{"id":659,"date":"2023-03-09T16:42:26","date_gmt":"2023-03-09T16:42:26","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/?post_type=chapter&p=659"},"modified":"2025-08-23T01:03:59","modified_gmt":"2025-08-23T01:03:59","slug":"integers-learn-it-2","status":"web-only","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/chapter\/integers-learn-it-2\/","title":{"raw":"Integers: Learn It 2","rendered":"Integers: Learn It 2"},"content":{"raw":"

Integers on a Number Line<\/h2>\r\n

Both positive and negative numbers can be represented on a number line. Recall that a number line created with only whole numbers starts at [latex]0[\/latex] and shows the counting numbers increasing to the right as shown in the number line below. The counting numbers [latex](1, 2, 3, \\ldots )[\/latex] on the number line are all positive. We could write a plus sign, [latex]+[\/latex], before a positive number such as [latex]+2[\/latex] or [latex]+3[\/latex], but it is customary to omit the plus sign and write only the number. If there is no sign, the number is assumed to be positive.<\/p>\r\n

\r\n[caption id=\"\" align=\"aligncenter\" width=\"332\"]\"This Figure 1. A positive number line[\/caption]\r\n<\/center>\r\n

 <\/p>\r\n

Now we need to extend the number line to include negative numbers. We mark several units to the left of zero, keeping the intervals the same width as those on the positive side. We label the marks with negative numbers, starting with [latex]-1[\/latex] at the first mark to the left of [latex]0,-2[\/latex] at the next mark, and so on. Refer to the number line below for reference.<\/p>\r\n

On a number line, positive numbers are to the right of zero. Negative numbers are to the left of zero. What about zero? Zero is neither positive nor negative.<\/p>\r\n

\r\n[caption id=\"\" align=\"aligncenter\" width=\"440\"]\"A Figure 2. A number line with positive and negative numbers[\/caption]\r\n<\/center>\r\n

 <\/p>\r\n

The arrows at either end of the line indicate that the number line extends forever in each direction. There is no greatest positive number and there is no smallest negative number.<\/p>\r\n

Plot the numbers on a number line:\r\n\r\n
    \r\n\t
  1. [latex]3[\/latex]<\/li>\r\n\t
  2. [latex]-3[\/latex]<\/li>\r\n\t
  3. [latex]-2[\/latex]<\/li>\r\n<\/ol>\r\n

    [reveal-answer q=\"692742\"]Show Solution[\/reveal-answer]
    \r\n[hidden-answer a=\"692742\"]
    \r\nDraw a number line. Mark [latex]0[\/latex] in the center and label several units to the left and right.<\/p>\r\n

    1. To plot [latex]3[\/latex], start at [latex]0[\/latex] and count three units to the right. Place a point as shown in the number line below.<\/p>\r\n

    \"A<\/center>\r\n

     <\/p>\r\n

    2. To plot [latex]-3[\/latex], start at [latex]0[\/latex] and count three units to the left. Place a point as shown in\u00a0the number line below.<\/p>\r\n

    \"A<\/center>\r\n

     <\/p>\r\n

    3. To plot [latex]-2[\/latex], start at [latex]0[\/latex] and count two units to the left. Place a point as shown in the number line below.<\/p>\r\n

    \"A<\/center>\r\n

     <\/p>\r\n\r\n[\/hidden-answer]<\/section>\r\n

    Order Positive and Negative Numbers<\/h2>\r\n

    We can use the number line to compare and order positive and negative numbers. Going from left to right, numbers increase in value. Going from right to left, numbers decrease in value. See the number line below.<\/p>\r\n

    \r\n[caption id=\"\" align=\"aligncenter\" width=\"440\"]\"A Figure 3. Orders on a number line[\/caption]\r\n<\/center>\r\n

     <\/p>\r\n

    Just as we did with positive numbers, we can use inequality symbols to show the ordering of positive and negative numbers.<\/p>\r\n

    Remember that we use the notation [latex]a < b[\/latex] (read [latex]a[\/latex] is less than<\/em> [latex]b[\/latex] ) when [latex]a[\/latex] is to the left of [latex]b[\/latex] on the number line. We write [latex]a > b[\/latex] (read [latex]a[\/latex] is greater than<\/em> [latex]b[\/latex] ) when [latex]a[\/latex] is to the right of [latex]b[\/latex] on the number line.<\/section>\r\n

    The number [latex]3[\/latex] is to the left of [latex]5[\/latex] on the number line. So [latex]3[\/latex] is less than [latex]5[\/latex], and [latex]5[\/latex] is greater than [latex]3[\/latex].<\/p>\r\n

    \r\n[caption id=\"\" align=\"aligncenter\" width=\"294\"]\"A Figure 4. A number line showing the order of 3 and 5[\/caption]\r\n<\/center>\r\n

     <\/p>\r\n

    The numbers lines to follow show a few more examples.<\/p>\r\n

    \r\n[caption id=\"\" align=\"aligncenter\" width=\"440\"]\"A Figure 5. A number line with 1 and 4 labeled[\/caption]\r\n<\/center>\r\n

     <\/p>\r\n

    [latex]4[\/latex] is to the right of [latex]1[\/latex] on the number line, so [latex]4>1[\/latex].
    \r\n[latex]1[\/latex] is to the left of [latex]4[\/latex] on the number line, so [latex]1<4[\/latex].<\/p>\r\n

     <\/p>\r\n

    \r\n[caption id=\"\" align=\"aligncenter\" width=\"440\"]\"A Figure 6. A number line with -2 and 1 labeled[\/caption]\r\n<\/center>\r\n

     <\/p>\r\n

    [latex]-2[\/latex] is to the left of [latex]1[\/latex] on the number line, so [latex]-2<1[\/latex].
    \r\n[latex]1[\/latex] is to the right of [latex]-2[\/latex] on the number line, so [latex]1>-2[\/latex].<\/p>\r\n

     <\/p>\r\n

    \r\n[caption id=\"\" align=\"aligncenter\" width=\"440\"]\"A Figure 7. A number line with -3 and -1 labeled[\/caption]\r\n<\/center>\r\n

     <\/p>\r\n

    [latex]-1[\/latex] is to the right of [latex]-3[\/latex] on the number line, so [latex]-1>-3[\/latex].
    \r\n[latex]-3[\/latex] is to the left of [latex]-1[\/latex] on the number line, so [latex]-3<-1[\/latex].<\/p>\r\n

    [ohm2_question hide_question_numbers=1]3172[\/ohm2_question]<\/section>","rendered":"

    Integers on a Number Line<\/h2>\n

    Both positive and negative numbers can be represented on a number line. Recall that a number line created with only whole numbers starts at [latex]0[\/latex] and shows the counting numbers increasing to the right as shown in the number line below. The counting numbers [latex](1, 2, 3, \\ldots )[\/latex] on the number line are all positive. We could write a plus sign, [latex]+[\/latex], before a positive number such as [latex]+2[\/latex] or [latex]+3[\/latex], but it is customary to omit the plus sign and write only the number. If there is no sign, the number is assumed to be positive.<\/p>\n

    \n
    \"This
    Figure 1. A positive number line<\/figcaption><\/figure>\n<\/div>\n

     <\/p>\n

    Now we need to extend the number line to include negative numbers. We mark several units to the left of zero, keeping the intervals the same width as those on the positive side. We label the marks with negative numbers, starting with [latex]-1[\/latex] at the first mark to the left of [latex]0,-2[\/latex] at the next mark, and so on. Refer to the number line below for reference.<\/p>\n

    On a number line, positive numbers are to the right of zero. Negative numbers are to the left of zero. What about zero? Zero is neither positive nor negative.<\/p>\n

    \n
    \"A
    Figure 2. A number line with positive and negative numbers<\/figcaption><\/figure>\n<\/div>\n

     <\/p>\n

    The arrows at either end of the line indicate that the number line extends forever in each direction. There is no greatest positive number and there is no smallest negative number.<\/p>\n

    Plot the numbers on a number line:<\/p>\n
      \n
    1. [latex]3[\/latex]<\/li>\n
    2. [latex]-3[\/latex]<\/li>\n
    3. [latex]-2[\/latex]<\/li>\n<\/ol>\n