Estimate and guess on the numbers

Guess and Estimate is an important ability in math.

Sometimes you do not know the exact number, but you can guess at it by some clues (for example, the last digit, or the divisors of the number); and you can pin down the number if there are some estimate available to give you the upper / lower bound.

Let us look at a couple of problems as possibly to apply Guess and Estimate:

(1) Suppose n2 = 572 + 282 – 82
We know n is an integer, 57 < n < 65.

What is the value of n?

(2) Suppose that n is an integer,

42 sqrt 5 – 33 < n < (21 + 48 ⁄ 7) sqrt 5

What are the possible values of n?

The clues are as follows:

For (1), pay attention to the last digit (the units digit for n2 is 9).

For (2), to facilitate an estimate, it is helpful to know that sqrt 5 is slightly less than 9 ⁄ 4.

This estimate will require certain fraction calculations, though.

Another approach to (1) is to notice that:

n2 – 572 = 28 – 8 2 so
(n + 57) (n – 57) = (28+8) (28-8) = (36) (20)

The answers to the problems (1) and (2) are as follows: (your task is to quickly figure out WITHOUT USING A CALCULATOR; and better NOT USING INVOLVED CALCULATION)

(1) Integer n = 63; (2) Possible values of n is 62 or 63.

Lastly, you might find the following estimate on the radicals pretty interesting (and helpful in some situations):

sqrt 2 = (7 ⁄ 5) +,

sqrt 3 = (7 ⁄ 4) , or sqrt 3 = (5 ⁄ 3) +, and

sqrt 5 = (9 ⁄ 4)

where + means slightly larger, and means slightly smaller.