What is the difference between squaring a number and doubling a number

In this activity, students look for patterns in square tile arrangements. Showing differences with tiles, multilink cubes, or drawings on squared paper is likely to help the students see the various patterns. In question 1, the tile arrangements relate to the differences between consecutive square numbers. When 32, represented by a 3 by 3 square, is compared with 2 2 a 2 by 2 square , the difference is 5 squares, represented as an L-shaped design.

The differences between two consecutive square numbers, for example, 2 2 — 1 2 , 3 2 — 2 2 , or 4 2 — 3 2 , can be shown as follows:. By considering the equations or the diagrams, the students may see a simple rule for these calculations, in which the difference between the numbers being squared is 1.

The rule is: add the two numbers that are being squared. The students may see a simple rule for these calculations, in which the difference between the numbers that are squared is 2. The rule is: double the sum of the two numbers that are squared.

Once again, the diagrams above will help the students to understand geometrically why the rule works. In question 3, the pattern for the difference between any square number and the third following square number can be shown as follows:. The students may see a simple rule for these calculations, in which the difference between the numbers that are squared is 3. The rule is: treble the sum of the two numbers that are squared. An exponential curve. Doubling time is the time it takes for something, growing at a steady rate, to double in size.

The reason it is so important is because doubling or halving is a much easier concept for us to comprehend whereas the exponent itself may not be. Hold down Alt and key in and let go of Alt.

A superscript 2 will appear. Incidentally, if you needed 'cubed' instead of 'squared' then type and you'll get a superscript 3. In fact, this will work anywhere in Windows or online — even in Word. Does doubling mean squaring? Asked by: Wilhelm VonRueden. How do you type 2 squared? What is the formula for doubling? Is a increase doubling? What is a doubling sequence?

Why is doubling cubes impossible? Is a perfect square? What are the first 3 square numbers? How do you teach doubling and halving? Here is an attempt to define the reverse process, finding square root, using the word "itself":. But I hope you get the drift.

Since this question hinges directly on some fundamental ideas of math, this answer attempts to explicate those ideas in a similarly fundamental way. Squaring a number can be thought of as a procedure. The particular procedure for squaring a number can use a template like the following:. Notice that these two boxes must each contain the same number.

Finally, we perform the indicated multiplication and write the result in the last box on the right:. To take a square root, we want to reverse the procedure, that is, work it backwards. Now we have to decide what to put in the two boxes in the middle.

Then we have:. Now we just need to deduce what number was in the leftmost box. So we have. We may later learn how to find square roots in a way that does not rely so much on making a lucky guess. But that's a matter of an algorithm for calculating a square root, not the definition of a square root. The name "square" is very consistent. And units are squared too! You can better interpret this as different dimensions, combined as powers to numbers and units.

When taking the root, you divide the power by two. Another question is: can one imagine a square with negative sides? Because its area is the same as the one with positive sides. In this form it has nothing to do with division. In fact, we don't even need to know what division is, to formulate it. Addition: So to explicitly answer the question - no, we should not mean to divide a number by itself when taking a square root, because it won't satisfy solve that equation.

Consider the following question, which is just yours with simpler operations substituted. Then you will hopefully understand. If doubling a number means adding that number to itself, then shouldn't halving a number mean to subtract a number from itself? And if you've got this, here is some more to contemplate. Suppose your boss proposes to raise your salary by any percentage you like, with as only condition that at the end of the year it will be lowered again by the same percentage; which percentage would you choose?

How so? The problem you have noticed is not uncommon at all. It is the problem of specifying the logic that looks plausible for one particular case but that cannot be generalized. You agree that by the squaring a square root, you need to come back to the original number. Obviously if we define it your way, you will not. Multiplication and division are not the opposite operations. What would my number be? You have to distinguish between ordinary language and technical language.

For convenience, ordinary language is mixed with technical language, but you must guard against being misled by ordinary language. For example, is land travel the opposite of sea travel? Applied from left-to-right, the distributive property is called multiplication, but applied from right-to-left, it is called factoring. The worst or, best example of a disconnect between ordinary language and mathematical language concerns divisors of 0. According to the definition of a divisor, 23 is a divisor of 0, as is Therefore, based on ordinary language, the real number system has divisors of 0, but according to mathematical language the real number system does NOT have divisors of 0.

Another good bad example of the disconnect between ordinary language and technical language is the difference in meaning between the formula for simple interest and the formula for compound interest: the formula for simple interest gives you exactly what it says, but the formula for compound interest gives you the total growth amount so, to get the amount of compound interest, you have to subtract the principal from it.

This example of the disconnect between ordinary language and technical language has the advantage of not requiring familiarity with ring-theoretic considerations. When you 'square' a number, you multiply a number by itself. But, when you derive the 'square root' of a number, you essentially find a number which, when 'squared', will give the number we're taking the 'square root' of. I think geometrical analogy can help you conceptualise. Think of 'squaring' as finding the area of a square with a certain length of sides while 'taking square root' will refer to finding the length of the sides of a square with a certain area.

It seems that in general, operations like multiplication and inverse operations like division are only valid inverses of each other when dealing with a given reoccurring value.

The exact same value will have to be a given operand to both operations. Therefore, in this context where there is no such reoccurring operand the inverse relationship is no more valid. Mathematics is entirely built on definitions. That's not a square root because it by definition is not a square root. In its most simple terms, a mathematical expression is a statement that can represent a single number.

In the latter example, the expression contains a variable, n. We say that this expression is a ' function of n', which we can write as f n or g n , or h n , or myfunction n , etc. An equation is a statement of equality between two different mathematical expressions. If an equation contains one unknown, like the latter two examples above, then there should be one or more values that this unknown can take in order for the equation to be satisfied i.

These are called the ' roots ' of the eqaution. By definition, the act of squaring a number, n, can be written as a function let's call this 'square'.