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Millie
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we can prove that √5 is irrational through contradiction and same applies for √3.
but can we prove that √5 + √3 is irrational by contradiction?
but can we prove that √5 + √3 is irrational by contradiction?
Not at all, but:HallsofIvy said:? Are you saying that if x+ y is rational then x- y must be rational? What if [itex]x= \sqrt{5}[/itex] and [itex]y= -\sqrt{5}[/itex]?
Jeff Ford said:Is it possible for two irrational numbers to be added together to get a rational number?
Definitely, HallsofIvy gave an example of that.Jeff Ford said:Is it possible for two irrational numbers to be added together to get a rational number?
And what might that be ?apmcavoy said:I don't know about that, but if you divide pi by phi you get a rational (they share a common irrational factor).
Gokul43201 said:And what might that be ?
apmcavoy said:I don't know about that, but if you divide pi by phi you get a rational (they share a common irrational factor).
apmcavoy said:I mean phi2.
http://members.ispwest.com/r-logan/fullbook.html
Jeff Ford said:Is it possible for two irrational numbers to be added together to get a rational number?
HallsofIvy said:I looked at the paper referred to above. There are no proofs at all. The author refers to φ2 as a "composite" number with no explanation of what "composite" is supposed to mean for a non-integer number. All he does is show that 15 decimal place computations come out the same, then spends the rest of the paper defining "fraction", "rational", "irrational", etc.
Millie said:we can prove that √5 is irrational through contradiction and same applies for √3.
but can we prove that √5 + √3 is irrational by contradiction?
Millie said:we can prove that √5 is irrational through contradiction and same applies for √3.
but can we prove that √5 + √3 is irrational by contradiction?
Siyabonga said:show me how by contradiction
The contradiction arises when you try to proof that sqrt(15) is rational.Siyabonga said:show me how by contradiction
An irrational number is a real number that cannot be expressed as a simple fraction. This means that it cannot be written in the form a/b, where a and b are both integers.
There are various methods for proving that a number is irrational. One common method is to assume that the number is rational and then use proof by contradiction to show that this assumption leads to a contradiction. This proves that the number cannot be rational and therefore must be irrational.
No, not all irrational numbers can be proved. There are some irrational numbers, such as pi and e, that are considered to be "unprovable" because they cannot be expressed as an exact value or a repeating decimal. However, we can still use mathematical methods to approximate these numbers to any desired degree of accuracy.
Yes, irrational numbers are very important in mathematics. They are used to solve problems in geometry, trigonometry, and other branches of mathematics. They also have important applications in physics, engineering, and other sciences.
Irrational numbers and rational numbers are different in that irrational numbers cannot be expressed as a simple fraction, while rational numbers can. Additionally, irrational numbers have decimal expansions that neither terminate nor repeat, while rational numbers have decimal expansions that either terminate or repeat.