james asked in Science & MathematicsMathematics · 5 months ago

# What is the purpose of arithmetic with signed numbers? What is the most confusing aspect of the process?

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• Dixon
Lv 7
5 months ago

The most confusing part is that positive and negative use the same symbols (+, -) as addition and subtraction but they mean different things.

Addition and subtraction are processes that take in two numbers and return a single value.

Positive and negative operate on a single number and (potentially) modify it.

So any attempt to generally understand what "+" means and what "-" means is destined to failure because neither has a single meaning, the meaning has to be deduced from context. There is a related problem in that the words plus and minus embrace this ambiguity of meaning, ie they are used to mean either case. This can be avoided by sticking to add and subtract, positive and negative.

Note that when writing positive and negative numbers the symbol is right next to the number with no space, -3.

Whereas addition and subtraction are indicated by having a space either side of the symbol, 3 + 5, 3 - 5.

Don't write 3+5 because it is bad style and technically either meaningless or 3(+5) = 15

For clarity, here is what all these things actually mean (at an everyday level).

Numbers have a direction which you can think of as an arrow,

positive (+) → and negative (−) ← and they are understood to be opposites, meaning the opposite or reverse of positive direction is negative direction, and vice versa.

A plain number such 3 defaults to the positive direction ie 3 = +3

Positive applied to a number leaves its direction unchanged.

Negative applied to a number reverses its direction. This is the single most important rule.

-(+3) = -3

-(-3) = +3

-(-(-3)) = -3

[ We intuitively understand how to add positive quantities in everyday life but this leads us to think we understand what is going on mathematically when actually we don't. The failure to understand and learn what addition really means is what leads to confusion when we add and subtract with negative numbers. ]

Adding takes in two numbers (eg 3 + 5 takes in 3, then 5) and stacks them head to tail , ie starting at the tail of the first arrow you go to its head, then from the head of the first number arrow to the tail of the second and then to the head of the second arrow, like following a arrows along a path. This is true even when a number itself is negative (-3 + 5) and points in the negative direction. The result of addition is simply the single number that represents where you finish with respect to where you started, ie your equivalent single number.

Subtraction.

OK, this is the one:

Subtraction is defined as adding the two numbers but with the direction of the second number reversed.

So 3 - 5 is really a shorthand way of writing 3 + (-5).

And famously, if you have 3 - (-5) then when you add and reverse the direction of -5 it becomes +5, which we can just write as 5, giving 3 + 5

• 5 months ago

In math we first learn that the symbol + tells us to add while the symbol - tells us to subtract.

When these signs are attached to numbers, however, they show us whether the value of that number is greater than 0 (+1) or less than 0 (-1). For example, when you see 4 or +4 (a.k.a. "positive four"), that number is 4 points greater than 0 on the number line below. On the other hand, a minus sign in front of a 4 makes it -4 (a.k.a. "negative four"). That number has a value that is 4 points less than 0 on the number line below.

• 5 months ago

Signed numbers are easier to understand if you use an 'elevator' example.

Signs can mean a location and/or a direction.

Imagine you have an elector at a building that has parking below ground.

Location:

+3 (or just Floor 3) means you're on the third floor

-3 means you're in the Parking Garage 3 floors below ground.

Direction:

Let's say you're on the 7th floor and want to go down.

-3 means you go down to the 4th floor

((7 -3 = 4))

Hope that helps.

You can also use money if you want.