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# 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). Study guides

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## A number a power of a variable or a product of the two is a monomial while a polynomial is the of monomials

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