Let's explore a different way of thinking about multiplication and square roots. Along the way, we'll discover a deeper meaning to the role of imaginary numbers.

We can think of every real number as being situated somewhere along a \(1\)-dimensional number line which reaches infinitely to the right (toward positive infinity) and infinitely to the left (toward negative infinity). The position at \(0\) is a point of reference which divides the real number line into a positive side, comprised of everything to the right of \(0\), and a negative side, comprised of everything to the left of \(0\). Note that \(0\) itself is neither positive nor negative; it is signless.

We can think of every real number as being situated somewhere along a \(1\)-dimensional number line which reaches infinitely to the right (toward positive infinity) and infinitely to the left (toward negative infinity). The position at \(0\) is a point of reference which divides the real number line into a positive side, comprised of everything to the right of \(0\), and a negative side, comprised of everything to the left of \(0\). Note that \(0\) itself is neither positive nor negative; it is signless.

We can use the real number line to view multiplication as a scaling operation. If we start with a real number, \(a\), and multiply it by a second real number, \(b\), we can consider the multiplication to be a scaling of \(a\) by \(b\) relative to \(a\)'s distance from \(0\). Let's start at the most basic position: \(1\).

Any number can be seen as a scaling of \(1\) because scaling by \(1\) preserves the input value; \(c = 1\times c\) for any real number \(c\). So, for instance, we can interpret \(3\) as a scaling of \(1\) by \(3\), ie. to reach the number \(3\) on the number line, the distance from \(1\) to \(0\) is stretched out by a factor of \(3\).

Similarly, \(3\times 2\) is a scaling of \(3\) by \(2\). If we start at \(3\) on the number line, multiplying by \(2\) will scale, or stretch, our position relative to \(0\) by a factor of \(2\). So from \(3\), we then go a second distance of \(3\), which lands us at \(6\). \(3\times 2\) will therefore have a distance of \(6\) away from \(0\).

We see this algebraically as \(3\times 2 = 6\), but rather than thinking of multiplication in a strictly numerical context, let's adopt the perspective of multiplication as a scaling which stretches values along the real number line. Multiplying \(12\) and \(\frac{1}{3}\), for example, scales \(12\) by \(\frac{1}{3}\). In this case, the scaling is a compression rather than a stretching, and we squeeze the \(0\) to \(12\) distance down to one third of its length, which is the distance \(0\) to \(4\). Hence \(12\times \frac{1}{3} = 4\). In this way, we see that division, which can be thought of as multiplication by the numbers between 0 and 1, is just a scaling which produces shorter distances instead of longer ones, eg. \(\frac{12}{3} = 12\times \frac{1}{3} = 4\).

Given a moment of thought, it's easy to see that any scaling of a positive number by another positive number always yields a positive result because the stretching can never pass to the left of \(0\), as \(0\) is our reference point for the scaling. We can stretch any positive real number out to any larger positive real number by multiplying by a scale factor which is greater than \(1\), and we can compress any positive real number down to a smaller positive real number by multiplying by a scale factor which is between \(0\) and \(1\). So how do we reach the negative numbers?

Let's consider another way to move around the number line: reflection. Multiplying by \(-1\) is the same as reflecting the position across the point at \(0\). Let's start back at \(1\).

Given a moment of thought, it's easy to see that any scaling of a positive number by another positive number always yields a positive result because the stretching can never pass to the left of \(0\), as \(0\) is our reference point for the scaling. We can stretch any positive real number out to any larger positive real number by multiplying by a scale factor which is greater than \(1\), and we can compress any positive real number down to a smaller positive real number by multiplying by a scale factor which is between \(0\) and \(1\). So how do we reach the negative numbers?

Let's consider another way to move around the number line: reflection. Multiplying by \(-1\) is the same as reflecting the position across the point at \(0\). Let's start back at \(1\).

Both \(1\) and \(-1\) have a distance of \(1\) from the reference point at \(0\). Multiplying any number by \(1\) preserves the distance of that number, while multiplying by \(-1\) simply reflects the position across the \(0\) point while also preserving the distance from \(0\). So, multiplying \(1\) by \(−1\) reflects the distance to \(1\) across the \(0\) point and lands at \(−1\).

The operation of multiplying by \(-1\) can therefore be viewed as a reflection transformation. \(4\times (-1) = -4\). The distances \(0\) to \(4\) and \(0\) to \(-4\) have the same length, but the two numbers are located on opposite sides of the number line. Multiplying \(4\) by \(-1\) flips our position from a distance of \(4\) to the right of \(0\) to a distance of \(4\) to the left of \(0\). Similarly, \((-4)\times (-1)\) takes \(-4\) to the position on the other side of \(0\) that is the same distance away from \(0\), in this case: \(4\). Thus, multiplying by \(-1\) simply flips to the corresponding number on the other side of \(0\). It's as if when we multiply a number by \(-1\), we are folding the real number line at the point \(0\) and landing wherever our original position touches the other side of the line.

Let's continue exploring. We've just seen that the operations of multiplication and division can be explained as combinations of two types of transformations: the scaling of distances along the number line and the reflection of positions across the reference point located at \(0\). Let's now consider the operation of taking square roots. Taking the square root of any number asks the question: which number, when multiplied with itself, produces the number under the square root symbol? On our number line, we can consider this question in the context of which transformation, when performed twice, produces the number under the square root symbol?

Let's start at \(1\). What operation can we perform twice to land back at \(1\)? If we multiply by \(1\), we're still at \(1\). Multiply by \(1\) a second time, and our position hasn't changed. \(1\times 1 = 1\), so \(\sqrt{1} = 1\). However, it is also true that, starting from \(1\), if we multiply by \(-1\) and then multiply by \(-1\) a second time, we land back at \(1\). \(1\times (-1)\times (-1) = (-1)\times (-1) = 1\), so \(\sqrt{1} = -1\).

Hence, \(\sqrt{1} = \pm 1\). (In fact, the square root of any positive real number has two real number answers: one answer on each side of the real number line, and both answers are equidistant from \(0\).)

As another example, and starting again from \(1\), scaling by \(2\) and then again scaling by \(2\) lands us at \(4\).

Let's start at \(1\). What operation can we perform twice to land back at \(1\)? If we multiply by \(1\), we're still at \(1\). Multiply by \(1\) a second time, and our position hasn't changed. \(1\times 1 = 1\), so \(\sqrt{1} = 1\). However, it is also true that, starting from \(1\), if we multiply by \(-1\) and then multiply by \(-1\) a second time, we land back at \(1\). \(1\times (-1)\times (-1) = (-1)\times (-1) = 1\), so \(\sqrt{1} = -1\).

Hence, \(\sqrt{1} = \pm 1\). (In fact, the square root of any positive real number has two real number answers: one answer on each side of the real number line, and both answers are equidistant from \(0\).)

As another example, and starting again from \(1\), scaling by \(2\) and then again scaling by \(2\) lands us at \(4\).

We can also reach \(4\) by scaling by \(-2\) and then again by \(-2\).

\(1\times 2\times 2 = 2\times 2 = 4\) and \(1\times (-2)\times (-2) = (-2)\times (-2) = 4\), so \(\sqrt{4} = \pm 2\). Notice that, because \(-2 = 2\times (-1)\), multiplying by \(-2\) can be seen as scaling by a factor of \(2\) while also reflecting to the other side of the real number line. So we can evaluate \((-2)\times (-2)\) by starting at \(-2\) and then scaling by a factor of \(2\) (doubling the distance from \(0\)) and then multiplying by \(-1\) (flipping across to the other side of the real number line). This lands us at \(4\).

What about \(\sqrt{-1}\)? Can we find some transformation that, when performed twice, lands us at \(-1\)?

We saw previously that although multiplicative scaling could allow us to reach any positive real number, in order to reach the negative numbers we had to incorporate a new type of transformation: reflection about \(0\) via multiplication by \(-1\). The same will be true here. Reaching the square roots of negative real numbers will require a new type of transformation.

Why is that?

Let's start, as usual, at \(1\). We saw that scaling by \(-1\) will land us at \(-1\), but for a valid square root transformation, we need to be able to perform the transformation twice in order to land on our goal. Scaling a second time by \(-1\) lands us back on \(1\), so we've missed our mark. We could scale by \(-\frac{1}{2}\), which looks like it gets us halfway to \(-1\), but scaling again by \(-\frac{1}{2}\) ends up landing us at \(\frac{1}{4}\).

What about \(\sqrt{-1}\)? Can we find some transformation that, when performed twice, lands us at \(-1\)?

We saw previously that although multiplicative scaling could allow us to reach any positive real number, in order to reach the negative numbers we had to incorporate a new type of transformation: reflection about \(0\) via multiplication by \(-1\). The same will be true here. Reaching the square roots of negative real numbers will require a new type of transformation.

Why is that?

Let's start, as usual, at \(1\). We saw that scaling by \(-1\) will land us at \(-1\), but for a valid square root transformation, we need to be able to perform the transformation twice in order to land on our goal. Scaling a second time by \(-1\) lands us back on \(1\), so we've missed our mark. We could scale by \(-\frac{1}{2}\), which looks like it gets us halfway to \(-1\), but scaling again by \(-\frac{1}{2}\) ends up landing us at \(\frac{1}{4}\).

In fact, we notice that if we scale by a positive number and then again by a positive number, we will always land on a positive number. And if multiply by a negative number and thus reflect across \(0\), we'll have to reflect back into the positive numbers during our second multiplication by that negative number, as the application of a valid square root transformation requires that we perform the operation exactly twice. So we cannot reach the negative numbers through any identical double application of scalings or reflections.

As noted above, we will have to find a new transformation in order to reach them. To do that, we'll need to break out of the first dimension and enter into the world of the second dimension. And we'll need to bring in an entirely new kind of number: the imaginary numbers.

The number \(i\) is the quintessential unit of the imaginary numbers. \(i\) is algebraically defined it to be the square root of \(-1\). When it was first conceptualized nobody knew quite what to think of it, and so it was deemed to be "imaginary". But even though its nature wasn't understood, it was a useful tool in certain calculations, and so it was birthed as an idea in order to make equations more solvable. Let's try to make sense of the mysterious nature of \(i\).

Just as the real number line housed the real numbers, the complex plane is the space in which both real numbers and imaginary numbers exist. Here is the complex plane:

As noted above, we will have to find a new transformation in order to reach them. To do that, we'll need to break out of the first dimension and enter into the world of the second dimension. And we'll need to bring in an entirely new kind of number: the imaginary numbers.

The number \(i\) is the quintessential unit of the imaginary numbers. \(i\) is algebraically defined it to be the square root of \(-1\). When it was first conceptualized nobody knew quite what to think of it, and so it was deemed to be "imaginary". But even though its nature wasn't understood, it was a useful tool in certain calculations, and so it was birthed as an idea in order to make equations more solvable. Let's try to make sense of the mysterious nature of \(i\).

Just as the real number line housed the real numbers, the complex plane is the space in which both real numbers and imaginary numbers exist. Here is the complex plane:

Along the horizontal axis we find our familiar real number line. But now we've added a vertical axis whose basic unit is \(i\) instead of \(1\). \(0\) is still the central point of reference, and \(i\) is located at a vertical distance of \(1\) away from \(0\), just as \(1\) is found at a horizontal distance of \(1\) away from \(0\). \(2i\) is located at a distance of \(2\), \(3i\) is located at a distance of \(3\), and so forth.

By incorporating the second dimension, we've opened up a whole wealth of new possibilities. We're no longer restricted to the linear world of the number line. We now have a new transformation at our disposal: rotation.

Let's start, as usual, at \(1\).

By incorporating the second dimension, we've opened up a whole wealth of new possibilities. We're no longer restricted to the linear world of the number line. We now have a new transformation at our disposal: rotation.

Let's start, as usual, at \(1\).

Just as we defined multiplication by \(-1\) to be a reflection across the point at \(0\), let us now define multiplication by \(i\) to be a \(90^{\circ}\) counterclockwise rotation around the complex plane. So \(1\times i = i\) brings us to the vertical position located at a distance of \(1\) away from \(0\).

A second multiplication by \(i\) will rotate our position again by \(90^{\circ}\).

And there it is. We've just found a way to perform the same transformation exactly twice (our criterion for a valid square root transformation) and land at \(-1\). Thus, \(i\) is a square root of \(-1\).

Just as in the case for square roots of positive numbers, square roots of negative numbers also come in pairs. The other square root of \(-1\) is \(-i\). Convince yourself that this is true by applying the same technique that we used in evaluating \((-2)\times(-2)\) in order to evaluate \((-i)\times (-i)\), but in this case, each time you multiply by \(-i\), first rotate and then reflect instead of scaling and then reflecting. Thus, \(\sqrt{-1} = \pm i\).

Let's look at another example. We saw that \(2\times 2 = 4\) and \(i\times i = -1\). Intuitively, \(2\times 2\times i\times i = 2i\times 2i\) should land us at \(4\times (-1) = -4\).

We'll start at \(1\).

Just as in the case for square roots of positive numbers, square roots of negative numbers also come in pairs. The other square root of \(-1\) is \(-i\). Convince yourself that this is true by applying the same technique that we used in evaluating \((-2)\times(-2)\) in order to evaluate \((-i)\times (-i)\), but in this case, each time you multiply by \(-i\), first rotate and then reflect instead of scaling and then reflecting. Thus, \(\sqrt{-1} = \pm i\).

Let's look at another example. We saw that \(2\times 2 = 4\) and \(i\times i = -1\). Intuitively, \(2\times 2\times i\times i = 2i\times 2i\) should land us at \(4\times (-1) = -4\).

We'll start at \(1\).

\(2i = 2\times i\), so to multiply by \(2i\) we'll first scale by a factor of \(2\) (the transformation associated with multiplying by \(2\)) and then we'll rotate our position by \(90^{\circ}\) (the transformation associated with multiplying by \(i\)).

Performing the transformation a second time by multiplying again by \(2i\), we indeed land at \(-4\).

Similarly, \((-2i)\times (-2i) = -4\). And so \(\sqrt{-4} = \pm 2i\).

We have thus uncovered the transformation which illustrates square roots of negative numbers, just as we did for the square roots of positive numbers. Using our new tools of scaling, reflection across the \(0\) point, and \(90^{\circ}\) rotation, we can now touch down upon any point along either axis of the complex plane. But there are ways to push even further. If we considered addition and explored the transformation associated with that operation, we would find that our reach would step off of the horizontal and vertical axes and expand into all directions, throughout the infinite expanse of the complex plane. We would then have grasp, not just over the real and the imaginary numbers, but over the entirety of the complex numbers.

The imaginary numbers, quite literally, open up a new dimension of operational freedom. So rather than seeing \(i\) and its imaginary friends simply as contrived numerical devices, we can consider them as legitimate, tangible pieces within the greater framework of numbers. In this way, the imaginary number line is just as "real" as the real number line. We just have to tilt our heads a bit to see it.

- - - - - - - - - -

Are there other ways to look at the operations we explored? Can you find different transformations which still validly reproduce the arithmetic results? Can you develop methods for reinterpreting operations which weren't mentioned, such as exponentiation and logarithms?

Are all numbers simply constructs: human creations brought about in order to facilitate organization? Are numbers real?

It is easier to conceptualize \(1\) than it is to intuit the nature of \(i\). We can hold \(1\) in our hand. Is there a way to hold \(i\) or \(\pi\) or \(\sqrt{2}\)? \(1\) and \(i\) both exist at a distance of \(1\) away from \(0\). Does \(1\) have a broader nature?

The imaginary numbers break us out of the first dimension and into the second. They give us an additional dimension of opportunity -- a new way to expand outward and perform operations which were previously inaccessible. By opening up our space, we could move in ways that allowed us to reach previously unattainable goals. Where can you find room to rotate into fresh modes of processing, reacting, and interacting?

How can you open up your own new dimensions? Can you develop personal transformations which break out of a linear world?

Consider potential. Leave restriction and impossibility within the limitations of narrowed perspective, and focus on possibilities. If you find an impasse within your current paradigm, widen your paradigm. The world is beautiful and flowing with undiscovered potential. Let yourself expand and imagine into these new and wondrous spaces.

We have thus uncovered the transformation which illustrates square roots of negative numbers, just as we did for the square roots of positive numbers. Using our new tools of scaling, reflection across the \(0\) point, and \(90^{\circ}\) rotation, we can now touch down upon any point along either axis of the complex plane. But there are ways to push even further. If we considered addition and explored the transformation associated with that operation, we would find that our reach would step off of the horizontal and vertical axes and expand into all directions, throughout the infinite expanse of the complex plane. We would then have grasp, not just over the real and the imaginary numbers, but over the entirety of the complex numbers.

The imaginary numbers, quite literally, open up a new dimension of operational freedom. So rather than seeing \(i\) and its imaginary friends simply as contrived numerical devices, we can consider them as legitimate, tangible pieces within the greater framework of numbers. In this way, the imaginary number line is just as "real" as the real number line. We just have to tilt our heads a bit to see it.

- - - - - - - - - -

Are there other ways to look at the operations we explored? Can you find different transformations which still validly reproduce the arithmetic results? Can you develop methods for reinterpreting operations which weren't mentioned, such as exponentiation and logarithms?

Are all numbers simply constructs: human creations brought about in order to facilitate organization? Are numbers real?

It is easier to conceptualize \(1\) than it is to intuit the nature of \(i\). We can hold \(1\) in our hand. Is there a way to hold \(i\) or \(\pi\) or \(\sqrt{2}\)? \(1\) and \(i\) both exist at a distance of \(1\) away from \(0\). Does \(1\) have a broader nature?

The imaginary numbers break us out of the first dimension and into the second. They give us an additional dimension of opportunity -- a new way to expand outward and perform operations which were previously inaccessible. By opening up our space, we could move in ways that allowed us to reach previously unattainable goals. Where can you find room to rotate into fresh modes of processing, reacting, and interacting?

How can you open up your own new dimensions? Can you develop personal transformations which break out of a linear world?

Consider potential. Leave restriction and impossibility within the limitations of narrowed perspective, and focus on possibilities. If you find an impasse within your current paradigm, widen your paradigm. The world is beautiful and flowing with undiscovered potential. Let yourself expand and imagine into these new and wondrous spaces.