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Integers - Ordering of Integers

Grade 6CBSE

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

🔑Concepts

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Integers Definition: The set of integers includes all whole numbers and their negative counterparts, represented as Z={…,−3,−2,−1,0,1,2,3,… }Z = \{\dots, -3, -2, -1, 0, 1, 2, 3, \dots\}. This set can be visualized as an infinite sequence of points extending in both directions from zero.

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The Number Line Layout: Visually, integers are represented on a horizontal line with zero (00) at the center. Positive integers (1,2,3,…1, 2, 3, \dots) are marked at equal distances to the right of zero, while negative integers (−1,−2,−3,…-1, -2, -3, \dots) are marked at equal distances to the left of zero.

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The Rule of Direction: On the number line, the value of integers increases as we move from left to right and decreases as we move from right to left. Therefore, any integer that lies to the right of another integer is always greater.

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Zero as a Benchmark: Zero is the neutral element. It is greater than every negative integer (because it is to their right) and smaller than every positive integer (because it is to their left). For example, 0>−50 > -5 and 0<50 < 5.

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Comparing Positive and Negative Integers: Every positive integer is greater than every negative integer. Regardless of the digits involved, a positive sign always indicates a value further to the right on the number line than a negative sign. Example: 1>−1001 > -100.

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Ordering Negative Integers: For negative integers, the integer with the smaller numerical value (ignoring the sign) is actually the greater integer. For example, −2>−10-2 > -10 because −2-2 is closer to zero and lies to the right of −10-10 on the number line.

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Successor and Predecessor: The successor of an integer is the number one unit to its right, calculated as n+1n + 1. The predecessor is the number one unit to its left, calculated as n−1n - 1. For example, the successor of −4-4 is −3-3, and its predecessor is −5-5.

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Ascending and Descending Order: Arranging integers in ascending order means listing them from the smallest (leftmost) to the largest (rightmost). Descending order means listing them from the largest (rightmost) to the smallest (leftmost).

📐Formulae

If a>b, then −a<−b\text{If } a > b, \text{ then } -a < -b

Successor of n=n+1\text{Successor of } n = n + 1

Predecessor of n=n−1\text{Predecessor of } n = n - 1

Positive Integer>0>Negative Integer\text{Positive Integer} > 0 > \text{Negative Integer}

⋯<−3<−2<−1<0<1<2<3<…\dots < -3 < -2 < -1 < 0 < 1 < 2 < 3 < \dots

💡Examples

Problem 1:

Arrange the following integers in ascending order: −7,5,0,−2,3,−10-7, 5, 0, -2, 3, -10

Solution:

  1. Identify the negative integers: −7,−2,−10-7, -2, -10. Comparing them, −10-10 is the smallest because it is furthest to the left, followed by −7-7, then −2-2.
  2. Identify zero: 00 is greater than all negative integers but smaller than all positive integers.
  3. Identify the positive integers: 3,53, 5. Comparing them, 3<53 < 5.
  4. Combine them in order from smallest to largest: −10,−7,−2,0,3,5-10, -7, -2, 0, 3, 5.

Explanation:

Ascending order requires moving from the leftmost point on the number line to the rightmost point. We start with the most negative value and end with the largest positive value.

Problem 2:

Find the successor and predecessor of the integer −15-15.

Solution:

  1. To find the successor: Add 11 to the integer. −15+1=−14-15 + 1 = -14
  2. To find the predecessor: Subtract 11 from the integer. −15−1=−16-15 - 1 = -16 So, the successor is −14-14 and the predecessor is −16-16.

Explanation:

On the number line, −14-14 is one step to the right of −15-15 (making it the successor), and −16-16 is one step to the left of −15-15 (making it the predecessor).