded off number (both the positive and the negative) will be equal only to this larger neighbor.

Rounding digit. Let's call the digit of the position place that is intended to round off to as the 'rounding digit'. The digit is 6: 566.5555555559

In a positive number, if the digit to the right of the 'rounding digit' is equal to 5 and it is not the last digit that is different from zero, then the number is rounded to the larger neighbor.

The digit is 5: 566.5555555559 ≈ 567 rounded to the nearest one (1 whole place).

Or, even simpler.
In any number rounded to whole places, if the digit to the right of the 'rounding digit' is 5 (but it is not the last non-zero digit to the right) or more than 5, ie. 6, 7, 8 or 9, then increase the 'rounding digit' by 1 and replace all the digits to the right of it with zeros:

566.5555555559 ≈ 567 rounded to the nearest one (1 whole place)

Half Up, Half Down, Half Away From Zero, Half Towards Zero, Half to Even, Half to Odd. To Ceiling, To Floor.

Number Round Half Up:

Numbers that are halfway between two neighbors are rounded off to the larger neighbor. 566.5555555559 is not halfway between its neighbors.

566.5555555559 ≈ 567

- 566.5555555559 ≈ - 567

rounded to the nearest one (1 whole place)

Number Round Half Down:

Numbers that are halfway between two neighbors are rounded off to the smaller neighbor. 566.5555555559 is not halfway between its neighbors.

566.5555555559 ≈ 567

- 566.5555555559 ≈ - 567

rounded to the nearest one (1 whole place)

Number Round Half Away From Zero:

Numbers that are halfway between two neighbors are rounded off to the neighbor which is farther away from zero. 566.5555555559 is not halfway between its neighbors.

566.5555555559 ≈ 567

- 566.5555555559 ≈ - 567

rounded to the nearest one (1 whole place)

Number Round Half Towards Zero:

Numbers that are halfway between two neighbors are rounded off to the neighbor which is closer towards zero. 566.5555555559 is not halfway between its neighbors.

566.5555555559 ≈ 567

- 566.5555555559 ≈ - 567

rounded to the nearest one (1 whole place)

Number Round Half to Even:
(Gaussian Rounding or Banker's Rounding)

Numbers that are halfway between two neighbors are rounded off to the neighbor with an even rounding digit. 566.5555555559 is not halfway between its neighbors.

566.5555555559 ≈ 567

- 566.5555555559 ≈ - 567

rounded to the nearest one (1 whole place)

Number Round Half to Odd:
(Gaussian Rounding or Banker's Rounding)

Numbers that are halfway between two neighbors are rounded off to the neighbor with an odd rounding digit. 566.5555555559 is not halfway between its neighbors.

566.5555555559 ≈ 567

- 566.5555555559 ≈ - 567

rounded to the nearest one (1 whole place)

Number Round Ceiling:

All the numbers that are between two neighbors are always rounded off to the larger neighbor.

566.5555555559 ≈ 567

- 566.5555555559 ≈ - 566

rounded to the nearest one (1 whole place)

Number Round Floor:

All the numbers that are between two neighbors are always rounded off to the smaller neighbor.

566.5555555559 ≈ 566

- 566.5555555559 ≈ - 567

rounded to the nearest one (1 whole place)