Directed Numbers - SS1 Mathematics Lesson Note
When all natural numbers are placed in an ascending or descending order, each number will occupy a unique position in the sequence. This is a number line.
There is a starting or central or reference point at zero (0) from which all other numbers derive their reference. On either side of zero, the number line extends to infinity, that is forever. Numbers to the left of zero are negative numbers (-) and numbers to the right of zero are positive numbers (+). Zero is neither a negative nor positive number.
RULES GUIDING OPERATIONS ON DIRECTED NUMBERS
ADDITION AND SUBTRACTION
When the signs are the same, the numbers are added but when the signs are different the numbers are subtracted and the sign of the greater number is carried by the result.
Example 2 Simplify the following: (a) \(- 6\ - 9\) (b) \(+ \ 9 + 7\) (c) \(- 12 + 8\) (d) \(- 8 + 18\)
Solution (a) \(- 6 - 9 = \ - (6 + 9) = \ - (15) = \ - 15\)
(b) \(+ \ 9 + 7 = \ + \ (9\ + \ 7) = \ + (15) = \ + 15\)
(c) \(- 12 + 8 = \ - (12 - 8) = \ - (4)\)
(d) \(- 8 + 18 = \ + (18 - 8) = \ + (10) = \ + 10\)
MULTIPLICATION AND DIVISION
The following rules apply to directed numbers under the operations of multiplication and division:
(i) \(+ \times + = +\)
(ii) \(- \times - \ = \ +\)
(iii) \(+ \times - \ = \ -\)
(iv) \(- \ \times \ + \ = \ -\)
(v) \(+ \div + = +\)
(vi) \(- \div - \ = \ +\)
(vii) \(+ \div - \ = \ -\)
(viii) \(- \ \div \ + \ = \ -\)
Example 3 Simplify the following: (a) \(+ 4 \times + 3\) (b) \(- 3 \times - 7\) (c) \(+ 8 \times - 10\) (d) \(- 4 \times + 6\)
(e) \(+ 100 \div + 10\) (f) \(- 81 \div - 9\) (g) \(+ 25 \div - 5\) (h) \(- 144 \div + 12\)
Solution (a) \(+ 4 \times + 3 = \ + 12\) (b) \(- 3 \times - 7 = \ + 21\) (c) \(+ 8 \times - 10 = \ - 80\)
(d) \(- 4 \times + 6 = \ - 24\) (e) \(+ 100 \div + 10 = \ \frac{+ 100}{+ 10} = \ + 10\) (f) \(- 81 \div - 9 = \ \frac{- 81}{- 9} = \ + 9\)
(g) \(+ 25\ \div \ - 5 = \ \frac{+ 25}{- 5} = \ - 5\) (h) \(- 144\ \div \ + 12 = \ \frac{- 144}{+ 12} = \ - 12\)