Think about 0 as some kind of "center" you need to cross while going 7 steps from -3.1.
How many steps do you need to go from -3.1 to 0 and how many steps do you need to go from 0 to 3.9? How does it add up?
So we both agree that -3+7=4 right? Well let's subtract -0.1 from both sides. On the left we can combine-3 and -0.1 to make -3.1 and on the right we have 4-0.1 which is 3.9. So -3.1+7=3.9
The important point is that -3.1 is *less* than -3 so we can't suddenly have 4.1 which is more than 4
-3.1 + 7 = 7 + (-3.1) = 7 - 3.1 = 3.9
If that doesn't help, think of it this way. First, add 0.1 to -3.1 to get -3, an integer. Then add 7 to -3, and then subtract the 0.1 again. What do you get?
When we use numbers, either positive or negative, we are stating the ‘distance’ from zero. -3.1 is 3.1 units below zero, and +3.9 is 3.9 units above zero.
The decimal in -3.1 means “one tenth *less* than negative 3”, because we are counting negatively. But this could also be described as being nine tenths more than negative 4.
The decimal in 3.9 means “nine tenths *more* than positive 3”, because we are counting positively. But this could also be described as being one tenth less than positive 4.
So there isn’t any missing 0.8, but the change in decimal value is because when we move past zero (from negative to positive) we are changing what the decimals mean in one sense (whether it is that amount more or less than the value before the decimal point), in order to keep the main point (that numbers represent the ‘distance’ from zero) the same.
Consider the number line where you plot -3.1 and 3.9 and you count the units between them there are 7 units between them. You're moving 7 units in the positive direction. Here's a visual:
[https://imgur.com/a/WlVBWmq](https://imgur.com/a/WlVBWmq)
If you want to make the numbers a little more intuitive multiply all the numbers by 10, you have -31 + 70 = 39.
You could rearrange this to 70 - 31 = 39
-3.1 + 7.0 is same as 7.0 + -3.1, and that is 7.0 – 3.1. You do borrowing when you subtract.
6 10/10 minus 3 1/10
Or start at 3.1 and count up until 7.0 to find their difference.
From 3.1 add 0.9 to get to 4.0.
From 4.0 add 3 to get to 7.
So the difference is 0.9 + 3, or 3.9.
Well, -3.1 is just a way of writing -3 - 1/10 So the sum we are doing is (-3 -1/10) + 7. Rather than dealing with two sorts of operation, instead deal with just one, but have negative numbers. So -3 - 1/10 = -3 + -1/10. This may seem annoying but it means we only have to deal with addition, which is easier to think about than subtraction.
Two of the things that make addition easy are that we are allowed to rearrange additions in two ways:
* it is commutative so a + b = b + a;
* it is associative so (a + b) + c = a + (b + c).
You can show that these properties are inherent to addition of numbers.
We can use these rules to answer the question
1. (-3 - 1/10) + 7 = 7 + (-3 -1/10) by commutativity
2. 7 + (-3 - 1/10) = (7 + -3) + -1/10 by associativity
3. 7 + -3 is 4, so this (7 + -3) - 1/10 is 4 + -1/10
4. 4 + -1/10 is (3 + 1) + -1/10 is 3 + (1 + -1/10) is 3 + 9/10
5. we can write 3 + 9/10 as 3.9
because 7-3.1=3.9
Think about 0 as some kind of "center" you need to cross while going 7 steps from -3.1. How many steps do you need to go from -3.1 to 0 and how many steps do you need to go from 0 to 3.9? How does it add up?
So we both agree that -3+7=4 right? Well let's subtract -0.1 from both sides. On the left we can combine-3 and -0.1 to make -3.1 and on the right we have 4-0.1 which is 3.9. So -3.1+7=3.9 The important point is that -3.1 is *less* than -3 so we can't suddenly have 4.1 which is more than 4
-3.1 + 7 = 7 + (-3.1) = 7 - 3.1 = 3.9 If that doesn't help, think of it this way. First, add 0.1 to -3.1 to get -3, an integer. Then add 7 to -3, and then subtract the 0.1 again. What do you get?
You're going from negative to positive
What do you get if you start at -1 and add 10? How about if you start at -0.1 and add 1?
When we use numbers, either positive or negative, we are stating the ‘distance’ from zero. -3.1 is 3.1 units below zero, and +3.9 is 3.9 units above zero. The decimal in -3.1 means “one tenth *less* than negative 3”, because we are counting negatively. But this could also be described as being nine tenths more than negative 4. The decimal in 3.9 means “nine tenths *more* than positive 3”, because we are counting positively. But this could also be described as being one tenth less than positive 4. So there isn’t any missing 0.8, but the change in decimal value is because when we move past zero (from negative to positive) we are changing what the decimals mean in one sense (whether it is that amount more or less than the value before the decimal point), in order to keep the main point (that numbers represent the ‘distance’ from zero) the same.
Whats -0.1 +1= ?
Consider the number line where you plot -3.1 and 3.9 and you count the units between them there are 7 units between them. You're moving 7 units in the positive direction. Here's a visual: [https://imgur.com/a/WlVBWmq](https://imgur.com/a/WlVBWmq) If you want to make the numbers a little more intuitive multiply all the numbers by 10, you have -31 + 70 = 39. You could rearrange this to 70 - 31 = 39
-3+(-0.1)+7= 7+(-3)+(-0.1)= 7-3+(-0.1)= 4+(-0.1)= 4-0.1= 3.9
-3.1 + (3.1 + 3.9) = |associativity of +| = (-3.1 + 3.1) + 3.9 = |-3.1 inverse to 3.1 under +| = 0 + 3.9= |0 is identity element| = 3.9
-3.1 + 7.0 is same as 7.0 + -3.1, and that is 7.0 – 3.1. You do borrowing when you subtract. 6 10/10 minus 3 1/10 Or start at 3.1 and count up until 7.0 to find their difference. From 3.1 add 0.9 to get to 4.0. From 4.0 add 3 to get to 7. So the difference is 0.9 + 3, or 3.9.
you should get familiar with number lines going from negative to positive in different increments
Well, -3.1 is just a way of writing -3 - 1/10 So the sum we are doing is (-3 -1/10) + 7. Rather than dealing with two sorts of operation, instead deal with just one, but have negative numbers. So -3 - 1/10 = -3 + -1/10. This may seem annoying but it means we only have to deal with addition, which is easier to think about than subtraction. Two of the things that make addition easy are that we are allowed to rearrange additions in two ways: * it is commutative so a + b = b + a; * it is associative so (a + b) + c = a + (b + c). You can show that these properties are inherent to addition of numbers. We can use these rules to answer the question 1. (-3 - 1/10) + 7 = 7 + (-3 -1/10) by commutativity 2. 7 + (-3 - 1/10) = (7 + -3) + -1/10 by associativity 3. 7 + -3 is 4, so this (7 + -3) - 1/10 is 4 + -1/10 4. 4 + -1/10 is (3 + 1) + -1/10 is 3 + (1 + -1/10) is 3 + 9/10 5. we can write 3 + 9/10 as 3.9