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Knowing Our Numbers - Comparing Numbers

Grade 6CBSE

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

🔑Concepts

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Numbers with different digits: When comparing two numbers, the number with the greater number of digits is always the larger number. For example, 1,2341,234 (4 digits) is greater than 998998 (3 digits). Visualize this as a longer line of digits being 'heavier' or larger than a shorter line of digits.

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Numbers with the same number of digits: If two numbers have the same number of digits, start comparing from the leftmost digit (highest place value). If the digits at that place are the same, move to the next digit to the right and continue until you find a difference. Imagine a vertical alignment of the numbers where you scan from left to right to find the first mismatch.

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Ascending Order: This is the arrangement of numbers from the smallest to the largest value. Think of it as climbing up a staircase where each step represents a higher number value, such as 12<45<89<12012 < 45 < 89 < 120.

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Descending Order: This is the arrangement of numbers from the largest to the smallest value. Visualize this as walking down a staircase where each step represents a lower number value, such as 500>340>210>15500 > 340 > 210 > 15.

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Place Value Chart Comparison: Using a Place Value Chart helps in visual comparison. Imagine a grid with columns for Thousands (10001000s), Hundreds (100100s), Tens (1010s), and Ones (11s). To compare 4,5674,567 and 4,5214,521, you place them in the grid; the thousands and hundreds match, but in the tens column, 6>26 > 2, making 4,5674,567 the larger number.

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Forming Numbers from Digits: To form the greatest number from a set of digits, arrange them in descending order. To form the smallest number, arrange them in ascending order. Note: If one of the digits is 00, it cannot be placed at the highest place value to maintain the digit count; it must be placed at the second position from the left.

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Use of Comparison Symbols: Symbols are used to show the relationship between numbers. The symbol >> means 'greater than', << means 'less than', and == means 'equal to'. The 'mouth' of the symbol always opens towards the larger number.

📐Formulae

Number of digits in A>Number of digits in B  ⟹  A>B\text{Number of digits in } A > \text{Number of digits in } B \implies A > B

Ascending Order: n1<n2<n3<⋯<nk\text{Ascending Order: } n_{1} < n_{2} < n_{3} < \dots < n_{k}

Descending Order: n1>n2>n3>⋯>nk\text{Descending Order: } n_{1} > n_{2} > n_{3} > \dots > n_{k}

Greatest Number using digits {d1,d2,…,dn}=Arrange digits such that dfirst≥dsecond≥⋯≥dlast\text{Greatest Number using digits } \{d_1, d_2, \dots, d_n\} = \text{Arrange digits such that } d_{first} \ge d_{second} \ge \dots \ge d_{last}

💡Examples

Problem 1:

Compare the numbers 9,2109,210 and 9,2859,285 and identify which is greater.

Solution:

Step 1: Count the digits. Both numbers have 4 digits. Step 2: Compare the leftmost digit (Thousands place). Both have 99. Step 3: Compare the next digit to the right (Hundreds place). Both have 22. Step 4: Compare the next digit (Tens place). The first number has 11 and the second number has 88. Since 8>18 > 1, then 9,285>9,2109,285 > 9,210.

Explanation:

When the number of digits is the same, we compare place values starting from the highest (leftmost) until a difference is found.

Problem 2:

Form the smallest and greatest 4-digit numbers using the digits 5,0,3,95, 0, 3, 9 without repeating any digit.

Solution:

  1. Greatest Number: Arrange the digits in descending order: 9,5,3,09, 5, 3, 0. The number is 9,5309,530. 2. Smallest Number: Arrange the digits in ascending order: 0,3,5,90, 3, 5, 9. Since a 4-digit number cannot start with 00, we swap 00 with the next smallest digit, which is 33. The number is 3,0593,059.

Explanation:

The greatest number is formed by placing the largest digits in the highest place values. For the smallest number, we use the smallest digits in the highest places, ensuring the leftmost digit is non-zero.