Negative numbers have a long and sometimes contentious history. Mathematicians on the Indian subcontinent had been using negative numbers for a thousand years before Europeans got around to accepting the idea. If you go back far enough, you'll find that some Europeans didn't even necessarily think that 1 was a number.
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(We owe our current so-called "Arabic" number system, including the number zero, to Indian mathematicians, too. Previously, Europeans had previously been stuck using Roman numerals for everything. There was a period during which using zero, or the "cipher", could get you jailed or executed.)
Chinese mathematicians beat the Europeans by two millennia in using negative numbers. (Humorous note: When the Indians were writing, they used the "plus" sign, "+", to indicate negative numbers.)
Europeans were not alone in being ignorant of, or dismissive of, or even morally opposed to, negative numbers. Egyptians, nearly two thousand years ago, regarded negative numbers as being ridiculous. Europeans, like the Egyptians, used a mathematics based on geometrical concepts such as area, which is always positive. This retarded their mathematical progress, because they were thinking of numbers in an unhelpful way. However, when European scholars started translating Arabic texts obtained from North Africa, they were finally exposed to new ways of thinking, and Europe started catching up to the Indians, Chinese, and North-African Arabs.
When you first learned your numbers, way back in elementary school, you started with the counting numbers: 1, 2, 3, 4, 5, 6, and so on. Your number line looked something like this:
Later on, you learned about zero, fractions, decimals, square roots, and other types of numbers, so your number line started looking something like this:
...but what if you had "5 −"9"? You just couldn't do this subtraction, because there wasn't enough "space" on the number line to go back nine units:
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You can solve this "space" problem by using negative numbers. The "whole" numbers start at zero and count off to the right; these are the positive integers. The negative integers start at zero and count off to the left:
Note the arrowhead on the far right end of the number line above. That arrow tells you the direction in which the numbers are getting bigger. In particular, that arrow also tells you that the negatives are getting smaller as they move off to the left. For example, −5 is smaller than −4.
For starters, by using the number line that has been expanded to the left into negative numbers, we can now do the subtraction "5 − 9". From zero, we go five units to the right, and then we subtract nine units to the left, as shown below:
We end up four units to the left of zero, so we now know that 5 − 9 = −4.
Negative numbers might seem a bit weird at first, but that's okay; negatives take some getting used to. Let's look at a few inequalities, to practice your understanding. Refer to the number line below, as necessary.
Look at the number line above. Since 6 is to the right of 3, then 6 is larger, so the correct inequality is:
3 < 6
Look at (or think of) the number line again. Every positive number is to the right of every negative number, so the correct inequality is:
−3 < 6
Think of the number line again. Since −6 is to the left of −3, then −3, being further to the right, is actually the larger number. So the correct inequality is:
−3 > −6
Zero is less than any positive number, so:
0 < 1
Zero is greater than any negative number, so:
0 > −1