Revision Notes Number System - Class 6 ICSE PDF Download

What is a Number?

• Number - a value used to count, measure or label things.
  For example, 5 apples or 10 books.
• Numeral - a symbol or group of symbols that represents a number.
  For example, the numeral 31 represents the number thirty-one.
• Numeration - writing a number in words.
  For example, the number 31 is written as “thirty-one”.

Hindu–Arabic Numeration System

• This system uses digits 0 to 9 to form all numbers.
• Each digit’s value depends on its position in the number (place-value system).
• Examples of numbers in this system: 98, 76, 54, 321.

The place values in this system are shown here:

Example: The place values in the number 54,321 are:

• 5 is in the ten-thousands place: 5 × 10,000 = 50,000
• 4 is in the thousands place: 4 × 1,000 = 4,000
• 3 is in the hundreds place: 3 × 100 = 300
• 2 is in the tens place: 2 × 10 = 20
• 1 is in the ones place: 1 × 1 = 1

International Numeration System

• This system is used in many countries and also uses digits 0 to 9, but the place values are grouped in threes (thousands, millions, billions ...).
• For example, 987,654,321 is read as 987 million, 654 thousand, 321 in the International system.

The place values are:

Example: The place values in the number 987,654,321 are:

• 9 is in the hundred millions place: 9 × 100,000,000 = 900,000,000
• 8 is in the ten millions place: 8 × 10,000,000 = 80,000,000
• 7 is in the millions place: 7 × 1,000,000 = 7,000,000
• 6 is in the hundred thousands place: 6 × 100,000 = 600,000
• 5 is in the ten thousands place: 5 × 10,000 = 50,000
• 4 is in the thousands place: 4 × 1,000 = 4,000
• 3 is in the hundreds place: 3 × 100 = 300
• 2 is in the tens place: 2 × 10 = 20
• 1 is in the ones place: 1 × 1 = 1

Place Value and Face Value

• Place value - the value of a digit depending on its position in the number.
  For example, in 4,603, the place value of 6 is 6 × 100 = 600 (hundreds place).
• Face value - the value of the digit itself, irrespective of position.
  For example, in 4,603, the face value of 6 is 6.

Approximation (Rounding Off)

• Approximation means finding a number that is close to the original number but easier to use in calculations.
• We round off numbers based on the digit in the place immediately to the right of the place we are rounding to.

Rule for rounding:

• If the digit to the right is 5 or more, round up (add 1 to the digit in the place you are rounding to).
• If the digit to the right is less than 5, round down (keep the digit in the place you are rounding to the same).

Example: Rounding 12,345 to different places is done using the rule above.

Number System

• Real numbers (R): All rational and irrational numbers; includes negative and positive numbers, zero, and fractions - for example, -2, 5, 0, 2/3, -4/5.
• Integers (Z): Whole numbers and their negatives - for example, -3, -1, 0, 1, 2, 3.
• Whole numbers (W): Non-negative integers - 0, 1, 2, 3, ...
• Natural numbers (N): Counting numbers - 1, 2, 3, 4, ...

Tests of Divisibility

• Divisible by 2: A number is divisible by 2 if its last digit is even (0, 2, 4, 6, 8).
  Example: 52 is divisible by 2 because the last digit 2 is even.
• Divisible by 3: A number is divisible by 3 if the sum of its digits is divisible by 3.
  Example: 192 → 1 + 9 + 2 = 12; 12 is divisible by 3, so 192 is divisible by 3.
• Divisible by 4: A number is divisible by 4 if the number made by its last two digits is divisible by 4.
  Example: 172 → last two digits 72; 72 ÷ 4 = 18, so 172 is divisible by 4.
• Divisible by 5: A number is divisible by 5 if its last digit is 0 or 5.
  Example: 65 and 90 are divisible by 5.
• Divisible by 10: A number is divisible by 10 if its last digit is 0.
  Example: 1120 is divisible by 10.
• Divisible by 6: A number is divisible by 6 if it is divisible by both 2 and 3.
  Example: 186 is divisible by 2 (last digit 6 is even) and by 3 (1+8+6 = 15 is divisible by 3), so 186 is divisible by 6.
• Divisible by 8: A number is divisible by 8 if the number made by its last three digits is divisible by 8.
  Example: 1,024 → last three digits 024 = 24; 24 ÷ 8 = 3, so 1,024 is divisible by 8.
• Divisible by 9: A number is divisible by 9 if the sum of its digits is divisible by 9.
  Example: 1,458 → 1 + 4 + 5 + 8 = 18; 18 is divisible by 9, so 1,458 is divisible by 9.
• Even natural numbers (E): Natural numbers divisible by 2 - E = {2, 4, 6, 8, 10, ...}.
• Odd natural numbers (O): Natural numbers not divisible by 2 - O = {1, 3, 5, 7, 9, ...}.
• Prime natural numbers (P): Natural numbers greater than 1 that have exactly two distinct positive divisors: 1 and the number itself.
  Example: P = {2, 3, 5, 7, 11, ...}.

Estimation

• Estimation is a method for finding an approximate value quickly when an exact answer is not required.
• It is useful to check the reasonableness of answers in addition, subtraction, multiplication and division.
• Example: If you have 47 apples you might estimate it as 50 to make calculations easier.

Rounding Off Numbers

Rounding off simplifies numbers to make them easier to work with. The same rules described earlier apply depending on which place value you round to.

General rounding rules

• If the digit to the right of the place you are rounding to is 5 or more, round up.
• If the digit to the right is less than 5, round down.

Examples:

• Round 6,789 to the nearest ten: 6,789 → 6,790 (because the units digit 9 ≥ 5).
• Round 6,789 to the nearest hundred: 6,789 → 6,800 (because the tens digit 8 ≥ 5).
• Round 6,789 to the nearest thousand: 6,789 → 7,000 (because the hundreds digit 7 ≥ 5).

Estimation in Multiplication

To estimate a product, round each factor to a convenient place value and multiply the rounded numbers.

Example: Estimate 47 × 23

• Round 47 to 50 (nearest ten).
• Round 23 to 20 (nearest ten).
• Multiply the rounded values: 50 × 20 = 1,000.
• So, 47 × 23 ≈ 1,000.

Roman Numerals

• Roman numerals are a numeral system used in ancient Rome, represented by letters: I, V, X, L, C, D, M.
• They are used occasionally in clocks, book chapters, outlines, or to show order.

Basic Roman numerals

Rules for Roman numerals

• A symbol is not repeated more than three times in succession (e.g., III is valid, but IIII is not).
• Symbols V, L, D are never repeated.
• If a smaller-value symbol is to the right of a larger-value symbol, add the values (e.g., VI = 5 + 1 = 6; XII = 10 + 1 + 1 = 12).
• If a smaller-value symbol is to the left of a larger-value symbol, subtract it (e.g., IV = 5 − 1 = 4; XL = 50 − 10 = 40).
• Symbols V, L, D are never subtracted.
• I can only be subtracted from V or X.
• X can only be subtracted from L or C.

Example conversions:

• VI = 5 + 1 = 6
• XII = 10 + 2 = 12
• LXV = 50 + 10 + 5 = 65
• IV = 5 − 1 = 4
• XC = 100 − 10 = 90

Metric System Conversions

The metric system measures length, mass and capacity. Common prefixes show how many times larger or smaller a unit is compared to a base unit.

Common prefixes

Metric conversions

• 1 kilometre = 1,000 metres
• 1 metre = 100 centimetres
• 1 metre = 1,000 millimetres
• 1 centimetre = 10 millimetres

Example: Convert 2.5 kilometres to metres.

Ans: 2.5 km × 1,000 = 2,500 metres

Additional notes and quick recap

• Use place value to read and write large numbers clearly (International grouping uses thousands, millions; many Indian contexts use thousands, lakhs, crores).
• Practice rounding and estimation to check the reasonableness of results quickly.
• Memorise divisibility tests for fast mental arithmetic and factor checking.
• Recognise the most common Roman numerals and their subtraction/addition rules for conversion tasks.