The p in PS means post script and you write it afer leters for things you forget.
If you are referring to the old expression, "mind your p's and q's" it stands for pints.
The plural form is Ps and Qs.The plural possessive form is Ps and Qs'.Example: Your Ps and Qs' training seems lacking.
PS stands for postscript. It is an additional remark or information added after the main body of a letter has been written.
The English letter 'p' does not exist in Japanese, though there are syllables that incorporate the sound. It does not stand for anything in Japanese.
There is none in English, not counting compound words.
In the word "psychology," the "P" is silent because it is derived from the Greek word "psyche." The English language borrowed this word along with its silent "P." In words like "pneumonia" or "psychiatry," the "P" is silent as well, following the same etymological pattern.
The plural form is Ps and Qs.The plural possessive form is Ps and Qs'.Example: Your Ps and Qs' training seems lacking.
Suppose p/q and r/s are rational numbers where p, q, r and s are integers and q, s are non-zero.Then p/q + r/s = ps/qs + qr/qs = (ps + qr)/qs. Since p, q, r, s are integers, then ps and qr are integers, and therefore (ps + qr) is an integer. q and s are non-zero integers and so qs is a non-zero integer. Consequently, (ps + qr)/qs is a ratio of two integers in which the denominator is non-zero. That is, the sum is rational.
Suppose p/q and r/s are rational numbers where p, q, r and s are integers and q, s are non-zero.Then p/q + r/s = ps/qs + qr/qs = (ps + qr)/qs.Since p, q, r, s are integers, then ps and qr are integers, and therefore (ps + qr) is an integer.q and s are non-zero integers and so qs is a non-zero integer.Consequently, (ps + qr)/qs is a ratio of two integers in which the denominator is non-zero. That is, the sum is rational.Also p/q * r/s = pr/qs.Since p, q, r, s are integers, then pr and qs are integers.q and s are non-zero integers so qs is a non-zero integer.Consequently, pr/qs is a ratio of two integers in which the denominator is non-zero. That is, the sum is rational.
Suppose x and y are two rational numbers. Therefore x = p/q and y = r/s where p, q, r and s are integers and q and s are not zero.Then x - y = p/q - r/s = ps/qs - qr/qs = (ps - qr)/qsBy the closure of the set of integers under multiplication, ps, qr and qs are all integers,by the closure of the set of integers under subtraction, (ps - qr) is an integer,and by the multiplicative properties of 0, qs is non zero.Therefore (ps - qr)/qs satisfies the requirements of a rational number.
Suppose x and y are rational numbers.That is, x = p/q and y = r/s where p, q, r and s are integers and q, s are non-zero.Then x + y = ps/qs + qr/qs = (ps + qr)/qsThe set of integers is closed under multiplication so ps, qr and qs are integers;then, since the set of integers is closed addition, ps + qr is an integer;and q, s are non-zero so qs is not zero.So x + y can be represented by a ratio of two integers, ps + qr and qs where the latter is non-zero.
Suppose A and B are two rational numbers. So A = p/q where p and q are integers and q > 0 and B = r/s where r and s are integers and s > 0. Then A - B = p/q - r/s = ps/qs - qr/qs = (ps - qr)/qs Now, p,q,r,s are integers so ps and qr are integers and so x = ps-qr is an integer and y = qs is an integer which is > 0 Thus A-B can be written as a ratio of two integers, x/y where y>0. Therefore, A-B is rational.
to be careful how you behave
The term Keeping up with your Ps and Qs is generally quoted as Minding your Ps and Qs. This is an old term, which means to Mind your Pints and Quarts, which means to mind your own business, basically, or to take care of a task.
If the two rational numbers are expressed as p/q and r/s, then their sum is (ps + rq)/(qs)
Yes.Suppose a and b are two positive rational numbers. Then a can be expressed in the form p/q where p and q are positive integers, and b can be expressed in the form r/s where r and s are positive integers.Then b - a = r/s - p/q = (qr - ps)/qs.Now, since p, q, r and s are integers, thenby the closure of the set of integers under multiplications, qr, ps and qs are integers;q and s are positive => qs is positive, andby the closure of the set of integers under addition (and subtraction), qr - ps is an integer.That is, b - a = (qr - ps)/qs is a ratio of two integers, where the denominator of the ratio is positive.
Mind your Ps and Qs means to use good manners.
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