Notes: Numbers | Mathematics & Pedagogy Paper 1 for CTET & TET Exams - CTET & State TET PDF Download

Understanding numbers is fundamental to mathematical proficiency, especially for those preparing for teacher-education and school-level examinations. Questions frequently test a candidate's numerical abilities, including numeration, place value, ordering, rounding and reading large numbers.

What is Numeration?

• Numeration is the process of using symbols (numerals) to represent numbers.
• When numbers are expressed in words, this is called word numeration.
• Numeration supports identification, ordering and counting - for example, race positions: 1st, 2nd, 3rd.

Number Systems

A number system defines the set of values and the rules used to represent quantities for counting and measuring.

• Indian Number System - groups digits in periods as: Ones, Tens, Hundreds | Thousands, Ten Thousands | Lakhs, Ten Lakhs | Crores, Ten Crores.
• Example comma placement (Indian): 1,23,45,678.
• International Number System - groups digits as: Ones, Tens, Hundreds | Thousands, Ten Thousands, Hundred Thousands | Millions, Ten Millions, Hundred Millions.
• Example comma placement (International): 123,456,789.

Counting Three-Digit Numbers

A three-digit number starts from 100 and goes up to 999. These numbers are formed by hundreds, tens and ones places.

• 10 tens = 100 = 1 hundred
• 20 tens = 200 = 2 hundreds
• 30 tens = 300 = 3 hundreds
• 40 tens = 400 = 4 hundreds
• 50 tens = 500 = 5 hundreds
• 60 tens = 600 = 6 hundreds
• 70 tens = 700 = 7 hundreds
• 80 tens = 800 = 8 hundreds
• 90 tens = 900 = 9 hundreds
• 100 tens = 1000 = 10 hundreds = 1 thousand

Do you know?

• 1000 is the smallest 4-digit number.
• 999 is the largest 3-digit number.
• 100 is the smallest 3-digit number.
• 99 is the largest 2-digit number.

Place Value and Face Value

Face Value

• The face value of a digit is the digit itself, irrespective of its position in the number.
• Example: In the numeral 569, the face value of 5 is 5, of 6 is 6, and of 9 is 9.

Place Value

• The place value of a digit depends on where it is placed in the number.
• Example: In the numeral 3482:

• The place value of 3 is 3000.
• The place value of 4 is 400.
• The place value of 8 is 80.
• The place value of 2 is 2.

Expanded Form of Numbers

• Writing a number as a sum of its place values is called its expanded form.
• Example: 78,543 = 70,000 + 8,000 + 500 + 40 + 3.

Comparison of Numbers

1. Different number of digits

• A number with more digits is always larger than a number with fewer digits.
• Example: 54,207 > 8,964.

2. Same number of digits

• When two numbers have the same number of digits, compare them from the leftmost digit and move right until you find a differing pair of digits.
• Example: Compare 528 and 536.

Hundreds digits are equal; compare tens digits: 2 < 3. Hence 528 < 536.

• Example: Compare 782 and 789.

Hundreds and tens digits are equal; compare ones digits: 9 > 2. Hence 789 > 782.

Ascending and Descending Order

• Ascending order means arranging from smallest to largest (small → big).
• Example: Arrange 498, 567, 834, 715 in ascending order: 498, 567, 715, 834.
• Descending order means arranging from largest to smallest (big → small).
• Example: Arrange 826, 736, 582, 914 in descending order: 914, 826, 736, 582.

Successor and Predecessor

• The successor of a whole number is the number obtained by adding 1 to it.
• The predecessor of a whole number is the number obtained by subtracting 1 from it.
• Example: Successor of 0 is 1; predecessor of 1 is 0.

Even and Odd Numbers

1. Even numbers

Numbers that can be grouped fully into pairs (sets of 2) are even.

Note: Even numbers have 0, 2, 4, 6 or 8 in the ones place.

2. Odd numbers

Numbers that cannot be fully grouped into pairs (one left over) are odd.

Note: Odd numbers have 1, 3, 5, 7 or 9 in the ones place.

How to Read and Write Five-Digit Numbers

The largest 4-digit number is 9,999. Adding 1 gives the smallest 5-digit number, 10,000 (read as ten thousand), which introduces the ten thousands place.

• Example with books: Ten books each of 1,000 pages total 10,000 pages = ten thousand.
• An abacus representation helps to visualise ten thousands.

Forming Five-Digit Numbers - Instructions

• 10,000 + 1 = 10,001 (ten thousand one).
• 10,001 + 1 = 10,002 (ten thousand two).
• 99,998 + 1 = 99,999 (ninety-nine thousand nine hundred ninety-nine).
• 99,999 is the greatest 5-digit number.
• 99,999 + 1 = 1,00,000 (one lakh = 100 thousand), the smallest 6-digit number.

Edurev Tips:

• Ones period: Ones, Tens, Hundreds
• Thousands period: Thousands, Ten Thousands

Writing and Reading Five-Digit Numbers

• Split a number into periods from the right to make it easier to read: for a 5-digit number the rightmost three digits form the ones period and the next two form the thousands period.

• Example: 18,730 can be written as 18,730 or 18 730.
• When reading, read all digits in the same period together and then the name of the period (except the ones). Example: 36,207 = thirty-six thousand two hundred seven.

We can represent the above numbers on the abacus, as shown below:

How to Read and Write Six-Digit Numbers

The smallest 6-digit number is 1,00,000 (one lakh). When writing with commas in the Indian system, put a comma after three digits from the right and then after every two digits.

• 1 is in the lakhs place in 1,00,000, so it is read as one lakh.

Instructions and Examples (Six-Digit Numbers)

• Counting in lakhs can be practised on an abacus.
• Forming numbers: 1,00,000 + 1 = 1,00,001 (one lakh one); 1,11,110 + 1 = 1,11,111 (one lakh eleven thousand one hundred eleven).
• 9,99,999 is the greatest 6-digit number.
• For a 6-digit number: the rightmost three places are the ones period, the next two (fourth and fifth) are the thousands period, and the sixth place is the lakhs period.
• Commas or spaces may be used: 4,00,000 or 4 00 000.

Comparison and Ordering of Numbers - Summary Rules and Examples

Case 1: Numbers with different number of digits

Rule: A number with more digits is larger than a number with fewer digits.

Example: 54,207 > 8,964 (5 digits > 4 digits).

Example 1: Which is the greatest: 3,815 or 567 or 36,812?

Sol:

If these are placed on a number line from left to right they appear as 567, 3,815, 36,812.

Therefore, 36,812 is the greatest.

Case 2: Numbers with the same number of digits

Rule: Compare digits from the leftmost place; the first differing digits determine which number is greater.

Example: Which is greater: 32,719 or 45,989?

Sol:

Both are 5-digit numbers. Ten-thousands digits: 3 (for 32,719) and 4 (for 45,989).

Since 4 > 3, 45,989 > 32,719.

Example: Which is greater: 7,32,612 or 7,32,545?

Sol:

Both are 6-digit numbers with the same lakh and thousand parts (7 lakh 32 thousand).

Compare the hundreds: 6 (in 7,32,612) and 5 (in 7,32,545).

Since 6 > 5, 7,32,612 > 7,32,545.

How to Form Numbers Using Given Digits

1. Smallest Number

• To form the smallest number from given digits, arrange digits in ascending order.
• If 0 is one of the digits, place the smallest non-zero digit first and then 0 next (so that the number does not start with 0).
• If digits repeat, write equal digits next to each other.

2. Greatest Number

• To form the greatest number from given digits, arrange digits in descending order.
• If digits repeat, write equal digits next to each other.

Edurev Tip: Zero occupies the second smallest place when forming the smallest number (it cannot start the number).

Example: Use 6, 2, 1, 8 to form the greatest and the smallest 4-digit numbers.

Sol:

Greatest number: arrange digits in descending order to get 8,621.

Smallest number: arrange digits in ascending order to get 1,268.

Example: Use 1, 5, 0, 8, 3 to form the greatest and the smallest 5-digit numbers.

Sol:

Greatest number: 85,310.

Smallest number: 10,358.

Rounding Off Numbers (Approximation)

Rounding off replaces a number by another number that is easier to work with and is close to it, e.g., to the nearest ten, hundred or thousand.

Rounding to the Nearest Ten

Look at the digit in the ones place.

• If ones digit is 1, 2, 3 or 4 → round down to the lower ten.
• If ones digit is 5, 6, 7, 8 or 9 → round up to the higher ten.

Examples:

• 57 → 60 (ones = 7, round up)
• 512 → 510 (ones = 2, round down)
• 1,965 → 1,970 (ones = 5, round up)
• 12,785 → 12,790 (ones = 5, round up)

Rounding to the Nearest Hundred

Look at the digit in the tens place.

• If tens digit is 1, 2, 3 or 4 → round down to the lower hundred.
• If tens digit is 5, 6, 7, 8 or 9 → round up to the higher hundred.

Examples:

• 729 → 700 (tens = 2, round down)
• 1,550 → 1,600 (tens = 5, round up)
• 24,874 → 24,900 (tens = 7, round up)
• 84,214 → 84,200 (tens = 1, round down)

Rounding to the Nearest Thousand

Look at the digit in the hundreds place.

• If hundreds digit is 1, 2, 3 or 4 → round down to the lower thousand.
• If hundreds digit is 5, 6, 7, 8 or 9 → round up to the higher thousand.

Examples:

• 8,958 → 9,000 (hundreds = 9, round up)
• 16,349 → 16,000 (hundreds = 3, round down)
• 29,500 → 30,000 (hundreds = 5, round up)

International Numeral System - Place Values and Relations

Place values proceed: Ones, Tens, Hundreds, Thousands, Ten Thousands, Hundred Thousands, Millions, Ten Millions, and so on.

In 12,345,678, the place values are:

• 8 - Ones
• 7 - Tens
• 6 - Hundreds
• 5 - Thousands
• 4 - Ten Thousands
• 3 - Hundred Thousands
• 2 - Millions
• 1 - Ten Millions

Key relations:

1. 1 hundred = 10 tens
2. 1 thousand = 10 hundreds = 100 tens
3. 1 million = 1000 thousands
4. 1 billion = 1000 millions

Indian Numeral System - Periods and Examples

In the Indian system place values are grouped as:

• Ones period: Ones, Tens, Hundreds
• Thousands period: Thousands, Ten Thousands
• Lakhs period: Lakhs, Ten Lakhs
• Crores period: Crores, Ten Crores

Example: For 75,80,72,608 the parts are:

• 75 - Crores
• 80 - Lakhs
• 72 - Thousands
• 6 - Hundreds
• 0 - Tens
• 8 - Ones

Comparison Between Indian and International Systems

• 100 thousand = 1 lakh.
• 1 million = 10 lakhs.
• 10 millions = 1 crore.
• 100 millions = 10 crores.

Comma placement:

• Indian: first comma after hundreds, then every two digits (for example, 1,23,45,67,890).
• International: first comma after hundreds, then every three digits (for example, 1,234,567,890).

All About One Crore

Understanding one crore requires knowing one lakh:

• One lakh = 100,000 (one hundred thousand). It is a 6-digit number.
• One crore = 10,000,000 = 100 lakhs.
• Examples: 50,000,000 = fifty million = five crore (Indian reading would be 5,00,00,000 read as five crore).

• The smallest 6-digit, 4-digit, 3-digit, 2-digit and 1-digit numbers are useful reference points.

Examples and playful reinforcement are common in early teaching: one crore is 10,000,000; 100 lakh = 1 crore.

Do you know? The Moon is approximately 384,400 km away from the Earth - a very large number that helps illustrate place value and large-number reading.

In the image Earth and Moon look very near but it in reality it isn't so.

Greatest- and Smallest-Digit Numbers

• Greatest 5-digit number examples and the method to obtain the smallest next higher digit number are often shown using 9s.

Question: Which is the greatest 8-digit number and how do we get the smallest 9-digit number from it?

• The greatest 8-digit number is 99,999,999, read as "nine crore ninety-nine lakh ninety-nine thousand nine hundred ninety-nine".
• Adding 1 gives the smallest 9-digit number: 100,000,000.

• 10,000,000 is read as one crore in the Indian system.
• Examples: 200,000,000 = twenty crore; 400,000,000 = forty crore.

Another example: the greatest 9-digit number is 999,999,999, read as "ninety-nine crore ninety-nine lakh ninety-nine thousand nine hundred ninety-nine". Adding 1 gives the smallest 10-digit number, 1,000,000,000, read as "hundred crore" or "one arab" in some conventions.

How to Read Large Numbers - Practical Rules

• Divide a large numeral into periods starting from the right (Indian periods: ones, thousands, lakhs, crores).
• To separate periods, place a comma after three digits from the right, then after every two digits (Indian style), or use a short space between periods.
• Example: 75,80,72,608 can also be written as 75 80 72 608 (spaces between periods).
• While reading, read digits in the same period together and then state the period name (except for the ones period).
• Do not use the word "and" between period names; do not pluralise period names (write seventy-five crore, not seventy-five crores).

Example: The population of Japan in 2018 was about 12,71,85,332. A place-value chart helps to explain the number's meaning.

Roman Numerals

• The Roman system of numeration was developed about 2000 years ago using alphabetic symbols. It is different from the Hindu-Arabic numerals commonly used today.
• Roman numerals use combinations of letters: the basic symbols and their values are shown in typical charts (I, V, X, L, C, D, M).
• Note: The Roman system has no symbol for zero.

Seven Basic Roman Numerals

How to Write Hindu-Arabic Numbers in Roman Numerals

Roman Numerals Chart 

Note: There is no symbol for zero in Roman numerals.

Roman Letters

• Roman letters correspond to most English alphabets, but three English letters J, U, W are not used in the classic Roman numeral set.
• Classic Roman letters are: A, B, C, D, E, F, G, H, I, K, L, M, N, O, P, Q, R, S, T, V, X, Y, Z. In Roman numerals only a subset (I, V, X, L, C, D, M) carry numeric values.
• Example: Year 2022 is written as MMXXII.

Large Roman Numerals and Charts

With charts and bars for multiplication (a bar placed over a numeral typically indicates multiplication by 1,000), larger Roman numerals can be written.

Charts of Roman Numerals from 1 to 10,000 ​

Rules to Write and Read Roman Numerals

Rule 1: When a letter is repeated, its value is added each time.

Examples: II = 1 + 1 = 2; XXX = 10 + 10 + 10 = 30; MMM = 1000 + 1000 + 1000 = 3000.

Tips:

• The same symbol cannot be repeated more than three times in succession.
• The symbols V, L, D are never repeated.

Check out the correct way of writing 45 in Roman Numerals!

Rule 2: If a smaller-value symbol is written to the right of a larger-value symbol, add the values.

Examples: VII = 5 + 1 + 1 = 7; XII = 10 + 1 + 1 = 12; LVII = 50 + 5 + 1 + 1 = 57.

Rule 3: If a smaller-value symbol is written to the left of a larger-value symbol, subtract the smaller from the larger.

Examples: IV = 5 - 1 = 4; IX = 10 - 1 = 9; XL = 50 - 10 = 40; CM = 1000 - 100 = 900.

Tips:

• V, L and D are never subtracted.
• I may be subtracted only from V and X and only once.
• X may be subtracted only from L and C and only once.
• C may be subtracted only from D and M and only once.

Examples converting numbers to Roman numerals:

45 = 40 + 5 = XL + V = XLV.

99 = 90 + 9 = XC + IX = XCIX.

92 = 90 + 2 = XC + II = XCII.

78 = 50 + 20 + 5 + 3 = L + XX + V + III = LXXVIII.

181 = 100 + 80 + 1 = C + LXXX + I = CLXXXI.

Examples converting Roman numerals to Hindu-Arabic numbers:

LXV = L + X + V = 50 + 10 + 5 = 65.

LIII = L + III = 50 + 3 = 53.

CDXCIX = CD + XC + IX = 400 + 90 + 9 = 499.

Rule 4: A symbol cannot be repeated more than three times in succession.

Rule 5: A bar (a line over a symbol) can be used to indicate multiplication by 1,000 (commonly used for very large numbers in Roman notation).

Rule 6: Mixed forms of Roman numerals can be determined and converted into integers by applying the rules above.

Roman Numerals to Hindu-Arabic Numbers - Using Rules for Operations

Rule for addition:

Step 1: Convert each Roman numeral to its Hindu-Arabic value.

Step 2: Add the Hindu-Arabic numbers.

Step 3: Convert the resulting Hindu-Arabic number back into Roman numerals (if required).

Example: Add LV and XV.

Solution:

(i) LV = L + V = 50 + 5 = 55.

(ii) XV = X + V = 10 + 5 = 15.

55 + 15 = 70.

70 = 50 + 10 + 10 = L + X + X = LXX.

Therefore, LV + XV = LXX.

Rule for subtraction:

Step 1: Convert each Roman numeral to Hindu-Arabic numbers.

Step 2: Subtract the Hindu-Arabic numbers.

Step 3: Convert the result back into Roman numerals (if required).

Example: Subtract XXVII from LXXXVIII.

Solution:

XXVII = X + X + V + I + I = 10 + 10 + 5 + 1 + 1 = 27.

LXXXVIII = L + X + X + X + V + I + I + I = 50 + 10 + 10 + 10 + 5 + 1 + 1 + 1 = 88.

88 - 27 = 61.

61 = 50 + 10 + 1 = L + X + I = LXI.

Final notes: The study of numeration and place value is essential for early mathematics teaching. Use place-value charts, abacuses and a variety of examples (including real-life large numbers and Roman numerals) to build understanding and fluency. Practice ordering, rounding and conversions often to develop speed and accuracy.