In comparing numbers, all you need to understand is the decimal number system itself and how it’s been constructed. When you understand that system, everything made in it including numbers, can be dealt with. We talked extensively about the decimal number system previously, so now, let’s get to some examples. Continue reading “Comparing Numbers”
The decimal number system is an elegant way of representing quantities using a system of digits and place values. When you understand how the system is put together, you’ll be able to use it in different ways. Continue reading “How Decimal Number System Works”
We mentioned before that when you square a number, you’ll get a square number or a perfect square. You can use the perfect square to find out what numbers were multiplied together to get the perfect square. The process of using the perfect square to get back to those numbers is finding the square root of a number. For example, 3 * 3 = 3^2 = 9. 9 Here is perfect square. You can take the square root of 9 to get back to 3. You can see that squaring a number and taking the square root of the result are actually inverse processes. Continue reading “How to Find the Square Root of a Number”
If you multiply any integer except for 0 by itself, you’ll get a positive perfect square. For example, 3 * 3 = 9 or you can write the same expression as 3^2 = 9. Now, if you have the number 9, knowing that 4 is a perfect square and want to know what number was raised to the second power or multiplied by itself to get the 9, you need a function that takes you from the perfect square back to the number. That function is called the “square root function.” That means,
3 * 3 = 9 Or 3^2 = 9, then, sqrt(9) = 3. Continue reading “Square Root Function”
Pythagorean Triplets refers to three natural numbers in such a way that the square of one of them (the greatest one) is equal to the sum of squares of the others two. For example 3, 4 and 5 form a Pythagorean triplet. 5^2 = 3^2 + 4^2. Continue reading “Pythagorean Triplets”
Calculating the square of numbers like 8, 9, etc., is easy but calculating the square of a number like 105 without using a calculator.
There is an interesting way to calculate the square of large numbers like 255 without using a calculator. It’s probably not easy to do it in your mind but you can do it on a piece of paper. It’s essentially a way to calculate the square root of numbers that have a 5 in 1’s place value. It does not matter what sort of digits you might have in the other place values but the 1’s place value has to be a 5.
So the formula will be (x5)^2 = x(x+1)100 + 25
The formula above was derived for a 2-digit number but it can be used for any number with any number of digits provided that the number has a 5 in the 1’s place. For example, if you were to calculate 115^2, you’d simply use the formula as follows:
115^2 = 11(12)100 * 25 = 13225
You can learn how this formula was created and based on that, you can create any sort of formula for any kind of situation you might find yourself in.
It turns out that you can write almost any squared number in terms of the product of two odd or even integers plus one. As an example, let’s go through the following operation:
29 * 31 = (30 – 1)(30 + 1) = 30^2 – 1 because (a – b)(a + b) = a^2 – b^2
30^2 = 29 * 31 + 1
So as you can see here, you can write 30^2 as the product of two consecutive odd natural numbers, 29 and 31 and add 1 to the product. You can do this with almost every square number. I am using the word “almost” here since I have not taken the time to test every possible case not have I proved this nor have I seen a proof for it.
Wherever this happens to come in handy depends on the kind of problems you might be solving.
You can express any odd perfect square in terms of the sum of two consecutive natural numbers. For example, 3 squared or 9 can be expressed as (4 + 5). 5 squared or 25 can be expressed as (12 + 13) and so on.
What this means, is that we can express the square of any odd number as the sum of two consecutive natural numbers.
We cannot do this for the square of even numbers because the square of an even numbers is always an even number. An example would be 16. 16 is a perfect square and it can be expressed as (8 + 8). 8 and 8 are NOT two consecutive natural numbers but (12 + 13) are two consecutive natural numbers added together to get to 25 which is a perfect square.
One important thing to notice here is that the inverse of the rule above is not always true, so it’s not true at all. Meaning that, (12 + 13) would be 25 which is a perfect square but let’s pick to other consecutive natural number like (13 + 14) which is 27. 27 is not a perfect square. So while any odd square number can be expressed as the sum of two consecutive natural numbers, not any two consecutive natural numbers added together would result in an odd perfect square.
There is a relationship between the number of odd natural numbers starting from 1 added together and square numbers. That means, 1, the same things as 1 squared is the same thing as the first odd natural number. And you know that 1 is a perfect square. 4, which is the same thing as 2 squared, can be written as (1 + 3) which is the first two natural odd numbers starting from 1 added together. 9, which is the same thing as 3 squared, can be written as (1 + 3 +5), which is the same thing as the first three natural odd numbers starting from 1 added together and so on and so forth.
What this means is that if you add the first n natural odd numbers starting from 1 together, the result can be expressed as n squared.
You can use this observation in different ways in different situations to check whether a number a perfect square or not. For example, if you were to test whether 121 is a perfect square, you’d check whether 121 can be expressed as a sum of some odd natural numbers starting from 1. If you tried this, you’d find out that the number of those odd natural numbers, in this case, would be 11, namely 1 through 21 or 1, 3, 5, 7, 9, … , 21. If you add all those numbers together, you’d get 121. So you can conclude that 121 is perfect square. Moreover, sqrt(121) = 11.
It’s a simple but useful skill to be able to find the number some numbers between two numbers in a number set. This could take all shapes or forms. For example, sometimes, you might need to know the number of numbers from say a to b, including a and b. I’ll talk about two of such situations here. Remember that any other type of situation that comes your way, you can use simple logic and test some simple cases and generalize your findings to more complicated cases to find the rule you’re looking for.
Calculating the Number of Numbers from a to b, Including a and b
Now, let’s say, you want to find the number of numbers from a to b, including a and b. In that case, you could think of a simple case like all the numbers from 1 to 10. Those numbers would be, 1, 2, 3, 4 5, 6, 7, 8, 9, 10. If you count them, you’ll find out that the number of those numbers is 10. So now that you know how many numbers are actually there, you could find some algebraic relationship an test it against the actual number of numbers that you have. If I subtract the first number, 1, from the last number, 10, I’d have 10 – 1 = 9 but I have 10 numbers there, not 9. So I add 1 to 9 to get to 10. Test this for any other case starting from any number and ending anywhere and it will work.
So (i) if you are working with a set of numbers like natural numbers where it’s possible to know what numbers are there and how many numbers there are between any two given numbers, and if, (ii) the numbers under study are happening based on a specific rule, then you can use the above solution to figure out the number of those numbers under study.
What (i) means is that, for example in the set of natural numbers, you know that numbers start from 1 and go all the way up to infinity. At every step, 1 is added to every number to get the next number. So the set looks something like, 1, 2, 3, … Moreover, you know that there is no number between 1 and 2 for example in this set of numbers. If you talk about the set of rational numbers, then there’s no way to know how many numbers there are between 1 and 2. The number would actually be infinite. So there, the whole solution would be useless.
What (ii) means is, that for example the numbers under study start from 1 and go all the way up to 101 or the numbers start at 1 and go all the way up to 100 missing every other number, 1, 3, 5, 7, … , 101. Here the progression of numbers is happening based on some uniform rule and so it’s possible to figure out the exact number of numbers.
Getting back to our original problem, figuring out the number of numbers from a to b, numbers happening based on a rule similar to 1, 2, 3, … , b, you could write the number of those numbers as (b – a + 1)
Calculating the Number of Numbers Between a and b, Excluding a and b
This would be the exact same situation, except that you’d need to make some changes to the formula you got in the previous step. Since in this case, you have to subtract two numbers from that sum (you have to exclude a and b from the sum), you’d end up with (b – a + 1) -2 which is equal to (b – a – 1).