"Rationals" Example Sentences
1. The set of rationals includes all fractions that can be expressed as a ratio of integers.
2. Rationals can be added, subtracted, multiplied, and divided just like any other number.
3. To simplify a fraction, the numerator and denominator must be divided by their greatest common divisor, which must be a positive integer for rationals.
4. The sum of any two rationals is always a rational number.
5. Irrationals cannot be expressed as a ratio of integers and therefore are not included in the set of rationals.
6. The decimal expansion of some rationals is finite, while others are infinite and repeating.
7. When comparing two rationals with the same denominator, the one with the larger numerator is considered to be greater.
8. The set of rationals is denoted by the symbol Q.
9. The set of integers is a subset of the set of rationals.
10. A rational function is a function that can be expressed as the quotient of two polynomials, where the denominator is not equal to zero.
11. The product of any two rationals is always a rational number.
12. The reciprocal of a non-zero rational number is also a rational number.
13. The difference of two rationals is always a rational number.
14. The set of rationals is countable, meaning that it can be placed in a one-to-one correspondence with the set of natural numbers.
15. The order of rationals is preserved under addition and multiplication, meaning that if a > b and c > 0, then a + c > b + c and a c > b c.
16. The set of rationals is dense in the real number system, meaning that between any two real numbers there exists a rational number.
17. The sum of any finite set of rationals is always a rational number.
18. Some famous irrational numbers include pi and the square root of 2, which cannot be expressed as a ratio of integers.
19. Rationals are closed under addition, subtraction, and multiplication, but not division by zero.
20. The product of any finite set of rationals is always a rational number.
21. The opposite of a rational number is also a rational number.
22. The quotient of two non-zero rationals is always a rational number.
23. An improper fraction is a rational number where the numerator is greater than or equal to the denominator.
24. The sum of the reciprocals of two distinct rationals can never be a rational number.
25. The set of rationals includes both positive and negative numbers.
26. Any finite subset of the set of rationals has a rational number as its average value.
27. The greatest common divisor of the numerator and denominator of a rational number can always be reduced to 1.
28. The set of rationals is not closed under division, since dividing by 0 produces an undefined result.
29. Any positive integer can be expressed as a sum of distinct unit fractions, which are rational numbers where the numerator is 1.
30. There exist infinitely many distinct pairs of rationals whose sum is an integer.
31. An equivalent fraction is a rational number that can be obtained by multiplying the numerator and denominator of a given fraction by the same non-zero integer.
32. Rationals can be represented geometrically on a number line.
33. The sum of any set of rationals where the denominators have a common multiple is also a rational number.
34. The product of any set of non-zero rationals can be written in lowest terms by canceling out common factors in the numerator and denominator.
35. The exponential function with a rational base is always a positive real number.
36. The identity element for addition of the set of rationals is 0.
37. The identity element for multiplication of the set of rationals is 1.
38. The absolute value of a rational number is always a positive rational number.
39. Any non-zero rational number can be expressed as a product of its absolute value and a rational number with a magnitude of 1.
40. Any rational number can be written as a finite sum of positive and negative unit fractions.
Common Phases
1.
Rationals are individuals who base their decisions on reason, logic, and facts;
2. It is important to engage in rational thinking to reach an objective conclusion;
3.
Rationals value evidence over emotions or gut feelings;
4. A rational approach to problem-solving can lead to a more efficient and effective solution;
5.
Rationals tend to prioritize intellect and rationality over social norms or traditions;
6. The scientific method and critical thinking are essential tools for rational analysis.