You might have some gaps in your understanding of arithmetic operators. In case you didn’t know:
Operations
- The Foundation:
- Addition is a form of counting up. 2 + 3 = 1, 2, 3, 4, 5
- Subtraction is a form of counting down. 3 - 2 = 3 , 2 , 1
- Addition & Subtraction are the inverse of each other, they cancel each other out. So 5 - 3 = 2, 2 + 3 = 5
- The Next Level, Multiplication & Division:
- Multiplication is a form of repeated addition: 3 * 5 = (5 + 5 + 5) or (3 + 3 + 3 + 3 + 3)
- Division is a form of repeated subtraction where you count the operation counts until you go negative or get to zero: 6 / 2 = 6 - 2 - 2 - 2 = 3 operations
- Like addition and subtraction, they are both the inverse of each other and cancel each other out
- The 3rd Level, Exponents & Logarithms
- Exponents (or powers) is a form of repeated multiplication. 2^4 = 2 * 2 * 2 * 2 = 16 . Since an exponent is repeated multiplication, and multiplication is repeated addition, and addition is a form of counting, you can expand exponentiation into a lot of adding or counting.
- Logarithms are a form of repeated division. Also like division, the result of a logarithm is a count of operations done to achieve the end result of 1. Ex log_2(16) = 16 / 2 / 2 / 2 / 2 = 1 => 4 operations.
- Exponents and logarithms are the inverse of each other and cancel each other out, like with multiplication and division, and addition and subtraction.
- The 4th level?
- Since after the 3rd level, the concept of the next level operation being an iterated version of the previous operation, you can generalize the concept into hyperoperations, although this is not taught in high school and entry level college math.
- In a hand wavy metaphorical way you can kind of say that an integral is kind of like exponentiation and a derivative is kind of like a logarithm, but on algebraic expressions.
Number Systems
You’d probably notice that all of the examples were whole numbers that never went negative. I’m not going to explain how all the above operations change when you use them with negative, fractional, irrational, etc numbers. Or the number zero or infinity. But:
- Natural numbers are whole numbers. 1, 2, 3, 4, 5, … , ∞
- Integers are natural numbers, but also go in a negative direction: -♾, …, -3, -2, -1, 0, 1, 2, 3, …, ∞
- Fractional (or rational) numbers represent a partial amount of a number, and are expressed with integers. Ex: -1/2, …, 0, …, 1/2, …, 2/3, …, 1, etc
- Decimal point numbers can express fractional numbers, and are used to show approximations of irrational numbers. All finite decimal numbers can be expressed with a fraction. Ex:
12.3456 = 123456/10000 - Irrational numbers are numbers that cannot be expressed by a finite fraction, but exist, such as pi or e. Irrational numbers can be (but not always) be described via a finite description of a process that goes on forever and can be said to be between two rational numbers.
- The real numbers are the rational numbers and irrational numbers combined.
Since there is a rule with negative numbers that if 2 negatives are multiplied together, they create a positive number always, it is impossible with all the above number systems to create a multiplication of some sort that gives a negative result when both numbers are positive or negative. Complex Numbers (aka imaginary numbers) can let you do multiplication operations that give you negative results with 2 identical numbers, and thus let you get the square root of negative numbers.