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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).
Explain whether it is easier to estimate the product 13.72 x 47.28 by using compatible numbers or by rounding each factor to the nearest whole number?
Compatible numbers would be easier. Rounding gives you 14 x 47. Compatible numbers could be 13 x 50 which would be closer to the actual product.
Could this ever be the rule of a function For input x the output is the number whose square is x If so what is its domain and range?
Yes, the domain(input) would be all natural numbers (numbers greater or equal to zero). The range (output) would be all real numbers. -- Not only natural numbers would be considered part of this domain, all negative numbers are also reasonable inputs to this function, as any negative number multiplied by itself would produce a positive number..... The output (range) would therefore be all positive real numbers......
Are all whole numbers also natural numbers?
That depends on whom you ask. The idea of what is "natural" evolves over time. I would answer yes, others here would answer no.The science of Mathematics has many different branches, developed over thousands of years. There is disagreement in some of those branches as to the definition of natural numbers, specifically whether or not to include zero. Some authors begin the natural numbers with 0, corresponding to the non-negative integers 0, 1, 2, 3, …, whereas others start with 1, corresponding to the positive integers 1, 2, 3, …. Texts that exclude zero from the natural numbers sometimes refer to the natural numbers together with zero as the whole numbers, but in other writings, that term is used instead for the integers (including negative integers).For the sake of clarity, I use "counting numbers" to indicate the set of non-negative integers that doesn't include zero and avoid the term "natural numbers" altogether, but that won't help you on a test. Answer whatever your teacher says, but if you were marked wrong for answering "yes" you could inquire about the Peano axioms or the von Neumann ordinal construction although we won't be held responsible for the result of that conversation if your teacher hasn't heard of them.Yes
The difference between these two numbers equals 435?
The numbers could be 436 and 1.Or there are billions of other pairs that they could be..
