Finding the value of numbers has fascinated people for ages. How do you determine the value of a number? What does the number mean?
Numbers can be defined in many ways. A number can be how many objects there are, it can be a quantity with a specific value, or it can be a code or representation of something.
The last definition is one of the most interesting to me. How does a number represent something? How do we use numbers to identify things and understand their worth?
Numbers can represent dimensions, values, ratios, and so much more. These attributes of numbers make them very versatile and useful in identifying things and comparing them.
This problem solving question will ask you to identify a number based on its square and another number based on their sum.
Subtract 12 from 7 times the number
Now let’s talk about the last part of the equation: What is the number? This part is a little tricky, but we will break it down for you.
The number is whatever number you end up with when you subtract 12 from 7 times the number. So if the number was 10, then what is the number would be -2.
It sounds weird, we know, but it makes more sense when you do it.
Say you start with 10 and multiply it by 7. You would get 70 as your final number. Now, to find what is the number, take away 12 from that result. So 70 – 12 = 58 = what is the number.
Divide by 2
The next step is to divide the number by 2. In this case, that would be 12.
Divide the number you got in the last step by 2 and write down the remainder if there is one. In this case, 12/2=6 with a remainder of 2.
Now, repeat the last two steps until you get to 1. This will take three more steps!
Square root of 7 is 3, so write down the 3 and divide 1 by 2, which is 1 with no remainder. Now we are done!
The mystery number was 1! Try it again with a different number to see if you get a different answer.
Take the square root of both sides
Now, let’s look at the problem above. First, we have to take the square root of both sides.
We can do this by factoring both sides and taking the square root of each side, or we can use the Square Root Property of Equality to do it.
To use the Square Root Property of Equality, you have to first write out that property (a=b implies that b=a^2). Then, you have to write out both sides as squares and add them together. The side with the equal sign has to be a square number, so you have to adjust the other side if needed.
For this problem, we can either factor both sides or use the Square Root Property of Equality.
Confirm using algebra
Now that you know the number, let’s confirm your answer using algebra.
The square of 12 is 144, and 12 times 7 is 84. So, adding those together gives us 128.
We know the original number was a multiple of 8 because we divided by 8 to get the answer. This means the original number must have been an even number.
Since our original number was an even number, we can divide it by 2 to get its opposite odd number. We will use the square root symbol (√) to denote that we are taking the square root of the original number instead of our answer of 128.
128 = √OriginalNumber
Square both sides of the equation by multiplying both sides by 128.
OriginalNumber = √128
We can now confirm that our original number is 128!
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Try a few numbers to check your result
Now that you know the answer, try a few numbers to make sure your method works every time. For example, try 5, since 5² is 25. 5 × 7 is 35, which is 12 less than 50.
If you square any number from 1 to 50, the difference between the number and its square is always 12. That’s pretty cool!
By the way, if you square any number from 1 to 100, the difference between the number and its square is always 12. That’s even cooler!
Try it for 500 or 1,000 and you’ll get the same result. It works for any integer up to 1,000! It seems this mysterious number is 1,000.
What is the significance of this formula?
This formula shows that the square of any number is related to the number itself, to the number one, and to the number twelve. This makes sense, as the square of any number is the number formed by multiplying that number by itself.
The relation to one shows that any number has a value of one. The relation to twelve shows that there are twelve numbers that have a square value of one.
Any whole number can be expressed as a product of these twelve numbers: 1 × 2 × 3 × 4 × 5 × 6 × 7 × 8 × 9 × 10 × 11. For instance, 28 = 1×2×14, so 28 is made up of ones, twos, threes, fours, fifteens, and eights.
The relation to seven shows how we get the formula we did: by multiplying seven by seven by seven by seven by seven by seven by seven. By doing this process until you get down to one unit left, you get one thousand units.
What are some applications of this formula?
This formula can be applied in many different situations. For example, you can apply this formula to find the area of a quadrilateral (a rectangle plus two congruent triangles).
Since the length of the sides of the quadrilateral are given, you can set up the formula for the area using these sides and get an answer.
You can also use this formula to find the volume of a parallelepiped (a box-shaped figure with six faces that are all congruent).
Once again, set up the formula for the volume using given dimensions and you will get an answer.
This formula is very useful in mathematics and even has applications in physics. It is important to know it and understand it well.
How could you extend this formula to get a closer approximation of the square root?
By adding an additional factor to the equation, you could get a closer approximation. For example, you could try x/x + 1, where x is the number you are trying to find the square root of.
This is because x/x is the perfect square root of x, plus 1 is the number that times x is equal to x. So when you take the square root of both sides, you get the number you are looking for!
By doing this with all numbers between 1 and 100, you can create a chart of 100 squares that gets closer and closer to the true square root of 100.
