This is a great question! I'll be glad to help.
Subtraction: To subtract by adding, just add negative numbers and thus move left on the number line (subtracting) instead of moving right (adding).
Example: 2 - 3 = 2 + (-3) = -1 You start at 2 on a number line and move to the left three numbers.
Multiplication: To multiply by adding, you add the first number to itself the same number of times as the second number.
Example: 2 x 3 = 2 + 2 + 2 = 6. Notice that you add three 2's together which is the same as 2 x 3. The same works for multiplying by a negative number, in which case you will end up adding negative numbers (see subtracting above) instead of positive. Try it yourself!
Example two: The tricky situation is when you mutiply two negative numbers together, so let's try one. One negative number is easy: -2 x 3 = -2 + -2 + -2 = -6. BUT, look at this: -2 x -3 = -(-2) -(-2) -(-2) = 2 + 2 + 2 = 6. See how you subtract -2 instead of adding -2. We do this because we must do the opposite of what a positive 3 would tell us to do; since -3 is the opposite of 3 when we are using addition as our operator. As long as you remember that a negative number times a negative number makes a positive, this should make sense. If not, let me know and I'll explain this a different way.
Division: To divide, change the problem into a multiplication problem and then follow the mulitplication example above.
Example: 8 / 2 = 8 x (1/2) = (1/2) + (1/2) + (1/2) + (1/2) + (1/2) + (1/2) + (1/2) + (1/2) = 4. Notice how you add eight (1/2)'s together which is the same as 8 / 2. Again, this works when one of the numbers is negative as well.
Exponentiation: Similar to division, we are dealing with another disguise. Exponentiation is a multiplication problem in diguise, and as we know from above, multiplication is addition in disguise. (Below the carrot, "^", is used before any number used as an exponent)
Example: 2^3 = 2 x 2 x 2 = (2 x 2) x 2 = (2 + 2) x 2= 4 x 2 = 4 + 4 = 8 After some rewriting in steps two and three, step four multiplies 2 by 2 meaning you add two 2's together. Then you take the resulting 4 and multiply it by 2 meaning you add two 4's together.
If the base number is a negative, you will have to follow the second example of multiplication. But, the real tricky one here is when the exponent is negative.
Example 2: 2^-3. You have to change the negative power to a positive power by putting the base in the denominator: 2^-3 = 1/(2^3). After that, you should be able to follow other examples above: 1/(2^3) = 1/((2 x 2) x 2) = 1/((2 + 2) x 2) = 1/(4 x 2) = 1/(4 + 4) = 1/8 = 1 x (1/8) = 1/8. The last step you add 1/8 one time, meaning you essentially don't add anything and you stay at the same spot on the number line, at 1/8, which is the answer.
Please let me know if any of this is hard to understand. This question is great and I would welcome the chance to disucss it more!