1. Rounding off numbers: definition.

• Rounding a number means replacing it by another value that is 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.

2. How to round off a number to whole places?

• Identify the place value of the digit to be rounded - this is called the rounding digit.
• Identify the next digit to the right of the rounding digit, if there is any:
• • If there is no other digit to the right, then the number is not changing, it stays the same.
  • Rounding down. If there is a digit to the right of the rounding digit and this digit is less than 5 (ie: 0 to 4), then leave the rounding digit as it is and replace all the digits to the right of it with zeros. This is called rounding down.
  • Rounding up. If there is a digit to the right of the rounding digit and this digit is 5 or more (ie: 5 to 9), then increase the rounding digit by 1 and replace all the digits to the right of it with zeros. This is called rounding up.

• >> See in the article how to round off numbers to whole places, plus examples

3. How to round off a number to decimal places?

• Identify the place value of the digit to be rounded - this is called the rounding digit and it is the last digit to keep.
• Identify the next digit to the right of the rounding digit, if there is any:
• • If there is no other digit to the right, then the number is not changing, it stays the same.
  • Rounding down. If there is a digit to the right of the rounding digit and this digit is less than 5 (ie: 0 to 4), then leave the rounding digit as it is and drop all the decimal digits to the right of it. This is called rounding down.
  • Rounding up. If there is a digit to the right of the rounding digit and this digit is 5 or more (ie: 5 to 9), then increase the rounding digit by 1 and drop all the decimal digits to the right of it. This is called rounding up.

• >> See in the article how to round off numbers to decimal places, plus examples

4. Mathematical explanation of the rules used in numbers rounding.

• >> See in this article the mathematical explanation of the rules used in numbers rounding

5. Special cases. Examples.

• The next digit to the right of the rounding digit is 5 and it is also the last non-zero digit in that number.
• In these cases the numbers can either be rounded up or down, depending on the type of rounding, as shown below.
• Types of rounding:

• 5.1. Number Round Half Up.

  • Numbers that are halfway between two neighbors are rounded off to the larger neighbor.

  • >> See examples on numbers rounded half up

• 5.2. Number Round Half Down.

  • Numbers that are halfway between two neighbors are rounded off to the smaller neighbor.

  • >> See examples on numbers rounded half down

• 5.3. 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.

  • >> See examples on numbers rounded half away from zero

• 5.4. Number Round Half Towards Zero.

  • Numbers that are halfway between two neighbors are rounded off to the neighbor which is closer towards zero.

  • >> See examples on numbers rounded half towards zero

• 5.5. 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.

  • >> See examples on numbers rounded half to even (Gaussian rounding or Banker's rounding)

• 5.6. 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.

  • >> See examples on numbers rounded half to odd (Gaussian rounding or Banker's rounding)

• 5.7. Number Round Ceiling.

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

  • >> See examples on numbers rounded half to ceiling

• 5.8. Number Round Floor.

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

  • >> See examples on numbers rounded half to floor