Fix the mistake: Positive and Negative Numbers

Positive and Negative Numbers ① Meaning and Absolute Value

The numbers learned in elementary school（0, 1, 2, 3, …）werenumbers greater than or equal to 0. In middle school, we also work with “numbers smaller than 0.” These are callednegative numbersand are written with the minus sign “−”. For example, “−5℃（5 degrees below zero）,” “the third basement floor（−3rd floor）,” and “−500 yen（a debt of 500 yen）” naturally appear in daily life.

Positive and Negative Numbers

A number greater than 0 is apositive number, and a number smaller than 0 is anegative number. A positive number has “＋” and a negative number has “−”（the positive sign may be omitted）.0 is neither positive nor negative.

A line on which numbers are arranged in order of size is called anumber line. With 0 as the reference point, positive numbers are on the right and negative numbers are on the left.

　←─┼──┼──┼──┼──┼──┼──┼──┼──┼─→
　 −4　−3　−2　−1　 0　 +1　+2　+3　+4
Example ①
Arrange the following numbers in increasing order based on their positions on the number line.
　+2、−5、+1、−1、0
Answer：Since numbers farther to the left on the number line are smaller, −5 ＜ −1 ＜ 0 ＜ +1 ＜ +2。
Note that among negative numbers, “the greater the absolute value of the negative number, the smaller the number.”

Absolute Value

The number that representshow far a number is from 0on the number line is called itsabsolute value. Since absolute value represents distance, it is always greater than or equal to 0. The symbol |　| is used.

  • |+5| = 5
  • |−5| = 5
  • |0| = 0

Thus, the absolute value of a positive number is the number itself, and the absolute value of a negative number is the number with the “−” sign removed. Two numbers with the same absolute value and opposite signs are callednumbers with opposite signs（for example, +3 and −3）.

Example ②
Answer the following questions.
(1) Find |−8|.
(2) Give all numbers whose absolute value is 6.
Answer：
(1) |−8| is “the distance from −8 to 0,” so it is 8。
(2) The numbers whose distance from 0 is 6 are 6（+6）to the right of 0 and 6（−6）to the left. Therefore, +6 and −6。
Key Points
  • Greater than 0 → positive number（＋）／smaller than 0 → negative number（−）／0 is neither positive nor negative
  • On a number line, numbers farther right are greater, and numbers farther left are smaller
  • Absolute value = distance from 0. It is always greater than or equal to 0（a positive number or 0）
  • Two numbers with equal absolute values and opposite signs（+a and −a）are numbers with opposite signs

Practice Problems

  1. Arrange the following numbers in increasing order.
    　−3、+2、−7、0、+4、−1
  2. Find the following absolute values.
    　(1) |−12|　(2) |+3.5|　(3) |−1/4|
  3. How many integers have an absolute value of 9 or less?

Answers and Explanations

  1. Answer：−7 ＜ −3 ＜ −1 ＜ 0 ＜ +2 ＜ +4
    Explanation：Numbers farther left on the number line are smaller. Note that among negative numbers, the one with the greater absolute value is smaller（−7 is smaller than −3）.
  2. Answer：(1) 12　(2) 3.5　(3) 1/4
    Explanation：Absolute value is distance from 0, so it is the value with the sign removed. Even for a negative number, the value itself remains unchanged.
  3. Answer：19 numbers
    Explanation：Integers with absolute value 9 or less are −9, −8, …, −1, 0, +1, …, +8, +9. There are 9 negative numbers, 1 zero, and 9 positive numbers, so 9+1+9 = 19 numbers.

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