Rounding Numbers

An estimation is a form of rounding off. Estimation is utilized in daily life as well as in topics such as mathematics and physics. Many physical values, such as the quantity of money, the distance traveled, the length measured, and so on, are calculated by rounding the actual figure to the closest whole number. 

Every measurement has an inaccuracy. Precision and accuracy are two elements that influence measurement. These two criteria aid in identifying the presence of physical quantities as well as their comparison and measurement. Let’s describe Precision and Accuracy as:

• Accuracy: The proximity of a measured value to a standard value is defined as accuracy. e.g. If you weigh a box and note 5 kg, but its known weight is 6 kg, your measurement is inaccurate.
• Precision: Precision is defined as the proximity of two or more measured values to one another. If you weigh the identical box five times and receive close values such as 4.11, 4.2, 4.22, 4.4, and 40., your measurements are accurate.

Precision and accuracy are two distinct concepts. You can be extremely exact but imprecise, or vice versa. Unit measurements are all about precision and accuracy. A measurement’s outcome should be exact. A physical amount is measured using two digits. The digits are classified into two types: reliable and uncertain. Now, it’s time to understand what are the significant figures.

What are the Significant Figures?

The measurements of physical commodities are a combination of a set of reliable as well as non-reliable, that is, an uncertain number of digits. Significant bits of the number comprised the first uncertain digit and all the reliable digits. They are also termed significant figures. The number of figures that are known with some degree of reliability is termed as significant digits.

For instance, the weight of a person is 90.46kg. In this case, the digits, 9,0 and 4 comprise significant digits and 6 is the non-significant one. 

Rules for considering Significant Figures:

• Non-zero digits are always significant.
• Final zeros to the right-hand side of the decimal are considered to be significant.
• Zeros written as placeholders are not significant.
• Zeros find between two significant digits are significant. For instance, 0.00065 has two significant digits.
• Zeros to the left of the first non-zero digit are not significant. For example, 0.0000466 has three significant digits.
• For non-decimal numbers, the trailing zeros are not significant. For example, 63000 has two significant digits.

Significant figures prove to be an indicator of the precision of measurement of a quantity. It is dependent on the least count of measuring instruments. In case of preservation of the number of significant figures while interconversion between units of the same commodity, the quantity is denoted in the scientific notation of the form, X × 10^Y, where X is the number between 1 and 10 which denotes the number of significant digits. 

Rounding off the Digits

Rounding off for digits is a phenomenon used for estimation. Estimation forms an important aspect for the accurate and precise measurement of objects and commodities in daily life. It also reduces the uncertainty while working with quantities. It is necessary to decrease the number of insignificant figures necessary for adhering to the rules of arithmetic operations with significant figures.

Rounding off implies simplifying a number by retaining its value but moving it closer to the next number. It is performed for whole numbers as well as decimals at various spots such as hundreds, tens, tenths, and so on. 

Numbers are rounded off to retain important digits. The number of important figures in a result simply refers to the number of figures that are understood with some degree of certainty.

There are three significant numbers in the number 21.3. Non-zero digits are always meaningful. The number 21.3149 has six significant digits (all the numbers give you useful information). As a result, the number 76 has two significant digits, but the number 76.3 has three significant digits.

Rules for Rounding off the digits

A standard convention is followed while rounding off the digits: 

• If the digit to be rounded off is less than 5 in the specified number, then the preceding digit is left unmodified.

          e.g. 9.81 is rounded off to 9.8, since the digit to be dropped is less than 5, and the preceding digit is left unchanged.

• If the digit to be rounded off is greater than 5 in the specified number, then the preceding digit is modified and raised by one.

          e.g. 9.88 is rounded off to 9.9, since the digit to be dropped is more than 5, and the preceding digit is incremented by one.

• If the digit to be dropped is equivalent to 5 and if followed by other non-zero digits, then the preceding digit of the specified number is raised by one.

          e.g. the number 9.755 has to be rounded off considering the tens digits, then it is rounded off to 9.8 since it is followed by other non-           zero digits. 

• If the digit to be dropped is equivalent to 5 or if 5 is followed by other zero digits, then the preceding digit of the specified number remains unmodified, in case it is even.

          e.g. the number 9.850 has to be rounded off considering the tens digits, then it is rounded off to 9.8 since it is followed by other zeros.

• If the digit to be dropped is equivalent to 5 or if 5 is followed by other zero digits, then the preceding digit of the specified number is incremented by one, in case it is odd.

          e.g. the number 9.750 has to be rounded off considering the tens digits, then it is rounded off to 9.8 since it is followed by other zeros.

The first non-significant digit is at the (n+1)th position from the leftmost place. 

Rules to Remove Ambiguities in Determining the Number of Significant Figures

Some of the important rules to remove Ambiguities in determining the number of Significant Figures are:

• Change in units should not have an impact on the number of significant digits of the number.

          e.g. 5.900m = 590.0 cm = 5900 mm. The first two numbers are 4 significant digits, and the last one has 2 digits respectively. 

• Scientific notation can be used to report measurements of numbers.
• Multiplication or division of exact numbers can have an infinite number of significant digits.

Rounding Rules for Whole Numbers

• A smaller place value for the specified whole number is chosen.
• Use the next smaller place which is towards the right of the number that is being rounded off to. While rounding digits from tens place, a digit in the one’s place is looked for.
• Look for the magnitude of the digit. If the smallest place is less than 5, then the digit is left untouched.  Any number of digits after that number becomes zero which is termed as rounding down of the digit. However, if the smallest place is greater than or equal to 5, then the digit is added with +1. Any digits after that number become zero and are termed as rounding up of the digit.

Rounding Rules for Decimal Numbers

• Determine the rounding digits and evaluate the right-hand side.
• If the digits on the right-hand side are less than 5, they are considered to be equivalent to zero. If greater than or equal to 5, then add +1 to that digit and consider all other digits as zero.

In many situations, obtaining an estimate is easier, less expensive, and takes less time than performing an actual count. If the scenario does not call for a precise count, an estimate will suffice. Let’s look at several estimating techniques.

1. Rounding off to the nearest Tens: Consider the following three numbers: 24, 25, and 26. These figures must be rounded to the nearest tens place. Consider putting these numbers on a scale. Is 24 closer to twenty or to Thirty? It’s closer to twenty, so we’ll round it up to twenty. Similarly, 26 is close to 30, so we may round it up to 30. 25 is the same distance as 20 and 30. It is common practice to round up, therefore 25 will be rounded up to 30 as well. Finally, numbers ending in 1, 2, 3, and 4 are rounded down, but numbers ending in 5, 6, 7, 8, and 9 are rounded up to the closest tens place.
2. Rounding off to Hundreds: The same idea applies here. We look at the number line to determine if it is nearer to the lower hundred or the upper one. With examples, we will have a deeper understanding. Completing the numbers 527 and 582. 527 is clearly closer to 500 in this case, so we round it up to 500. And 582 will be rounded up to the nearest hundred, resulting in 600. One thing to keep in mind is that 450, which is about between 400 and 500, is usually rounded up to 500. As an example, round 43 to the closest hundred. Because 43 is closer to 0 than to 100, we’ll round it up to 0.
3. Rounding off to Thousands: All values between 0 and 499 that are closer to 0 on the number line will be rounded to 0. Numbers ranging from 500 to 999 will be rounded to 1000. And the same idea will apply to all bigger integers. Numbers nearer to the lower thousand will be rounded down, while numbers over 500 will be rounded up.

Sample Problems

Problem 1: Explain the number of significant digits in 0.000650.

Solution: 

Since the leading zeros are not necessary for the significant digits. 

Therefore, there are three significant digits 650. 

Problem 2: Round these numbers to three significant figures each: (a) 9.845 and (b) 6.735.

Solution: 

(a) The number to be rounded off is equivalent to 5, so in the first case, the preceding digits are even, therefore remain unmodified. Therefore, the obtained number 9.84.

(b) The number to be rounded off is equivalent to 5, so in the first case, the preceding digits are odd, therefore are incremented by one. Therefore, the obtained number 9.83.

Problem 3: Does the change of units change significant digits?

Solution: 

The change of units does not change the significant digits. For example, in terms of length, 12 m = 1.2 × 10 m = 1.2 × 10³ cm where the significant digits is equivalent to 1. 

Problem 4: Write the number of significant figures in 0.410 m.

Solution:

Zeros to the left of a significant number that are not limited to the left by another significant figure are not significant, according to the general criterion for defining significant figures. Significant are zeroes put after other numbers but before a decimal point. As a result, for the provided value 0.410, the final three digits will be significant numbers. As a result, it has three important figures.

Problem 5: What are the numbers 4.845 and 4.835 on rounding off to 3 significant figures?

Solution:

Because the previous digit is even, the number 4.845 rounded to three significant digits yields 4.84. The number 4.835 rounded to three significant digits, on the other hand, becomes 4.84 since the previous digit is odd.