“Why does -3 times -4 equal 12?”
Some people may answer that by saying that the product of two negative numbers must be positive, so -3 times -4 is the same as 3 times 4, which equals 12.
However, that a negative number times a negative number is a positive number is only a result, not a cause.
Last time, I explained why 3 times 4 equals 12. Because three 4s, or four 3s, add up to 12.
But the same idea can hardly be applied to -3 times -4, or when multiplying any two negative numbers.
After all, what is the meaning of negaive three -4s?
However, this idea can still be used to multiply a positive number by a negative number. For example, 3 times -4 can be understood as the sum of three -4s.
Following the idea that multiplication is repeated addition, we can explain why 3 times -4 equals -12. Remember that this conclusion will be used later on.
Before explaining why -3 times -4 equals 12, we need another conclusion:
If the sum of two numbers is zero, then the numbers are the opposites to each other.
Although this conclusion seems simple and obvious, half of the students may not be able to explain why.
How can one prove that if the sum of two numbers is zero, then the two numbers must be opposites to each other?
a + b = 0 is an equation. The equation is like a balance in equilibrium. If weights of the same weight are added or subtracted from both ends at the same time, the balance shall remain. Therefore, if you add or subtract the same number from both sides of the equal sign, the equal sign still holds.
Therefore, if the sum of two numbers is zero, then the two numbers are the opposites of each other.
That’s all we need. Now let’s get to the point.
Since it is not possible to explain why -3 times -4 equals 12 directly, we have to use an alternative.
How?
By using 0 as a bridge.
First of all, 0 multiplied by -4 is the sum of zero -4s, and the result is zero.
Someone may ask, what is the use of such an obvious result?
Never underestimate the importance of number 0. In mathematics, 0 is a most special number.
In the famous Euler’s identity, 0 alone takes half side.
zero -4s is equivalent to (3-3) -4s, which is equivalent to three -4s, plus negative three -4s. so,
As stated earlier, 3 times -4 equals -12.
If the sum of two numbers is zero, they must be opposites to each other. The opposite of -12 is 12.
Therefore, -3 times -4 equals 12.
Note that in the proof, the number 0 is used as a bridge.
Using the same idea, it is not difficult to explain that the product of any two negative numbers must be a positive number, so we can conclude that negative times negative equals positive.
From why 3 times 4 equals 12, to why 3.3 times 4.4 equals 14.52, to why -3 times -4 equals 12, these three questions about multiplication gradually increase in difficulty.
To answer the first question, you need to know the definition of multiplication between two positive integers.
To answer the second question, you not only need to know the definition of multiplication, but also need to clearly understand the expressions of Arabic numerals.
To answer the third question, you need to understand the meaning of the equation and must use the special number 0 as a bridge.
Many children find it easy to learn primary school mathematics, but find it difficult to learn junior high school mathematics. Leaning high school mathematics is like learning Greek. A major reason is that they pay too much attention to numerical calculations and spend too little time studying mathematical principles, resulting in a poor foundation of mathematics.
Don’t be satisfied with knowing lots of ‘whats’. Work hard to explore more ‘whys’. Only then can you learn the essence of mathematics and climb higher and more steadily the great pyramid of mathematics.