Reducing FractionsDate: 03/17/97 at 21:19:24 From: SARAH WILLIAMS Subject: Make Sure Fraction is Reduced Dear Dr. Math, How do you check/verify that a fraction is in its simplest form? For example, how do you know that 5/7 is reduced? Thanks, Sarah Date: 03/18/97 at 08:23:23 From: Doctor Jerry Subject: Re: Make Sure Fraction is Reduced Hi Sarah, The answer is that you use what is often called The Fundamental Theorem of Arithmetic, which is: Every integer can be expressed as the product of one or more prime numbers (2,3,5,7,11,13,... are prime numbers; 4,6,8,9,10,12,... aren't) to a power. An integer is a prime if it is divisible (without remainder) by only itself and 1. So, 32, for example, can be written as 2^5 (this means 2*2*2*2*2); 100 is 2^2 * 5^2 and so on. To see if a fraction m/n is in its simplest form, apply the Fundamental Theorem of Arithmetic (FTA) to each of m and n (m is the top part of the fraction, n the bottom). If there are no primes in common to m and n, then m/n is in simplest form. Here are two examples: 49/1547 = 7^2/(7*13*17), not in simplest form. But FTA shows hows to put this fraction in simplest form. Just cancel the extra 7. 49/1547= 7/(13*17) = 7/221. 7/247 = 7/(13*19), no common factors and so in simplest form. Of course, for big numbers it's not so easy to factor a number into its prime factors. You can now see that 5/7 is in its simplest form. Both 5 and 7 are primes. -Doctor Jerry, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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