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What is the division of integers on a number line?

Published in Integer Division 3 mins read

The division of integers on a number line visually represents how many times one integer (the divisor) fits into another (the dividend). According to the reference material, it involves graphing the multiples of the divisor starting from 0 up to the dividend. By counting the number of "jumps" or intervals of the divisor's size it takes to reach the dividend, we determine the quotient, which is the answer to the division problem.

Understanding Division of Integers on a Number Line

Here’s a more detailed explanation:

  • Representation: The number line provides a visual aid to understand the concept of division, especially when dealing with integers (positive and negative whole numbers and zero).

  • Process:

    1. Start at 0 on the number line.
    2. Mark multiples of the divisor.
    3. Count how many "jumps" of the divisor's length are needed to reach the dividend. This count represents the quotient.

Examples

Let's illustrate this with a few examples:

  • Example 1: 6 ÷ 2

    • We want to find out how many times 2 fits into 6.
    • Start at 0.
    • Make jumps of size 2: 0 -> 2 -> 4 -> 6.
    • It took 3 jumps to reach 6. Therefore, 6 ÷ 2 = 3.

  • Example 2: (-6) ÷ 2

    • We want to find out what number, when multiplied by 2, gives -6.
    • Start at 0.
    • Make jumps of size 2 in the negative direction: 0 -> -2 -> -4 -> -6.
    • It took 3 jumps in the negative direction to reach -6. Therefore, (-6) ÷ 2 = -3.

  • Example 3: 6 ÷ (-2)

    • We want to find out what number, when multiplied by -2, gives 6.
    • Start at 0.
    • To reach 6, we must make jumps of size 2 in the negative direction from 0: 0, meaning each jump must be -2.
    • In effect we are asking how many -2s fit into +6. To show this on a number line, we are looking for the point 6 and counting the number of -2s that brings us there: from zero, -2 moves us to -2 on the line, another moves us to -4 on the line, and the third takes us to -6 on the line.
    • We need to take 3 of those movements/jumps to arrive to 6. But the jumps are on the negative side of 0. Since the question is how many times -2 fits into +6, we arrive at our answer: 6 ÷ (-2) = -3

Practical Insights

  • When dividing integers with the same sign (both positive or both negative), the result is positive.
  • When dividing integers with different signs (one positive and one negative), the result is negative.
  • Division by zero is undefined.

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