Directed Numbers

Lesson

We have the whole numbers $0,1,2,3,4,5,6,...$0,1,2,3,4,5,6,... which we can count up by, but these are only the positive ones.

An integer is a counting number that can be positive or negative:

$...,-6,-5,-4,-3,-2,-1,0,1,2,3,4,5,6,...$...,−6,−5,−4,−3,−2,−1,0,1,2,3,4,5,6,...

Any expression that can be simplified to an integer counts as an integer. For example, $\frac{14}{2}$142 doesn't look like an integer, but it equals $7$7, so it is an integer.

Zero ($0$0) is also an integer but it is not positive or negative.

## Numbers that ARE integers |
## Numbers that ARE NOT integers |
---|---|

$5$5 | $4.9$4.9 |

$-17$−17 | $-13.2$−13.2 |

$12.0=12$12.0=12 |
$\sqrt{13}=3.605$√13=3.605$\dots$… |

$4^2=16$42=16 | $\frac{17}{6}=2.833$176=2.833$\dots$… |

$\sqrt{100}=10$√100=10 | $15.5^2=240.25$15.52=240.25 |

$\frac{56}{8}=7$568=7 |
$\frac{1}{3}=0.\overline{3}$13=0.3 |

Which of the following numbers are integers?

$3,-2\frac{1}{2},\frac{12}{2},-100$3,−212,122,−100

**Think**:

$3$3 is a whole number, so it is an integer.

$-2\frac{1}{2}$−212 has a fraction in it, so it is not an integer.

$\frac{12}{2}$122 equals $6$6, which is a whole number, so it is an integer.

$-100$−100 is the negative of a whole number, so it is an integer.

**Solution**: The integers are $3,\frac{12}{2}$3,122 and $-100$−100.

We've seen before that the further an integer is to the right on a number line, the larger the integer is.

Let's look at the number line below.

We can see that $8$8 (or $+$+$8$8) is further to the right than $-10$−10.

So $8$8 is larger than $-10$−10, or we can say $-10$−10 is smaller than $8$8.

We can imagine this through two scenarios: having $\$8$$8 in our bank account or being in debt by $\$10$$10. We have more money if we have $\$8$$8 in the bank than if we owe $\$10$$10.

Identify the largest number: $2,13,-16$2,13,−16

Identify the smallest number: $9,-20,14$9,−20,14

We've discussed different ways of ordering integers in earlier chapters. But just to refresh, it is common to order numbers in:

*ascending order*(smallest to largest), or*descending order*(largest to smallest).

When we need to arrange integers in order, we can think about their relative positions on the number line.

To arrange numbers in ascending order, we would move from left to right along the number line.

To arrange numbers in descending order, we would move from right to left along the number line.

Arrange $0,7,-1$0,7,−1, and $6$6 in ascending order.

**Think:**

Here is the number line:** **

$-1$−1 is the smallest number as it is furthest to the left on the number line. $7$7 is furthest to the right, so it is the biggest.

**Do:** The numbers in ascending order are $-1,0,6,7$−1,0,6,7.

Arrange $-15,-2,-7$−15,−2,−7 and $-5$−5 in descending order.

**Think: **$-2$−2 is furthest to the right, so it is the biggest number. $-15$−15 is the smallest number as it is furthest to the left.

**Do:** $-2,-5,-7,-15$−2,−5,−7,−15

We can also find and complete number patterns by considering the relationship between the integers.

Complete the next three numbers in the pattern: $1,4,7,10$1,4,7,10, __ __, __ __, __ __.

**Think:** Notice that we are adding $3$3 to go from one number to the next.

$1+3=4$1+3=4 $\rightarrow$→ $4+3=7$4+3=7 $\rightarrow$→ $7+3=10$7+3=10

**Do:** The next three numbers in the pattern are $13,16,19$13,16,19.

Let's look at an example with negative numbers.

Complete the next three numbers in the pattern: $0,-2,-4$0,−2,−4, __ __, __ __, __ __.

**Think: **Notice that the numbers are decreasing by $2$2 each time.

$0-2=-2$0−2=−2 $\rightarrow$→ $-2-2=-4$−2−2=−4

**Do:** The next three numbers will be $-6,-8,-10$−6,−8,−10.

Complete the next three numbers in the pattern: $7,5,3$7,5,3, __ __, __ __, __ __.