A special instance of exponential operation for power of 2

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The basic rule of the combined operation of power exponent of 2 is presented in the article "the combined Operation instance of power of 2". At the time of merging the powers of 2, two rules were used, which I call the power of the 2 and the power of the 2 to halve the power of the operation. This is not a standard rule and only applies to a power of 2. Although there are rules for multiplication and division power, I have discovered the value of it in addition and subtraction exponentiation. I'll explain the rules and show the use cases.

Double power arithmetic rule for power of 2

Here is the double power rule that I call the power of 2:

2a + 2a = 2a+1

(2a+1 is 2a squared, which is double 2a.)

For example, 24 + 24 = 25, and 2-2 + 2-2 = 2-1.

In general, you do not have to consider arithmetic rules, but you need to add power. Doing its own addition equals doubling, equals multiplying by 2-is a power of 2. This rule is the hidden form of the quadrature rule of the power of 2. Here's the math process:

2a + 2a = 2a 2 = 2a 2a+1 =

Instance

An example is when you add a binary number that represents a power form of 2. For example, 101 and 1100 add up, you add 22 + 20 and 23 + 22, get 23 + 22 + 22 + 20. You want to combine a power of 2 to simplify the calculation. According to the power of 2 of the double power operation rule is known as 22 + 22 = 23. Now there are two 23, we have 23 + 23 = 24. Final Result 24 + 20 = binary 10001.

The power of 2 is halved by a power operation

Here is the rule that I call a power of 2 in the power of halving:

2a–2a-1 = 2a-1

(The square of 2a-1 is 2a; it is half of 2a.) )

For example, 24–23 = 23, and 2-2–2-3 = 2-3.

Similarly, you do not usually need to consider the subtraction rule of the power operation, but it is required here. Minus half is equal to half, multiplied by 1/2, or divided by 2-is a power of 2. This rule is the secret form of the quadrature rule or the quotient rule (respectively) of the power of 2. Here is the mathematical formula:

2a–2a-1 = 2A–2A/2 = (2 2a–2a)/2 = (2a (2-1))/2 = 2A/2 = 2a-1.

The above is a verbose form, in order to more clearly show the process of halving; The following is a concise form:

2a–2a-1 = 2a-1 (2-1) = 2a-1

Instance

I will use these rules when solving the following infinite geometric equations:

1/2−1/4 + 1/8−1/16 + 1/32–1/64 + ...

The solution for Wikipedia is to extract 1/2 first:

1/2−1/4 + 1/8−1/16 + 1/32–1/64 + ...

= 1/2 (1–1/2 + 1/4–1/8 + ...)

Then use the following formula to replace the sum:

Make R =-1/2, simplify to

And I'm more direct. I do not extract 1/2, I start by replacing the adjacent elements with a halving rule that uses the power of 2:

1/2−1/4 = 1/4, 1/8−1/16 = 1/16, 1/32–1/64 = 1/64, ...

Last remaining 1/4 + 1/16 + 1/64 + 1/256 + 1/1024, writing

This time use the following formula to simplify:

Make r = 1/4, get

I like my proof because I use positive r, which is easier to understand. You can interpret a calculation as a binary number (0. ).

A special instance of exponential operation for power of 2