Numeration, Place Value and Whole Numbers

Every arithmetic question rests on a clear sense of how numbers are written and what each digit is worth. The APTET syllabus begins with knowing about numbers: the Hindu-Arabic (Indian) and international systems of numeration, place value and face value, comparing and ordering numbers, the four fundamental operations and whole numbers. Questions here are quick, but traps such as “place value of zero” and “lakh vs million” are common.

Systems of numeration

Our number system is the Hindu-Arabic system. It uses ten digits (0 to 9), is a base-10 system, and uses place value with zero as a placeholder. Large numbers are grouped and read in two ways.

Commas: in the Indian system, the first comma comes after three digits from the right, then after every two digits: 5,62,34,817 (five crore sixty-two lakh thirty-four thousand eight hundred seventeen). In the international system, commas come after every three digits: 56,234,817 (fifty-six million two hundred thirty-four thousand eight hundred seventeen).

Memory aid: 10 lakh = 1 million, 1 crore = 10 million.

Place value, face value and comparing numbers

The face value of a digit is the digit itself. Its place value is the digit × the value of its place.

The place value of 0 is always 0, whatever its position. The expanded form writes a number as the sum of place values: 3,47,285 = 3,00,000 + 40,000 + 7,000 + 200 + 80 + 5.

Comparing and ordering

• A number with more digits is greater.
• With the same number of digits, compare from the left; the first place where the digits differ decides.
• Ascending order: smallest to largest; descending: largest to smallest.

Forming numbers from digits: with 3, 0, 7, 5 used once each, the largest 4-digit number is 7530 (digits in descending order) and the smallest is 3057 (smallest non-zero digit first, then 0, then the rest in ascending order).

Counting numbers of a given size: there are 9 one-digit numbers, 90 two-digit numbers (10 to 99), 900 three-digit numbers and 9,000 four-digit numbers.

Four operations and whole numbers

Division algorithm: Dividend = Divisor × Quotient + Remainder (remainder < divisor). Example: 434 ÷ 17 gives quotient 25 and remainder 9, because 17 × 25 + 9 = 425 + 9 = 434.

Natural numbers are 1, 2, 3, …; whole numbers are 0, 1, 2, 3, …. Zero is the smallest whole number; there is no largest. Every natural number is a whole number, but 0 is not a natural number.

Whole numbers are not closed under subtraction (3 − 5 is not a whole number) or division (3 ÷ 5 is not). Division by zero is not defined.

Exam focus

• Place value vs face value; place value of 0 is 0.
• Indian vs international commas and names: 10 lakh = 1 million; 1 crore = 10 million.
• Number of 2-digit numbers = 90; 3-digit = 900.
• Dividend = divisor × quotient + remainder.
• Whole numbers: closed under + and ×, not under − and ÷.

Example

Mrs Jyothi gives Class 4 a place-value chart and bundles of sticks (ones, tens, hundreds). Children show 347 with 3 bundles of hundred, 4 of ten and 7 single sticks, then write it in expanded form. Later, newspaper figures such as “the district population is 42,15,000” become reading practice in lakhs.

Key takeaways

• The Hindu-Arabic system is base 10, with ten digits, place value and zero.
• Indian commas: 3 digits, then every 2; international: every 3.
• Place value = face value × place; the place value of 0 is 0.
• Compare numbers by the number of digits, then from the left.
• Dividend = divisor × quotient + remainder.
• Whole numbers start at 0 and are closed under addition and multiplication only.

Practice

1. The place value of 6 in 3,46,215 is:
   • (A) 6
   • (B) 600
   • (C) 6,000
   • (D) 60,000
2. 1 crore is equal to:
   • (A) 10 million
   • (B) 100 million
   • (C) 1 million
   • (D) 1,000 million
3. The difference between the place value and the face value of 5 in 2,53,601 is:
   • (A) 50,000
   • (B) 0
   • (C) 4,995
   • (D) 49,995
4. The largest 4-digit number formed using the digits 4, 0, 9, 2 once each is:
   • (A) 9,240
   • (B) 9,420
   • (C) 4,920
   • (D) 2,049
5. How many 3-digit numbers are there?
   • (A) 999
   • (B) 900
   • (C) 899
   • (D) 1,000
6. 7 × (5 + 3) = 7 × 5 + 7 × 3 shows the:
   • (A) commutative property
   • (B) associative property
   • (C) closure property
   • (D) distributive property
7. In a division, the divisor is 17, the quotient is 25 and the remainder is 9. The dividend is:
   • (A) 434
   • (B) 425
   • (C) 443
   • (D) 416
8. Seventy million in the Indian system is:
   • (A) 70 lakh
   • (B) 7 lakh
   • (C) 7 crore
   • (D) 70 crore