How do I find the squareroot of one million

How do I find the squareroot of one million

Answers
First off, there are no easy ways to find the square root of a random number. Given that a square root is finding a number that multiplied by itself gives you the starting number you can get close(ish) by looking at the number of digits in your final number and looking at the ending digits of your final number. Let me explain...
walterthewhale
21 September 2011
When you square root a number you roughly halve the number of digits in it, e.g., the square root of 49 (2 digits) is 7 (1 digit), the square root of 2500 (4 digits) is 50 (2 digits). However, this is not prescise, e.g., the sqaure root of 625 (3 digits) is 25 (2 digits), the sqare root of 11025 (5 digits) is 105 (3 digits). So that gives you somewhere to start.
walterthewhale
21 September 2011
Now you need to narrow your search a little by looking at the last digit of your target number & deciding what number you need to multiply by itself to get that number. e.g., if your target number ends in 9 you either are squaring a 3 or a 7 (3x3=9, 7x7=49), if your target number ends in a 6 you are squaring a 6 (6x6=36), etc, this process depends upon you knowing your basic square & sqaure roots, (not a bad thing to learn).
walterthewhale
21 September 2011
Then, when you have an idea of where to start, you need to pick some likely looking numbers and start squaring them to see if they are too big or too small. This is called iteration, or trial & error, or trial & improvement, depending upon the level of maths that you are learning it at.
walterthewhale
21 September 2011
Sooooo, lets try it for 1000000 (7 digits) so halve to 3.5, round up (you usually round up in this operation) to 4. 1000000 ends in 0 so we must be squaring a 4 digit number ending in 0. We can be a bit more intelligent in this case because we can see that 1000000 has a lot of 0s, so lets try a 4 digit number with lots of 0s: 2000 squared is 4000000. So that's too big so lets try 1200 sqaured is 1440000. Still too big, so lets try 1000 sqaured, is 1000000. Yeh ! We've got it !
walterthewhale
21 September 2011
This is all assuming that the number has a whole number square root, if it doesn't (e.g., the square root of 8567 is 92.55808987...), then the method only works to a certain extent, ie 4 digits becomes 2 digits of whole numbers (the 92 part of the answer), but finding the decimal places is a hard slog with lots of trial & error. Depending upon the accuracy you need you can give an approximate answer, e.g., 92 sqaured is 8464 & 93 squared is 8649, so you know the answer is somewhere between 92 & 93 & since 8567 is roughly halfway between 8464 & 8649 you could even say that the answer is approximately 92.5, more accuracy will require more trial & error with your calculator. Luckily, greater accuracy is rarely required (or perhaps never required) on exam papers (because you'd waste loads of time on it).
walterthewhale
21 September 2011
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