Byron S. answered • 11/03/14
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Let's consider some concrete examples to help illustrate this idea.
Multiplication is repeated addition, and division splits the multiplication into pieces. All additions and subtractions start with zero as your reference point, you just ignore it in most cases. I included it to more clearly illustrate the issues with subtracting and negatives.
12 ÷ 3 = 4 and 12 ÷ 4 = 3 because
12 = 4 × 3 = 0 + 3 + 3 + 3 + 3
and 12 = 0 + 4 + 4 + 4
Multiplication a negative number by a positive number repeatedly adds negative numbers, which gives a negative result.
-12 ÷ -3 = 4 and -12 ÷ 4 = -3 because
-12 = 4 × -3 = 0 + -3 + -3 + -3 + -3
and -12 = 0 - 4 - 4 - 4 (adding 4 "negative three times" really means subtracting 4 three times)
When multiplying two negative numbers, the result is positive since you must repeatedly subtract negative numbers to get a positive result.
12 = -3 × -4 = 0 - (-3) - (-3) - (-3) - (-3)
and 12 = 0 - (-4) - (-4) - (-4)
So, 12 ÷ -3 = -4
and 12 ÷ -4 = -3
I hope this helps your understanding of the concept. Please comment if you still have questions.
Byron S.
Another way to consider multiplying and dividing negative numbers is by looking at the number line, and the directions.
If you think of a number as an arrow pointing from zero to that number, then for example 5 is an arrow pointing 5 units to the right. If you multiply by -1, 5*-1 = -5, and you get an arrow pointing 5 units to the left. Multiplying (or dividing) by a negative
number switches the direction the arrow points. Positive numbers point right, and negative numbers point left.
If you have a negative number, and reverse its direction by multiplying by a negative number, it switches back to the right, and is positive.
-5 * -1 = 5
(5 left) * (1× in the opposite direction) = (5 right)
If you keep multiplying by a negative, the direction switches back and forth as many times as you have negative signs.
When you multiply by a positive number, the direction the number points doesn't change, it just changes length.
5 * 2 = 10
(5 right) * (2× bigger) = (10 right)
-5 * 2 = -10
(5 left) * (2× bigger) = (10 left)
-5 * -2 = 10
(5 left) * (2× bigger, in the opposite direction) = (10 right)
No matter what sign you begin with, multiplying or dividing by a positive number
never changes its sign.
+ * + = +
- * + = -
No matter what sign you begin with, multiplying or dividing by a negative number
always changes its sign.
+ * - = -
- * - = +
If you are adding or subtracting, the rules are different. In those cases, it matters what the sign of the larger number is. I can give details and explain those if you like.
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11/04/14
Komal K.
lol they r so big concepts answer me small answer i am a student of class 6 i am just 11 years please kindly give me easy concepts
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11/06/14
Komal K.
11/04/14