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Which type of numbers can be written as a fraction plq where p and q are integers and q is not equal to zero?
rational numbers
Explain why every natural number is also a rational number but not every rational number is a natural number?
The natural numbers (ℕ) are the counting numbers {1, 2, 3, ...} (though some definitions also include zero: 0) which are whole numbers with no decimal part. Every rational number (ℚ) can be expressed as one integer (p) over another integer (q): p/q where q cannot be 0. The rational numbers can be converted to decimal representation by dividing the top number (p) by the decimal number (q): p/q = p ÷ q. When q = 1, this produces the rational numbers: p/1 = p ÷ 1 = p which is just an integer; it could be one of {[0,] 1, 2, 3, ...} - the natural numbers above: thus all natural numbers are rational numbers. When q = 2, and p = 1, this produces the rational number 1/2 = 1 ÷ 2 = 0.5 which is not one of the natural number above - so some rational numbers are not natural numbers, thus all rational numbers are not natural numbers. Thus ℕ ⊂ ℚ (the set of natural numbers is a proper subset of the set of rational numbers).
Difference between rational and irrational?
Rational numbers can be expressed in the form p/q (where q is not equal to zero). Irrational numbers cannot be expressed in this form. For example, the square root of 2 cannot be expressed as p/q.
Is it possible to divide one rational number by another to obtain an irrational number as a quotient?
No. Rational numbers are defined as fractions of whole numbers. Suppose we have two rational numbers A = m/n and B = p/q. Then their quotient is defined as A/B = (m*q) / (n*p). Since m,n,p and q are whole, the products m*q and n*p are whole as well, making A/B a rational number.
What letter is used to notate the set of a rational number?
ℚ (fancy capital Q) is the set of rational numbers.
