# How to Round Off Numbers to Whole or Decimal Places, Theory, Explanations, Examples. Special Cases

## How to round off numbers?

### 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.

### Example 1: round off the number 163.87 to the nearest ten.

• Identify the place value of the rounding digit, the digit of the tens, 6: 163.87.
• Identify the next digit to the right of the rounding digit, the digit of the units, 3: 163.87.
• The digit to the right of the rounding digit is 3, which is less than 5 - leave the rounding digit as it is and replace all the digits to the right of it with zeros:
163.87 ≈ 160.00 = 160.
• This was an example of rounding a number down.

### Example 2: round off the number 169.87 to the nearest ten.

• Identify the place value of the rounding digit, the digit of the tens, 6: 169.87.
• Identify the next digit to the right of the rounding digit, the digit of the units, 9: 169.87.
• The digit to the right of the rounding digit is 9, which is more than 5 - increment the rounding digit by 1 and replace all the digits to the right of it with zeros:
169.87 ≈ 170.00 = 170.
• This was an example of rounding a number up.

### 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.

### Example 1: round off the number 163.945 to the nearest tenth.

• Identify the place value of the rounding digit, the last digit to keep, the digit of the tenths, 9: 163.945.
• Identify the next digit to the right of the rounding digit, the digit of the hundredths, 4: 163.945.
• The digit to the right of the rounding digit, 4, is less than 5 - leave the rounding digit as it is and drop all the digits to the right of it:
163.945 ≈ 163.9.
• This was an example of rounding a number down.

### Example 2: round off the number 163.965 to the nearest tenth.

• Identify the place value of the rounding digit, the last digit to keep, the digit of the tenths, 9: 163.965.
• Identify the next digit to the right of the rounding digit, the digit of the hundredths, 6: 163.965.
• The digit to the right of the rounding digit, 6, is more than 5 - increment the rounding digit by 1 and drop all the decimal digits to the right of it:
163.965 ≈ 164.0 = 164.
• This was an example of rounding a number up.

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

• In order to understand and explain the rules that were used above, let's have an example. Let's round the number 4.38 to the nearest tenth.
• Identify the place value of the digit to be rounded, which is the tenth digit place, 3: 4.38.
• 4.38 is sitting on the axis of numbers between two 1 decimal place consecutive neighboring numbers:
4.3 < 4.38 < 4.4.
• 4.38 is going 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:
(4.3 + 4.4) ÷ 2 = 4.35.
• 4.38 is larger than 4.35, so it is closer to the larger neighbor, 4.4, which it will be rounded off to.
• 4.38 is larger than 4.35 because 8 is larger than 5.
This is where the rule of the digit to the right of the rounding digit started from.
• If the digit to the right of the rounding digit is 5 or more (5 to 9), as in our case, 8, then the number 4.38 is rounded off to the larger neighbor, 4.4, and for that the rounding digit is increased by 1 while the digits to the right of it are droped, if we round off to decimal places, as in our case, or are replaced with zeros, if we round off to whole places.
• If the digit to the right of the rounding digit is smaller than 5, for example for number 4.32, 2 is smaller than 5, then the number is rounded off to the smaller neighbor, 4.3, and so the rounding digit stays unchanged while the digits to the right of it are droped, if we round off to decimal places, as in our case, or are replaced with zeros, if we round off to whole places.

### 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.
• Examples:
• Case 1: 0.75 rounded to the nearest tenth.
The rounding digit is 7 and the next digit to the right is 5. 0.75 is equal to the middle of the interval between the two consequitive one decimal numbers, 0.7 and 0.8.
0.75 is egually close to the either neighbor.
Is 0.75 going to be rounded off to 0.7 or to 0.8?
• Case 2: -8,350 rounded to the nearest hundred.
The rounding digit is 3 and the next digit to the right is 5. Counting by hundreds -8,350 is equal to the middle of the interval between two consequitive numbers, -8,400 and -8,300.
-8,350 is egually close to the either neighbor.
Is -8,350 going to be rounded off to -8,400 or to -8,300?
• 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. Below there are examples of decimal numbers rounding half up to the nearest unit:
• 0.5 ≈ 1 // -0.5 ≈ 0 (not -1; 0 is larger than -1).
• If the number is not halfway between the two neighbors, then it must be rounded using normal rounding techniques:
• 0.4 ≈ 0 // -0.4 ≈ 0 // 0.6 ≈ 1 // -0.6 ≈ -1.

• #### 5.2. Number Round Half Down.

• Numbers that are halfway between two neighbors are rounded off to the smaller neighbor. Below there are examples of decimal numbers rounding half down to the nearest unit:
• 0.5 ≈ 0 // -0.5 ≈ -1 (not 0; -1 is smaller than 0).
• If the number is not halfway between the two neighbors, then it must be rounded using normal rounding techniques:
• 0.4 ≈ 0 // -0.4 ≈ 0 // 0.6 ≈ 1 // -0.6 ≈ -1.

• #### 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. Below there are examples of whole numbers rounding half away from zero to the nearest hundred:
• 150 ≈ 200 // -150 ≈ -200.
• If the number is not halfway between the two neighbors, then it must be rounded using normal rounding techniques:
• 140 ≈ 100 // -140 ≈ -100 // 160 ≈ 200 // -160 ≈ -200.

• #### 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. Below there are examples of whole numbers rounding half towards zero to the nearest unit:
• 7.5 ≈ 7 // -7.5 ≈ -7.
• If the number is not halfway between the two neighbors, then it must be rounded using normal rounding techniques:
• 7.4 ≈ 7 // -7.4 ≈ -7 // 7.6 ≈ 8 // -7.6 ≈ -8.

• #### 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. Below there are examples of whole numbers rounding half to even to the nearest thousand:
• 1,500 ≈ 2,000 // -1,500 ≈ -2,000.
• If the number is not halfway between the two neighbors, then it must be rounded using normal rounding techniques:
• 1,440 ≈ 1,000 // -1,440 ≈ -1,000 // 1,640 ≈ 2,000 // -1,640 ≈ -2,000.

• #### 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. Below there are examples of whole numbers rounding half to odd to the nearest thousand:
• 1,500 ≈ 1,000 // -1,500 ≈ -1,000.
• If the number is not halfway between the two neighbors, then it must be rounded using normal rounding techniques:
• 1,440 ≈ 1,000 // -1,440 ≈ -1,000 // 1,640 ≈ 2,000 // -1,640 ≈ -2,000.

• #### 5.7. Number Round Ceiling.

• All the numbers that are between two neighbors are always rounded off to the larger neighbor (no matter if the numbers are halfway or not between their neighbors). Below there are examples of decimal numbers rounding ceiling to the nearest unit:
• 0 = 0 // 0.1 ≈ 1 // -0.1 ≈ 0 // 0.9 ≈ 1 // -0.9 ≈ 0 // 1 = 1 // -1 = -1.
• 0.5 ≈ 1 // -0.5 ≈ 0 // 0.6 ≈ 1 // -0.6 ≈ 0.

• #### 5.8. Number Round Floor.

• All the numbers that are between two neighbors are always rounded off to the smaller neighbor (as above, no matter if the numbers are halfway or not between their neighbors). Below there are examples of decimal numbers rounding floor to the nearest unit:
• 0 = 0 // 0.1 ≈ 0 // -0.1 ≈ -1 // 0.9 ≈ 0 // -0.9 ≈ -1.
• 0.5 ≈ 0 // -0.5 ≈ -1 // 0.6 ≈ 0 // -0.6 ≈ -1.