I found this interesting article about negative numbers. It’s a quote from the paper titled The extension of the natural number domain to the integers in the transition from arithmetic to algebra by Aurora Gallardo. The quote was transcribed from the article Negative by D’Alembert (1717-1738) for Diderot Encyclopedia.
In order to be able to determine the whole notion, we must see, first, that those so called negative quantities, and mistakenly assumed as below Zero, are quite often represented by true quantities, as in Geometry where the negative lines are no different from the positive ones, if not by their position relative to some other line or common point. See CURVE. Therefore, we may readily infer that the negative quantities found in calculation are, indeed, true quantities, but they are true in a different sense than previously assumed. For instance, assume we are trying to determine the value of a number x which, added to 100, gives 50, Algebra tells us that: x + 100 = 50, and that: x = –50, showing that the quantity x is equal to 50, and that instead of being added to 100, it must be subtracted from that number. Consequently, the problem should have been stated in the following way: Find the quantity x which, subtracted from 100, gives 50. Thus, we would have: 100 – x = 50, and x = 50. The negative form for x would then no longer exist. Thus, the negative quantities really show the calculation of positive quantities assumed in a wrong position. The minus sign found in front of a quantity is meant to rectify and correct a mistake in the hypothesis, as clearly shown by the above example. (quoted in Glaeser, 1981, 323–324)
Interesting, isn’t it? Numbers are abstract ideas. They get their meanings from the context we apply them to. Of course from the school mathematics point of view we cannot start with this idea.
Here are the different meanings of the negative number that students should know before they leave sixth grade: 1) it is the result of subtraction when a bigger number is taken away from a smaller number; 2) it is the opposite of a counting/ natural number; 3) that when added to its opposite counting number results to zero; and 4) it represents the position of a point to the left of zero.
Likewise for the minus sign which indicates subtraction. Subtraction has three meanings: take away, find the difference, and inverse operation of addition. For further explanation read the post What exactly are we doing when we subtract?
Gauss proved in 1796 that (a + bi), with i = square root of -1, must exist to solve nth degree equations by listing n roots. Intuitive numeration, arithmetic and algebraic definitions before 1796 suspected n roots existed, but no one published a successful proof before Gauss. Hence, pertinent to this thread, since the square root of negative numbers exist, negative numbers also exist in the real world.
yesterday I tried to post that most intuitive views of numeration have failed to capture foundations of algebra and higher math. Concerning negative numbers and their square roots, they have formally existed to solve the fundamental theorem of algebra, published by Gauss in 1796 added complex numbers (a + bi) to our world of numbers. Everyone here should know i, the square root of -1 was and will always be needed to calculate n roots in n degree equations. Q.E.D.
The point that negative numbers were formally defined and accepted was 1796. Gauss proved in college the ‘fundamental theorem of algebra’ for the first time saying that complex numbers a + bi , with 1 = square root of minus 1, MUST exist for all rational second degree equation roots to be validated. Prior to 1796 fuzzy logic omitted the use of square roots of negative numbers.
Is -3 really to the left of zero?
Since the number line is not a segment, if you start at zero
and move to the right you will eventually get to -3.
I personally would also like students to understand that all of the concepts of subtraction you mention derive from the 4th one – that a negative number represents a position on the number line left of zero.
If we take this as a starting definition of “negative number” then the process of addition becomes movement on the number line. For instance, “3+4” means “start from zero, move 3 units to the right and then 4 more units to the right”. Addition of negative numbers indicates movement to the left. For instance, “3+ (-4)” means “start from zero, move 3 units to the right and then 4 units to the left”.
Similarly, the process of subtraction becomes a distance formula. For instance, 6 – 4 = 2 because 6 and 4 are 2 units apart. Similarly, 5 – (-2) = 7 because 5 and -2 are 7 units apart.
Both of these are examples of the larger concepts of vector arithmetic that students will see again in high school.
Also, it might be true that in a popular or, perhaps, philosophical sense numbers are abstract concepts. In a mathematical sense, however, numbers are very precisely defined. One definition comes from set theory. Another equivalent definition that’s more relevant to this discussion is that a number is defined simply as its position on the number line – that is, a number is simply a position or distance to the left or right of zero.
Finally, describing numbers and operations using the framework of the number line (rather than memorizing a list of rules or concepts) also gives a fine example of mathematics as an axiomatic system. Starting from clear definitions, the rules fall out naturally. This is another big concepts that students will see again in high school – usually in Geometry class.
Hello Erlina,
Interesting but I don’t think this is a lifeline to Greece or Italy