Inverses, Reciprocals, Identities, and Canceling
Inverse means the opposite. In general it means to invert or flip something vertically. Inverse and invert both have 'vers' in it which means to rotate. A book has chapters that are made of verses. You turn the page, you 'vers' it. Very old latin. You will learn that verse means to rotate in trigonometry. In math there are three types of inverses, the inverse of an operation, the inverse of a function, the inverse of a number:Inverse operations: These are called the reciprocal operations. They are the reverse operation that cancels another. An inverse operation works backwards from the answer to find the starting number. This can be used to check your answers. When working backwards if you arrive at the starting number then you have the right answer.
Inverse of a function: Functions are equations with many operations. The inverse of a function cancels another function. When two functions cancel what happens is they are multiplied and the answer is 1. 1 itself cancels when it is a factor in a multiplication operation, as we will see.
Inverse of a number: The inverse of a number is the multiplicative reciprocal of a number. The reciprocal of a number is when you write a number as a fraction and then invert the numerator with the denominator. In multiplication when two number multiplied result to 1 as the answer they are reciprocals. A reciprocal is the number you multiply to equal one. The purpose of this is to cancel the number or term in the equation.
When you cancel a number it is erased from the equation. There are three ways to cancel something:
Additive inverses: Additive inverses are zero sums. If two numbers are added together and the total is 0 then the numbers used are additive inverses. Adding opposite numbers cancels them both. Although canceling is not the proper term this is the easiest way to explain it. If 8+10–8 then we can change the order to 8–8+10 = 0+10. We see that 8 and –8 are opposite of each other. These are called additive inverses. Additive inverses are used in inverse operations. Additive means adding but can also mean adding a negative number like 4 + (–1) = 4 – 1. Likewise an additive inverse we can have 4 + (–4) = 4 – 4, or simply –4+4. In either case both –4+4 and 4–4 both equal zero. This cancels them out, or more properly sums to 0.
Reciprocals: This is the proper usage of canceling. When you multiply a number, be it an integer or a rational, and the product is 1 and the number is said to be canceled. Informally canceling is the act of reducing a fraction so that its quotient is 1. This is better explained under the fractions section. Reciprocals deal heavily with multiplication. Under property laws there exists the law of reciprocal multiplication.
Division: You can divide a number, term or entire equation by itself and the answer is 1. This is obvious for simple problems like 3÷3 = 1 but not more complicated problems like √2/√2 = 1 or (3x2+4x+7)/(3x2+4x+7) = 1.
Identities: Identities are formulas that make canceling easier. They are used in long equations with many parts to simplify the problem before or as you work.