Lesson 5 · from §15 of the 1910 text
Multiplying Signed Numbers
In arithmetic, multiplying meant taking one number (the multiplicand) as many times as there are units in another (the multiplier). That definition broke down for fractions, since you can't add 8 to itself \tfrac{2}{3} of a time, so it was extended: 8 \times \tfrac{2}{3} was defined to mean \tfrac{8 \times 2}{3}.
Algebra has negative numbers too, so we extend the definition again to cover four cases: +4 \cdot +3, \qquad -4 \cdot +3, \qquad +4 \cdot -3, \qquad -4 \cdot -3
- +4 \cdot +3 means (+4) + (+4) + (+4) = +12.
- -4 \cdot +3 means (-4) + (-4) + (-4) = -12.
- +4 \cdot -3 means 4 is subtracted three times. That's the same as subtracting 12 once, so +4 \cdot -3 = -12.
- -4 \cdot -3 means -4 is subtracted three times. That's the same as subtracting -12 once, and subtracting -12 is the same as adding +12. So -4 \cdot -3 = +12.
In general: +a \times +b = +ab \qquad -a \times +b = -ab \qquad +a \times -b = -ab \qquad -a \times -b = +ab
Law. The product of two numbers with like signs is positive. The product of two numbers with unlike signs is negative.
Quick check
-8 \cdot -7
With three or more factors, multiply two at a time: -2 \cdot -3 \cdot -4 = +6 \cdot -4 = -24. And anything times zero is zero.
Quick check
-1 \cdot -2 \cdot -3 \cdot -4
From the history of algebra
"Man is responsible for the rest." The German mathematician Leopold Kronecker (1823–1891) once said that "the good Lord made the positive integers, but man is responsible for all the rest of the numbers." We count objects from our earliest years, so the positive whole numbers feel natural. Other kinds of numbers only become necessary when we study mathematics. Negative numbers are a great convenience whenever oppositeness is involved, but their artificial character delayed their sensible use for many hundreds of years. The Hindus said that "the square of a negative is positive," but the statement probably meant nothing to those who read it. Not until after the time of Descartes were the rules for negative numbers understood, even by great mathematicians.
Exercises
How to type math
Use ^ for powers (a^2), / for fractions ((a+b)/2), sqrt(x) for square roots, and cbrt(x) for cube roots. You can leave out the multiplication sign: 3ab means 3 \cdot a \cdot b. Mixed numbers like 4 1/2 work too. The preview under each box shows how your answer was read.
Find the products.
+3 \cdot +4
+4 \cdot +12
-5 \cdot +4
+6 \cdot -6
-7 \cdot +8
-7 \cdot -3
-12 \cdot +9
+6 \cdot -4
-6 \cdot -6
+5 \cdot -10
+0 \cdot +4
-7 \cdot 0
+4 \cdot -5 \cdot +6
+4 \cdot -5 \cdot -6
-4 \cdot -5 \cdot -6
12 \cdot +0 \cdot -5
9 \cdot -10 \cdot -0
-4 \cdot +3 \cdot -6
-3 \cdot -2 \cdot -5
2 \cdot -3 \cdot +5