Approximation

Approximation and Estimation

• Approximation: A value that is close but not exact.
• Estimation: The process of sensibly guessing numbers close to their real values.

Rounding

Types of Roundings

Rounding numbers can be done in several ways, depending on the context and the desired precision. Here are some common types of rounding:

Standard Rounding (Round Half Up):

• If the digit to the right of the rounding place is 5 or greater, round up. If it’s less than 5, round down.
• Example: 3.46 rounds to 3.5; 3.44 rounds to 3.4.

Round Half Down:

• Similar to standard rounding, but if the digit is exactly 5, round down.
• Example: 3.45 rounds to 3.4.

Round Up (Ceiling):

• Always round up to the nearest number, regardless of the digits.
• Example: 3.01 rounds to 4.

Round Down (Floor):

• Always round down to the nearest number.
• Example: 3.99 rounds to 3.

Bankers Rounding (Round Half to Even):

Banker rounding, also known as Round Half to Even, is a method used to reduce bias in rounding, particularly in financial calculations. The idea is to round to the nearest even number when the number to be rounded is exactly halfway between two options (i.e., ends in 0.5). Here’s how it works:

Rounding Rules:

1. If the digit to the right of the rounding place is less than 5, round down.
2. If the digit to the right of the rounding place is greater than 5, round up.
3. If the digit to the right of the rounding place is exactly 5:
   • If the last kept digit is even, leave it unchanged.
   • If the last kept digit is odd, increase it by 1.

Examples:

• Rounding to One Decimal Place:

  • 2.35 → 2.4 (3 is odd, so round up)
  • 2.45 → 2.4 (4 is even, so stay the same)
  • 2.55 → 2.6 (5 is odd, so round up)
  • 2.65 → 2.6 (6 is even, so stay the same)

• Rounding to Two Decimal Places:

  • 3.275 → 3.28 (5 is odd, so round up)
  • 3.285 → 3.28 (8 is even, so stay the same)
  • 4.505 → 4.50 (5 is odd, so round up)
  • 4.515 → 4.52 (5 is odd, so round up)

Applications:

Banker rounding is often used in financial contexts to minimize rounding errors that can accumulate over many transactions. Rounding halfway cases to the nearest even number helps maintain a balance in overall calculations, especially in large datasets.

This method provides a fairer distribution of rounding results over time compared to always rounding up or down.

Rounding to Significant Figures:

Significant Figures (Sig Figs)

• Definition: Digits that convey meaningful information about a number's precision.

• Rules:
  1. All non-zero digits are significant (e.g., 46.7 has 3 sig figs).
  2. Leading zeros are not significant (e.g., 0.0045 has 2 sig figs).
  3. Zeros between non-zero digits are significant (e.g., 30.6 has 3 sig figs).
  4. Trailing zeros after a decimal are significant (e.g., 38.600 has 5 sig figs).
  5. Trailing zeros may or may not be significant for whole numbers without a decimal. Use scientific notation for clarity.

Counting Significant Figures

• Examples:
  • a) 4.30 x 10^4: 3 sig figs
  • b) 0.003011: 4 sig figs
  • c) 3.7: 2 sig figs

Rounding Rules

• If the leftmost digit to drop is less than 5, keep the preceding digit the same.
• If it is 5 or more, round the preceding digit up.
• Special Case: If the digit dropped is exactly 5 and no digits follow, round up only if the prior digit is odd.

Examples

1. Round 34.5736 to three significant figures:

   • Drop 7 (since 7 > 5), so 34.6.

2. Round 3.0025 to four significant figures:

• Drop 5 (preceding digit is even), so 3.002.

Rounding to Decimal Places

CASE A:

• Rule: If the first figure dropped is less than 5, the last kept figure remains unchanged.
• Example:
  • 3.423 → 3.42 (the last figure remains 2 because 3 < 5)

CASE B:

• Rule: If the first figure dropped is greater than 5, increase the last kept figure by 1.
• Example:
  • 5.487 → 5.49 (the last figure increases from 8 to 9 because 7 > 5)

CASE C:

• Rule: If the first figure dropped is 5, and all following figures are zero (or none), the last kept figure remains unchanged if it’s even.
• Example:
  • 8.4500 → 8.4 (the last figure remains 4 because it's even)

CASE D:

• Rule: If the first figure dropped is 5, and all following figures are zero (or none), the last kept figure increases by 1 if it’s odd.
• Example:
  • 6.7500 → 6.8 (the last figure increases from 7 to 8 because it's odd)

CASE E:

• Rule: If the first figure dropped is 5 and there are non-zero figures following it, increase the last kept figure by 1.
• Example:
  • 6.755 → 6.76 (the last figure increases from 5 to 6 because of 5 followed by non-zero digits)

Examples:

1. One Decimal:

   • 7.422 → 7.4 (Case A)
   • 5.6500 → 5.6 (Case C)
   • 9.6503 → 9.7 (Case E)

2. Two Decimals:

   • 6.4872 → 6.49 (Case B)
   • 7.485 → 7.48 (Case C)
   • 6.755000 → 6.76 (Case E)
   • 8.995 → 9.00 (Case B)
   • 7.4852007 → 7.49 (Case E)

• Follow the same general rounding rules as above when limiting to a certain number of decimal places.

Rounding to Nearest Ten, Hundred, etc.:

Rounding to the nearest ten, hundred, or other place values follows similar principles to standard rounding, but focuses on the specific digit in the desired place. Here’s how it works:

Rounding to the Nearest Ten

• Rule: Look at the digit in the ones place.
  • If it’s 5 or greater, round up.
  • If it’s less than 5, round down.
• Examples:
  • 34 → 30 (3 in the tens place; 4 < 5)
  • 36 → 40 (3 in the tens place; 6 ≥ 5)
  • 85 → 90 (8 in the tens place; 5 ≥ 5)

Rounding to the Nearest Hundred

• Rule: Look at the digit in the tens place.
  • If it’s 5 or greater, round up.
  • If it’s less than 5, round down.
• Examples:
  • 234 → 200 (2 in the hundreds place; 3 < 5)
  • 276 → 300 (2 in the hundreds place; 7 ≥ 5)
  • 850 → 900 (8 in the hundreds place; 5 ≥ 5)

Rounding to the Nearest Thousand

• Rule: Look at the digit in the hundreds place.
  • If it’s 5 or greater, round up.
  • If it’s less than 5, round down.
• Examples:
  • 1,234 → 1,000 (1 in the thousands place; 2 < 5)
  • 1,567 → 2,000 (1 in the thousands place; 5 ≥ 5)
  • 4,850 → 5,000 (4 in the thousands place; 8 ≥ 5)

General Steps for Rounding to a Specific Place Value:

1. Identify the digit in the place value you are rounding to (ten, hundred, thousand, etc.).
2. Look at the digit immediately to the right of it.
3. Apply the rounding rule based on that digit.

This method can be applied to any place value, allowing you to round to the nearest ten, hundred, thousand, or even more specific places like tenths or hundredths, by adjusting the digits you evaluate.

Rounding Error

When rounding numbers, there can be discrepancies between the rounded value and the original value, which can be quantified using percentage error. Here’s how to calculate it:

The rounding error is the difference between the actual value and the rounded value.

Formula for Rounding Error:

$$\text{Rounding Error} = \text{Actual Value} - \text{Rounded Value}$$

Percentage Error

Percentage error expresses the rounding error as a percentage of the actual value. It provides a way to assess the size of the error relative to the original quantity.

The formula for Percentage Error:

$$\text{Percentage Error} = \left( \frac{\text{Rounding Error}}{\text{Actual Value}} \right) \times 100$$

Steps to Calculate Percentage Error:

1. Determine the Actual Value: The original number before rounding.
2. Round the Value: Use your chosen rounding method.
3. Calculate the Rounding Error: Subtract the rounded value from the actual value.
4. Calculate the Percentage Error: Divide the rounding error by the actual value and multiply by 100.

Example:

Let's say you want to round the number 5.67 to one decimal place.

1. Actual Value: 5.67
2. Rounded Value: 5.7 (using standard rounding)
3. Calculate Rounding Error: $$\text{Rounding Error}=5.67-5.7=-0.03$$
   
4. Calculate Percentage Error: $$\text{Percentage Error}=\left(\frac{-0.03}{5.67}\right)\times 100\approx -0.53\mathrm{\%}$$
   

Interpretation:

A negative percentage error indicates that the rounded value is greater than the actual value. In this case, the rounding introduces a small error of approximately 0.53% relative to the original number.

Considerations:

• The significance of percentage error can vary depending on the context and the precision required.
• In fields like finance or engineering, even small rounding errors can have significant implications, making it crucial to track and manage them.

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