Answer:
When both rational and irrational numbers are written in decimal form, rational numbers can be written as finite decimals and infinite recurring decimals, such as 4=4.0, 4/5=0.8, 1/3=0.33333... and irrational numbers can only be written as infinite non-recurring decimals, such as √ 2=1.414213562…………According to this point, people define irrational numbers as infinite non-recurring decimals. 2. All rational numbers can be written as the ratio of two integers; irrational numbers cannot.
(Rational number): A number that can be accurately expressed as the ratio of two integers. Such as 3, -98.11, 5.72727272..., 7/22 are all rational numbers. Integers and fractions are all rational numbers. Rational numbers can also be divided into positive rational numbers, 0 and negative rational numbers. Irrational numbers refer to infinite and non-recurring decimals such as: π ·The difference between irrational and rational numbers: 1. When both rational and irrational numbers are written in decimal form, rational numbers can be written as finite decimals and infinite cyclic decimals. For example, 4=4.0, 4/5=0.8, 1/3=0.33333... and irrational numbers can only be written as infinite and non-cyclic decimals. For example, √2=1.414213562…………According to this point, people define irrational numbers as infinite and non-cyclic decimals. 2, all rational numbers can be written as the ratio of two integers; irrational numbers cannot. Based on this, some people suggested that irrational numbers should be removed from the "irrational" hat, and rational numbers should be called "comparative numbers" and irrational numbers should be called "irrational numbers". Originally, irrational numbers are not unreasonable, but people don't know much about it at first. Using the main difference between rational and irrational numbers, we can prove that √2 is irrational. Proof: Assume that √2 is not an irrational number, but a rational number.Since √2 is a rational number, it must be written as the ratio of two integers: √2 = p / q Since p and q do not have a common factor to be reduced, p / q can be considered to be the reduced fraction, the simplest form of fraction. Put √2 = p / q squared on both sides Get 2 = (p ^ 2)/(q ^ 2) Is 2 (q ^ 2) = p ^ 2 Since 2q ^ 2 is an even number, p must be an even number, so set p = 2m From 2 (q ^ 2) = 4 (m ^ 2) Get q ^ 2 = 2m ^ 2 Similarly, q must be an even number, let q = 2n Since p and q are both even numbers, they must have a common factor of 2, which contradicts the previous assumption that p/q is a reduced fraction. This contradiction is caused by the assumption that √2 is a rational number. Therefore √2 is an irrational number.
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