General Form of Numbers | Mathematics & Pedagogy Paper 2 for CTET & TET Exams - CTET & State TET PDF Download

The Place Value

Example of Place Value

We have previously studied place value for two- and three-digit numbers. Here we review and extend that idea to the general form of numbers so you can read, write and expand numbers clearly.

Example: Two-digit number 67.

Starting from the right, the place of 7 is called Ones and the place of 6 is called Tens.

For larger numbers the naming of places continues from right to left. For a three-digit number the places are Ones, Tens and Hundreds. For a four-digit number the places are Ones, Tens, Hundreds and Thousands. For a five-digit number the leftmost place is called Ten thousands.

Example: Suppose we have the three-digit number 345.

The value of each place is:

• Hundreds place value = 100
• Tens place value = 10
• Ones place value = 1

Place value formula: Place value = Place × Digit

The number 345 can be expanded as follows.
3 × 100 + 4 × 10 + 5 × 1
= 300 + 40 + 5
= 345

Similarly, the five-digit number 54672 is expanded as follows.
5 × 10000 + 4 × 1000 + 6 × 100 + 7 × 10 + 2 × 1

Large Numbers

To work with numbers having many digits we use named places and two common systems of numeration: the Indian system and the International system. These systems help us read and write large numbers correctly.

The Indian System of Numeration (general form)

• Places (left to right in order): Ten crores - Crore - Ten lakhs - Lakh - Ten thousands - Thousand - Hundreds - Tens - Ones

The International System of Numeration

• Places (left to right in order): Hundred million - Ten million - Million - Hundred thousand - Ten thousand - Thousand - Hundred - Tens - Ones

Using these place names we can expand and name 9-digit numbers and more.

Example: Expand and name the number 265478100.

265478100 = 2 × 100000000 + 6 × 10000000 + 5 × 1000000 + 4 × 100000 + 7 × 10000 + 8 × 1000 + 1 × 100 + 0 × 10 + 0 × 1

The number name in the Indian system is:

• Twenty-six crore fifty-four lakh seventy-eight thousand one hundred
• In the International system the name is: Two hundred sixty-five million four hundred seventy-eight thousand one hundred

Using Commas

Commas are used to make large numbers easier to read. Comma placement differs between the Indian and International systems.

• Indian system: commas mark thousands, lakhs and crores. Example: 72705062 → 7,27,05,062
• International system: commas mark thousands and millions. Example: 72705062 → 72,705,062

Comparing Numbers

To compare two numbers we look at their digits from the leftmost side.

• If two numbers have different numbers of digits, the one with more digits is larger. Example: Among 12, 123, 1234 and 12345, the greatest is 12345 because it has five digits.
• If two numbers have the same number of digits, compare digits one by one from the left until you find a difference. The number with the larger digit at the first differing place is greater. Example: To compare 4567 and 4578 compare thousands: both 4, hundreds: both 5, tens: 6 and 7. Since 7 > 6, 4578 > 4567.

Ascending Order and Descending Order

Once we can compare numbers, we can arrange a set of numbers in order.

• Ascending order - numbers arranged from smallest to largest (increasing order).
• Descending order - numbers arranged from largest to smallest (decreasing order).

Example: The set 12, 18, 45, 39, 20, 1, 5
(i) Ascending order: 1, 5, 12, 18, 20, 39, 45
(ii) Descending order: 45, 39, 20, 18, 12, 5, 1

Note: Ascending and descending orders are reverses of each other.

Solved Example

Ques: Arrange the following in ascending order:

847, 9754, 8320, 571

Ans: The numbers in ascending order are as follows:
571, 847, 8320, 9754

Estimation

In many real-life situations, it is not necessary to know an exact count; an estimate is sufficient, quicker and often easier to obtain. Estimation methods include rounding off to the nearest tens, hundreds, thousands, and then using those rounded numbers to estimate sums, differences or products.

1. Rounding off to the nearest Tens

To round to the nearest ten, look at the ones digit.

• If the ones digit is 0, 1, 2, 3 or 4, round down to the lower ten.
• If the ones digit is 5, 6, 7, 8 or 9, round up to the next ten. (By common convention 5 is rounded up.)

Examples:
14 rounds to 10.
16 rounds to 20.
15 rounds to 20.
47 rounds to 50, and 83 rounds to 80.

2. Rounding off to the nearest Hundreds

To round to the nearest hundred, look at the tens and ones combined (the number from 0 to 99).

If the number from 0 to 49, round down to the lower hundred. If it is from 50 to 99, round up to the next hundred. By convention 50 is rounded up.

Examples:
416 rounds to 400.
485 rounds to 500.
43 rounds to 0 when rounding to the nearest hundred, because 43 is closer to 0 than to 100.

3. Rounding off to the nearest Thousands

To round to the nearest thousand, look at the hundreds, tens and ones (the number from 0 to 999).

Values from 0 to 499 round down to the lower thousand; values from 500 to 999 round up to the next thousand.

Examples:
1234 → 1000
7399 → 7000
9845 → 10000
3500 → 4000

Estimating Sums and Differences

Estimate sums and differences by rounding the quantities to a suitable place value and then performing the arithmetic on the rounded numbers.

Example: Four friends collected amounts 435, 664, 410 and 239 rupees. Estimate the total by rounding to the nearest ten.
435 rounds to 440
664 rounds to 660
410 rounds to 410
239 rounds to 240

Estimated sum = 440 + 660 + 410 + 240
= 1750

Actual sum = 435 + 664 + 410 + 239
= 1748

The estimate 1750 is quick to compute and close to the actual total.

Estimating a difference: 5733 - 458

If we round both numbers to the nearest thousand:
5733 → 6000
458 → 0
Estimated difference = 6000 - 0 = 6000 (not accurate in this case)
If we round to the nearest hundred:
5733 → 5700
458 → 500
Estimated difference = 5700 - 500
= 5200 (a more reasonable estimate)

Estimating Product of Numbers

Round each factor to its greatest place (the leftmost place), then multiply the rounded factors.

Example: Estimate 199 × 31.

199 rounds to 200

31 rounds to 30

Estimated product = 200 × 30

= 6000

Actual product = 199 × 31

= 6169

Solved Examples (Multiple Choice and Short)

Ques 1: Round-off 4353 to nearest 100.
1. 4360
2. 4200
3. 4300
4. 4400

Ans: The correct answer is "D".
4353 is closer to 4400 than to 4300 on the number line.

Ques 2: (7268-2427) estimated to the nearest hundred is

1. 4800
2. 4900
3. 4841
4. 5000

Ans: The correct option is "A".
We have (7268 - 2427) = 4841.
Rounding 4841 to the nearest hundred gives 4800.