Analytical method and comprehensive method

In mathematical proof, if the direction of reasoning is from verification to known or from unknown to known, this way of thinking is called analysis. On the other hand, if the direction of reasoning is from known to verified, or from known to unknown, this thinking method is called synthesis method.

Example 1

It is known that a and b are unequal positive numbers. Prove:

a3+B3 & gt； a2b+ab2

Proof method one

analyse

Trying to prove

a3+B3 & gt；

a2b+ab2

Just trying to prove

(a+b)(a2-ab+b2)>ab(a+b).

∵a & gt； b，b & gt0，a+b & gt； 0,

∴ As long as the certificate A2-AB+B2 >; ab，a2-2ab+B2 & gt； 0,

Namely (a-b) 2 >; 0, and this is obviously true.

So a3+B3 >;

a2b+ab2 .

Syndrome method 2

synthetic method

∵a≠b，

∴a-b≠0,(a-b)2>； 0,

that is

a2-2ab+B2 & gt； 0，a2-a b+ B2 & gt； ab .

and

a & gt0，b & gt0，a+b & gt； 0,

∴

(a+b)(a2-ab+b2)>ab(a+b)，

therefore

a3+B3 & gt；

a2b+ab2

two

Direct evidence law and indirect evidence law

In mathematical proof, the method of proving the truth value of a proposition from the front is called direct proof. Anything that proves a proposition to be true through deduction is a direct proof. It is a common proof method in middle school mathematics. The method to prove the authenticity of a topic is not to directly prove the authenticity of the topic, but to prove that the negative proposition of the topic is untrue or its equivalent proposition is established, which is called indirect proof. Indirect proof methods mainly include reduction to absurdity and homology.

1.

reductio ad absurdum

By proving that the negative topic of the topic is untrue, the method to affirm the authenticity of the topic is called reduction to absurdity.

There are two kinds of reduction to absurdity and exhaustion.

The general steps of reduction to absurdity are as follows:

(1) Assume that the conclusion of the proposition is not established (that is, the negation of the conclusion is established);

(2) Starting from the negative conclusion, reasoning step by step, and drawing a conclusion that contradicts axioms or the aforementioned theorems, definitions or conditions, or contradicts temporary assumptions (that is, the conclusion cannot be denied);

(3) According to law of excluded middle, the original proposition was finally confirmed.

In the application of reduction to absurdity, if there is only one possible situation to deny the conclusion of a proposition, then as long as this situation is overturned, the conclusion can be affirmed. This reduction to absurdity is called reduction to absurdity (see Example 2). If there is more than one negative aspect of the proposition conclusion, then all possible negative aspects must be refuted one by one to determine the conclusion. This reduction to absurdity is called exhaustive method (see Example 3).

Example 2

Verification:

cos 10

This is an irrational number.

Proving hypothesis

cos 10

Remember, this is a rational number.

cos 10

=

(p and q are prime numbers),

Then cos30

=4cos3 10

-3cos 10

=4(

)3-3(

)

Is a rational number,

And cos30

=

Is an irrational number,

This contradicts the known assumption, so cos 10.

This is an irrational number.

Example 3

As shown in the figure, in △ABC, it is known that BE and CF are bisectors of B and C respectively, and BE=CF, and verification: AB=AC.

Proof: if AB≠AC, there is AB >；; AC or AB

AC, then ∠ ACB > ∠ABC，

∴∠bcf>； ∠CBE，BF & gtCE，

∵

BF=EG，

∴

EG & gtEC，∠ECG & gt； ∠EGC .

and

∠FCG=∠FGC，

∴∠fce<； FCE=∠FBE .

rule

∠ACB & lt； ∠ABC (contradicting the hypothesis),

that is

AB & gtAC, that's impossible.

(2)

The same can be proved, AB

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