General terms related to spherical mirrors: A mirror with the shape of a portion cut out of a spherical surface or substance is known as a...

General Terms Related to Spherical Mirrors

April 11, 2024**Comparing Decimals: **The word decimal comes from the Latin word “Decem”, which means \(10.\) Decimal numbers are the standard form of representing integers and non-integer numbers.

When you compare the decimals, you need to check the digits before the decimal point and check if they are smaller than or greater than the other number. Second, if the digits before the decimal point are identical, you have to compare the first digit after the decimal point, which is the tenth digit and identify which is greater or smaller. Let’s learn in detail how to compare decimals with solved examples.

Decimals are the extension of the number system. They can also be considered as fractions only when the denominators are \(10, 100, 1000\) etc. The numbers expressed in decimal forms are known as decimals.

**Example:** \(17.235, 0.149, 125.005, 2534.0\) etc. are the decimal numbers or decimals.

So, each decimal number or decimal has two parts, namely

- Whole number part
- Decimal part

These two parts are separated by a dot \((.),\) which is known as the decimal point.In the decimal number \(27.54,\) the whole number part is \(27,\) and the decimal part is \(54.\)

**Decimal Places: **The number of digits accommodated in the decimal part of a decimal number is known as the number of decimal places.

**Examples: **\(3.75\) has two decimal places, and \(85.325\) has three decimal places.

There are two types of decimals which are mentioned below, along with the examples:

- Terminating decimals
- Non-terminating decimals

These decimals have a finite number of digits after the decimal point. These are also called exact decimal numbers, as the digits after the point is countable.

Example: \(89.9856, 2.32, -1.89546, 3.37543\) etc., are examples of the terminating decimal numbers. They can also be written in the form of \(\frac{p}{q},\) and hence, they are rational numbers.

These decimals have an infinite number of digits after the decimal point. They are non-terminating decimals that repeat endlessly. These are further divided into two more numbers, and they are:

- Recurring decimal numbers
- Non-recurring decimal numbers

These decimals have an infinite number of digits after the decimal point, but these digits (single-digit or a group of digits) are repeated.

Examples: \(5.313131…., 7.89898989…., 9.1111111….,\) etc. these are recurring decimals as one or a group of digits at the decimal places are repeated. The recurring decimal numbers can also be expressed by putting a bar over the digit or the group of digits that are repeating.

Example: \(120.\overline {35} = 120.3535353535 \ldots \ldots ,\)

\(5.\overline {13} = 5.1313131313 \ldots \ldots ,\)

\(7.\overline {809} = 7.809809809809 \ldots \ldots \ldots \)

Recurring decimals also can be expressed as fractions.

Example: \(0.6666 \ldots .0.\bar 6 = \frac{2}{3},\)

\(0.88888 \ldots = 0.\bar 8 = \frac{8}{9}\)

This recurring decimal can also be pure periodic or ultimately periodic.

**Pure periodic decimals: **These decimals are those numbers in which a part is repeated endlessly. For example, the numbers \(1.3333…., 0.55555….,\) and \(1.99999…\) are examples of pure periodic decimals. They can also be written using signs over them like \(1.\bar 3,\,0.\bar 5\) and \(1.\bar 9,\) respectively.

**Ultimately period decimal numbers: **These decimals are those numbers in which the periodic part follows the non-periodic part. For example, \(34.126666…, 6.1788888….\) and \(45.9333333….\) are few examples of ultimately periodic decimals. These can also be written like \(34.12\bar 6,6.17\bar 8\) and \(45.9\bar 3,\) respectively.

These non-recurring decimal numbers are non-terminating and non-repeating decimal numbers. They have an infinite number of digits at the decimal places, and their decimal place digits don’t follow a specific order.

**Examples: **\(1.3687493043, 1.2376894…., 21.3749940…., 1.76368939….\) are few non-terminating, non-repeating decimal numbers.

Decimals having the same number of decimal places are known as like decimals, i.e., decimals having the same number of digits on the right of the decimal point are known as like decimals. Otherwise, the decimals are unlike decimals.

**Example:** \(5.25, 15.04, 273.89\) are like decimals as you have the same number of digits after the decimal point.

Now, \(9.5, 18.235, 20.0254\) etc., are unlike decimals as they have different numbers of digits after the decimal point.

It is important to note that annexing the zeros on the right side of the extreme right digit of the decimal part of a number does not alter the value of the number. Here, unlike decimals can be converted into like decimals by annexing the required number of zeros on the right side of the extreme right digit in the decimal part.

Example: \(7.4, 15.35, 49.105\) are unlike decimals. These decimals can be rewritten as \(7.400, 15.350. 49.105.\) Now, these are like decimals.

To compare the decimal numbers, we may follow the given steps:

- Obtain the decimal numbers.
- Compare the whole number parts of the decimal numbers. The number with the greater whole number part will be the greater. If the whole number parts are equal, go to the next step.
- Compare the extreme left digits of the decimal parts of the two numbers. The number with a greater extreme left digit is greater. If these digits are equal, then compare the following digits and so on.

Example-1: The given decimals have distinct whole number parts, so compare the whole parts only. In \(73.63,\) the whole number part is \(73.\)

In \(44.57,\) the whole number part is \(44.\)

\(∵73>44\)

\(∴73.63>44.57\)

Example-2: Let us take decimal numbers \(65.564\) and \(65.327\)

You can see that the digits before the decimal point are the same \(65=65.\)

Now, compare the tenths place digits that are \(5>3.\)

Thus, \(5\) is the greater number than \(3,\) so we can say \(65.564>65.327\)

We have to follow the given steps to compare the decimals by using the place value chart:

- First, you have to write the decimal numbers in the place value chart.
- Second, you have to compare the digits right from the left side digits in the place value chart.
- Finally, if the digits are similar, then compare the digits in the next place value by moving to the right side. Keep comparing the digits with the same place value until you find digits that are different.

**Example:** Compare \(0.67\) and \(0.87\)

Ones | Decimal Point | Tenths | Hundredths |

\(0\) | \(.\) | \(6\) | \(7\) |

\(0\) | \(.\) | \(8\) | \(7\) |

Now, compare the one’s digits, \(0=0;\) they are similar. Next, compare the tenths place digits \(6<8,\) you can see that the number \(8\) is greater than the number \(6.\)

So, \(0.67<0.87\)

** Q.1. Which is greater of** \(48.23\)

In \(48.23,\) the whole number part is \(48.\)

In \(39.35,\) the whole number part is \(39.\)

\(∵48>39\)

\(∴48.23>39.35\)

** Q.2. Compare and write the following decimals in ascending order:** \(5.64, 2.54, 3.05, 0.259\)

\(5.640, 2.540, 3.050, 0.259\) and \(8.320.\)

Clearly, \(0.259<2.540<3.050<5.640<8.320\)

Hence, given decimals in the ascending order are:

\(0.259, 2.54, 3.05, 5.64\) and \(8.32.\)

** Q.3. Compare the decimal numbers** \(23.798\)

You can see that the digits before the decimal point are the same \(23=23.\)

Now, compare the tenths place digits that is \(7>6.\)

Thus, \(7\) is the greater number than \(6,\) so we can say \(23.798>23.659\)

Hence, the required answer is given above.

** Q.4. Compare the decimal numbers** \(563.777\)

We compare the hundredth place digits (compare whole number part of the two decimal numbers) that is \(563.777\) and \(984.777,\) so, \(5<9.\)

Thus, \(5\) is a smaller number than \(9.\) We can say \(563.777<984.777\)

Hence, the required answer is given above.

** Q.5. Compare the decimal numbers** \(9999.88\)

\(9999.88\) and \(9999.88\)

You can see that all the digits (both before and after the decimal point) of the two decimal numbers are the same.

Now, \(9999.88=9999.88\)

Thus, \(9999.88\) is equal to \(9999.88\)

In the given article, we have discussed decimals with an example, and then we talked about types of decimals followed by like and unlike decimals. We glanced at comparing the decimal numbers and mentioned a few steps to follow to compare the decimals. Finally, we have provided you with solved examples along with a few FAQs.

*Q.1. How do you know which decimal is greater?*** Ans:** When you compare any decimal numbers, you have to compare the whole numbers as shown in the below example:

Example: The given decimals have distinct whole decimal number parts, so compare the whole parts only.

In \(79.63,\) the whole number part is \(79.\)

In 27.13, the whole number part is \(27.\)

\(∵79>27\)

\(∴79.63>27.13\)

*Q.2. How do you compare two decimal values?*** Ans:** To compare the decimal numbers, we may follow the given steps:

Compare the whole number parts of the decimal numbers. The number with the greater whole number part will be the greater. If the whole number parts are equal, go to the next step.

Compare the extreme left digits of the decimal parts of the two numbers. The number with a greater extreme left digit is greater. If these digits are equal, then compare the following digits and so on.

*Q.3. What should I do in comparing the Decimals?*** Ans:** When you are comparing decimals, start in the tenths place, the digit with the bigger value is bigger. In case they are the same, then compare the hundredths place and, if the values are still equal, then move to the right until you find one that is greater.

**Q.4. Is 0.6 bigger or 0.06?**** Ans:** In both the numbers after the decimal point, the digit in \(0.6\) is \(6\) and in \(0.06\) is \(0.\)

So, tenths place comes before hundredths place \(0.6>0.06.\)

*Q.5. What is the decimal place?*** Ans:** The number of digits accommodated in the decimal part of a decimal number is known as decimal places.

Examples: \(3.75\) has two decimal places, and \(85.325\) has three decimal places.