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Number System - Comparing and Ordering Large Numbers

Grade 5ICSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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Place Value Chart: Comparing large numbers requires understanding the position of each digit. In the Indian system, digits are grouped into periods: Ones, Thousands, Lakhs, and Crores. Imagine a table where each column to the left is 10 times larger than the one to its right. For example, in the number 5,43,2105,43,210, the digit 55 is in the Lakhs place, representing 5,00,0005,00,000.

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Rule of Digits: When comparing two numbers, first count the number of digits in each. The number with more digits is always the greater number. For instance, a 7-digit number like 10,00,00010,00,000 (Ten Lakhs) is visually longer and mathematically larger than a 6-digit number like 9,99,9999,99,999 (Nine Lakhs Ninety-nine Thousand Nine Hundred Ninety-nine).

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Left-to-Right Comparison: If two numbers have the same number of digits, compare them digit by digit starting from the extreme left (the highest place value). Visualize this as a 'scanner' moving from left to right across the numbers until you encounter two digits that are different. The number with the larger digit at that specific place value is the greater number.

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Ascending Order: This refers to arranging a set of numbers from the smallest value to the largest value. You can visualize this as climbing a staircase, where each step represents a number higher than the previous one (Smallest→LargestSmallest \rightarrow Largest).

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Descending Order: This is the arrangement of numbers from the greatest value to the smallest value. Imagine walking down a ladder from the highest rung to the lowest (Largest→SmallestLargest \rightarrow Smallest).

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Comparison Symbols: We use specific symbols to show the relationship between numbers. The 'Greater Than' symbol (>>) has its open 'mouth' pointing toward the larger number. The 'Less Than' symbol (<<) points its narrow vertex toward the smaller number. If the numbers are identical, we use the 'Equal To' (==) sign.

📐Formulae

If n(A)>n(B)n(A) > n(B), then A>BA > B (where nn is the number of digits)

Ascending Order: N1<N2<N3<⋯<NmaxN_{1} < N_{2} < N_{3} < \dots < N_{max}

Descending Order: N1>N2>N3>⋯>NminN_{1} > N_{2} > N_{3} > \dots > N_{min}

Place Value of a digit = Digit×Value of the place\text{Digit} \times \text{Value of the place}

💡Examples

Problem 1:

Compare the numbers 84,35,61984,35,619 and 84,39,21084,39,210 and use the correct symbol (>>, << or ==).

Solution:

Step 1: Count the digits. Both numbers have 7 digits. Step 2: Compare from the leftmost place (Ten Lakhs). Both have 88. Step 3: Compare the Lakhs place. Both have 44. Step 4: Compare the Ten Thousands place. Both have 33. Step 5: Compare the Thousands place. The first number has 55 and the second has 99. Step 6: Since 5<95 < 9, then 84,35,619<84,39,21084,35,619 < 84,39,210.

Explanation:

Since the numbers have the same number of digits, we compare each place value from left to right. The first point of difference occurs at the thousands place, determining the final result.

Problem 2:

Arrange the following numbers in ascending order: 5,62,1095,62,109; 56,21,09056,21,090; 45,99945,999; 5,62,8005,62,800.

Solution:

Step 1: Count the digits in each number.

  • 45,99945,999 (5 digits)
  • 5,62,1095,62,109 (6 digits)
  • 5,62,8005,62,800 (6 digits)
  • 56,21,09056,21,090 (7 digits) Step 2: The 5-digit number is the smallest: 45,99945,999. Step 3: Compare the 6-digit numbers 5,62,1095,62,109 and 5,62,8005,62,800. Both have 5,6,25, 6, 2 in the first three places. In the hundreds place, 1<81 < 8, so 5,62,109<5,62,8005,62,109 < 5,62,800. Step 4: The 7-digit number is the largest: 56,21,09056,21,090. Step 5: Final order: 45,999<5,62,109<5,62,800<56,21,09045,999 < 5,62,109 < 5,62,800 < 56,21,090.

Explanation:

To arrange in ascending order, we first group numbers by digit count and then compare numbers with the same digit count using place value logic.