Last week in Number Sense 033, we took a look at an equation that produced a parabola when it was plotted on Cartesian coordinates. This opens up the possibility of a situation having two solutions, or no solutions. This week I'd like to change course a bit, and begin a discussion about how we know mathematical things.
Let's suppose we add up the first five counting numbers, 1, 2, 3, 4 and 5. Here is a picture:
We can count the squares, there are 15 of them. 1 + 2 + 3 + 4 + 5 = 15
That seems to be an unusual number to get from an addition of five numbers. But we might notice that 15 is 5 times 3. The middle number in the series is 3, as well. We might also notice that if we leveled off our diagram, pushing the highest three squares into the empty spaces, we would get a rectangle that was three squares high, and five squares wide:
Our multiplication facts tell us that 3 x 5 = 15. So counting, addition and multiplication all agree that the number of squares in this particular case is 15.
Now let's try another number, say, 4.
Counting the squares give us 10. But there is no middle number, as there was when we worked with five. We can begin to level this pile of squares by taking the highest square and filling a gap, but
what do we do with the two on top now? Slide them to the other side?
That doesn't seem to help. Our pile of squares is four wide, and we are looking for 10 as the product of the width and the height. Our division facts tell us that 10/4 is 2 ½, not a whole number. If we want to work with whole numbers, we may want to multiply by two in the beginning, then divide by two at the end.
Let's see what multiplying by two gives us:
If we rotate one of these shapes, we can fit it onto the other and make a rectangle:
Now we have 4 x 5 = 20, then divide by 2 = 10. No fractions involved. Does this also work for our earlier example, using 1,2,3,4,and 5?
Multiply by 2
Rotate and fit together into a rectangle which is 5 x 6 = 30 squares. Divide by 2 to get 15.
Can we start with different numbers? Say, 2 instead of 1.
Let's look at the series 2 + 3 + 4 + 5
Here is our result: a rectangle that is four wide and seven high. 4 x 7 = 28, divide by 2 = 14.
Count the red squares, there are 14.
It seems we can determine the sum of a series of numbers by multiplying two numbers then dividing by two. The two numbers we multiply are, first, the amount of numbers in the series, and second, the sum of the first and last numbers.
How can we write a mathematical expression for that statement?
We can assign letters to represent quantities that we need to know: the first number shall be “a”.
Now, there are two more numbers mentioned: the amount of numbers in the series, and the last number. These are not independent: if I know the amount of numbers and the first number, I can calculate the last number. If I know the last number and the first number, I can calculate the amount of numbers in the series.
I'm going to choose the amount of numbers in the series. When mathematicians write about series of numbers, they generalize by using the amount of numbers in the series, rather than the last number. I follow that lead.
So, we have a series:
a + (a+1) + (a+2) + … + (a+n-2) + (a+n-1)
The last number is (a+n-1) because our beginning number is a, which adds an additional number to the series. If we ended the series with (a+n) we would have one too many numbers in it.
So, the sum is (n) times (a + (a+n-1)) all divided by 2 or
Let's see if that works for our examples: 1 + 2 + 3 + 4 + 5 and 2 + 3 + 4 + 5
First example a = 1, n = 5, last number = 1 + 5 – 1 which is 5.
Second example a = 2, n = 4, last number = 2 + 4 – 1 which is 5
So both our examples check out. Our formula is complicated. Let's simplify it.
Let's draw a picture of our simplified formula
The red squares on the bottom are a times n. The red and blue squares in the top rectangle are n squared. Then we subtract n (crossed out a line of blue squares) and divided what's left by two. Half are blue, half are red. Adding all the red squares gives us the sum of 2 + 3 + 4 + 5
So it appears that our formula for finding the sum of a series of counting numbers has a physical representation, using the colored squares.
Do we know, mathematically, that it's true?
Unfortunately, the answer is, not yet. We have demonstrated the formula for a couple of cases, and, given the way the physical representation fits the formula, our intuition tells us that it's likely true, but we haven't proved it mathematically.
In order to prove this, we must start out with some simple but general representation, something everyone can see is true, and then, step by step, show that beginning transformed into the formula. Each step, of course, must also be valid and true.
We will save that for next week, unless someone wants to take it on in the comments. Have fun.