Directed Numbers

Lesson

Let's write down some of the things we have discovered about addition and multiplying on the number line.

The absolute value of a number is a measure of its size or magnitude. On the number line the absolute value of a number is its distance from the the number $0$0. Distances have no sign attached to them, and neither do absolute values.

We use small vertical lines to surround a number to indicate its absolute value, so that the absolute value of minus three is written as $|-3|$|−3|. Its value is simply $3$3. The value of $|3|$|3| is also $3$3, because $3$3 and $-3$−3 are an equal distance away from $0$0 on the number line.

The meaning of the two absolute values $|-3|$|−3| and $|2|$|2| are illustrated here on this number line:

We will use the numbers $6$6, $-6$−6, $2$2, and $-3$−3 to illustrate what happens when we add, subtract or multiply using signed numbers in mathematical sentences.

The same rules apply for any number on the number line.

The sentence $3+(-3)$3+(−3) changes to $3-3$3−3 and so the answer is $0$0

The sentence $-3+(-3)$−3+(−3) changes to $-3-3$−3−3 and so the answer is $-6$−6

The sentence $6-(-6)$6−(−6) changes to $6+6$6+6 and so the answer is $12$12

The sentence $-6-(-6)$−6−(−6) changes to $-6+6$−6+6 and so the answer is $0$0

The sentence $6\times(-3)$6×(−3) becomes $-18$−18, because the product of a $+$+ and a $-$− is always a $-$−.

The sentence $-6\times(-3)$−6×(−3) becomes $18$18, because the product of a $-$− and a $-$− is always a $+$+.

The sentence $-6\times3$−6×3 becomes $-18$−18, because the product of a $+$+ and a $-$− is always a $-$−.

The sentence $6\times3$6×3 becomes $18$18, because, well, thats always been the answer!

Division works in the same way as multiplication. The division of a positive number by a negative number, or a negative number by a positive number is always negative.

so $6\div(-2)$6÷(−2) is $-3$−3 and $-6\div2$−6÷2 is $-3$−3.

Division of any negative number by a negative number is always positive, so $-6\div(-2)$−6÷(−2) becomes $3$3.

And, as usual, division of a positive number by a positive number is positive!

Of course, dividing two integers can lead to numbers that are not integers. So $-3\div-6$−3÷−6 becomes $\frac{1}{2}$12.

Mathematicians use an important convention when simplifying number sentences.

A convention is an agreed way of interpreting things so that we don't confuse meaning.

For example, we might think that a sentence like $2+(-3)\times6-(-2)$2+(−3)×6−(−2) is worked out from left to right so that we might simplify it as:

$2+(-3)\times6-(-2)$2+(−3)×6−(−2) | $=$= | $-1\times6-(-2)$−1×6−(−2) |

$=$= | $-6-(-2)$−6−(−2) | |

$=$= | $-6+2$−6+2 | |

$=$= | $-4$−4 | |

We tend to think this is correct, because we are used to reading words in a sentence from left to right.

But in mathematics, ... "*convention not is this the*".

In fact we have a strict and agreed order to reading certain sentences.

This includes working within brackets first, then simplifying divisions and multiplications from left to write, and then finally doing the additions and subtractions, also from left to right.

The sentence above, $2+(-3)\times6-(-2)$2+(−3)×6−(−2) is correctly worked out as:

$2+(-3)\times6-(-2)$2+(−3)×6−(−2) | $=$= | $2+(-18)-(-2)$2+(−18)−(−2) |

$=$= | $2-18+2$2−18+2 | |

$=$= | $-16+2$−16+2 | |

$=$= | $-14$−14 | |

Here is another interesting example involving a fraction where the line in the fraction (called a vinculum) acts as a bracket, so that the numbers in the numerator are simplified first:

$\frac{6-(-4)}{-5}$6−(−4)−5 | $=$= | $\frac{6+4}{-5}$6+4−5 |

$=$= | $\frac{10}{-5}$10−5 | |

$=$= | $-2$−2 | |

For the following expressions, $q$`q` is a negative number, $r$`r` is a negative number, and $t$`t` is a positive number.

We want to decide whether each expression simplifies to a positive or negative number, or if it is not possible to determine.

The sign of $\frac{q}{r-t}$

`q``r`−`t` is:Not possible to determine

APositive

BNegative

CNot possible to determine

APositive

BNegative

CThe sign of $q+t$

`q`+`t`is:Not possible to determine

APositive

BNegative

CNot possible to determine

APositive

BNegative

CThe sign of $r\left(q-t\right)$

`r`(`q`−`t`) is:Negative

APositive

BNot possible to determine

CNegative

APositive

BNot possible to determine

C

An investment loses $\$22200$$22200.

If this loss is shared equally among six, use a negative integer to describe the loss for each person.

Consider the following phrase:

The quotient of $-3$−3 and the sum of $7$7 and $6$6 .

Without simplifying the result, translate this sentence into a mathematical expression.

Evaluate the expression.

Understand operations on fractions, decimals, percentages, and integers