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Integers - Representation of Integers on a Number Line

Grade 6CBSE

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

🔑Concepts

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Definition of Integers: Integers are a collection of numbers comprising natural numbers {1,2,3,...}\{1, 2, 3, ...\}, their negatives {−1,−2,−3,...}\{-1, -2, -3, ...\}, and the number 00. This set is denoted by the symbol Z\mathbb{Z}.

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Structure of the Number Line: Visualize a straight horizontal line with a central point marked 00. Points at equal intervals to the right of 00 represent positive integers +1,+2,+3,...+1, +2, +3, ..., while points at equal intervals to the left represent negative integers −1,−2,−3,...-1, -2, -3, ....

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The Role of Zero: On the number line, 00 is known as the origin. It is neither positive nor negative. It separates the positive territory (right side) from the negative territory (left side).

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Ordering and Magnitude: For any two points on the number line, the number appearing to the right is always greater than the number appearing to the left. For example, −1>−5-1 > -5 because −1-1 is located to the right of −5-5.

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Movement and Operations: To represent addition of a positive integer, move towards the right. To represent subtraction (or addition of a negative integer), move towards the left. For example, to find 2−32 - 3, start at 22 and move 33 units to the left to land on −1-1.

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Successor and Predecessor: Every integer has a successor (the integer immediately to its right, calculated as n+1n + 1) and a predecessor (the integer immediately to its left, calculated as n−1n - 1). On the number line, the successor of −1-1 is 00.

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Opposites (Additive Inverses): Pairs of integers like +4+4 and −4-4 are the same distance from zero but in opposite directions. These are called opposites, and their sum is always 00 (4+(−4)=04 + (-4) = 0).

📐Formulae

Successor = n + 1

Predecessor = n - 1

a>b if a is to the right of b on the number linea > b \text{ if } a \text{ is to the right of } b \text{ on the number line}

a + (-b) = a - b

a - (-b) = a + b

💡Examples

Problem 1:

Represent the integer that is 44 units less than 11 on a number line and find its value.

Solution:

  1. Draw a horizontal number line and mark the origin 00.
  2. Locate the starting point at 11.
  3. Since the problem asks for '44 units less than', move 44 units to the left from 11.
  4. First move: 1→01 \rightarrow 0.
  5. Second move: 0→−10 \rightarrow -1.
  6. Third move: −1→−2-1 \rightarrow -2.
  7. Fourth move: −2→−3-2 \rightarrow -3.
  8. The final position is −3-3.

Explanation:

To find a value 'less than' a given number, we must move to the left on the number line. Starting at 11 and shifting 44 steps left results in the negative integer −3-3.

Problem 2:

Find the sum of −2-2 and 55 using the number line.

Solution:

  1. Start at the point −2-2 on the number line.
  2. To add 55 (a positive integer), move 55 units to the right.
  3. Step 1: −2→−1-2 \rightarrow -1.
  4. Step 2: −1→0-1 \rightarrow 0.
  5. Step 3: 0→10 \rightarrow 1.
  6. Step 4: 1→21 \rightarrow 2.
  7. Step 5: 2→32 \rightarrow 3.
  8. The result is 33.

Explanation:

Addition of a positive number corresponds to moving rightward along the number line. Starting from a negative position and moving right past zero leads to a positive result.