Seeing complex numbers

A number line is a picture that represents numbers as points on a straight horizontal line. As can be seen in the image, the numbers increase as you move to the right along the line and decrease as you move to the left. Zero is represented in the middle of the line, bisecting the positive and negative numbers. Fractions and irrational numbers are also representable on this line between the whole number points.

Operations such as addition, subtraction, multiplication and division are simply transformations of a number on this line. For example, to add 2 to the number 3 means to start at the number three point and then move two positions to the right along the line (in geometrical terms, this is called translation). To multiply 5 by 2 is to start at the number five point and then move right until you are twice the distance from the origin (geometrically, this is called scaling).

In the sixteenth century, a new kind of number called a complex number was introduced by Italian mathematician Gerolamo Cardano. It is expressed as a + bi where a and b are real numbers and i is termed the imaginary unit which satisfies the equation i ² = -1. As negative numbers have applications in areas such as temperature measurement and finance, so complex numbers apply in areas such as physics and fractal geometry.

But is it possible to have an intuitive understanding of complex numbers? Didn't we learn in school that any number (including a negative number) multiplied by itself is always a positive number? So how can i ² = -1 make any sense? What could i be?

The first step to understanding complex numbers is to extend our concept of the number line to a number plane by adding a second dimension that cuts through the number line origin. As shown in the image, this is called the imaginary dimension with i as the unit and it is perpendicular to the real dimension.

The second step is to introduce rotation around the origin as a new transformation operation. When we rotate a number on the real number line by 90^o in an anti-clockwise direction, we can see that it ends up on the imaginary number line. This operation is what it means to multiply by i. If we rotate a further 90^o in an anti-clockwise direction (that is, multiply by i again), then we end up back on the real number line. But we are now on the opposite side of the origin. That is, if we start at 3, multiply by i, then multiply by i again, we end up at -3. That is, 3i ² = -3. The effect of multiplying by i ² is simply to negate our original number by rotating 180^o to the other side of the origin. If we multiply by i ² again, then we pass through -3i and end up back where we started at 3.

Transforming a negative number into a positive number requires translation on the real number line. Transforming a complex number into a real number requires rotation on the real-imaginary plane. So what is i? The real number one rotated by 90^o onto the imaginary number line. That's all it means.

Seeing negative numbers

When I lived in New York City, I had the opportunity to tutor neighborhood children in subjects such as math and reading. On one occasion, the boy I was tutoring was struggling with adding negative numbers. The problem was that he lacked an intuitive grasp of what the operation meant.

The concept of adding positive numbers is easy to understand. We can take three apples, add two apples and see that we end up with five apples. We can also subtract two apples and end up with three again. But what happens if you subtract ten apples from five apples? Or add negative two apples? This no longer makes intuitive sense.

The solution is to find practical applications where the concept of negative numbers does make sense. I noticed that the floor was constructed of tiles. So I got my student to stand on a tile which we marked as the starting point (this was our number line origin). Taking a step forward to the next tile corresponded to adding the number one. Taking a step backward corresponded to subtracting the number one. It was then possible to step backwards until he was one or more tiles behind the starting point. Unlike with the apples scenario, it was now possible to "see" a negative number and meaningfully add and subtract with them.

Other practical applications of negative numbers include calculating budget deficits (as against surpluses) and measuring temperatures below freezing point. Essentially, negative numbers are applicable whenever the subject matter can be considered in terms of opposite magnitudes.

Interestingly, up until the eighteenth century, negative numbers were often considered absurd or meaningless. But as the above examples show, it is just a matter of applying them appropriately and any absurdity goes away.

One trillion dollars!

You may be able to just make out a person in the bottom left corner of the image standing next to 10,000 double-stacked standard pallets each holding a million $100 bills. That's what one trillion dollars looks like.

Humans have trouble intuitively conceptualizing large numbers. A billion or a trillion can seem incomprehensibly large when you only have ten fingers. This means that any analysis involving large figures will often just confuse us rather than inform us. So, for example, when we hear that the public debt of the United States is $12 trillion, or the deficit is almost $1 trillion, it sounds both overwhelming and unsustainable. And maybe it is. But drawing that conclusion without knowing the yearly GDP figures or the historical record is unjustified.

So I thought I'd show two charts that help provide a basis for drawing conclusions. Both charts provide information as percentages instead of large dollar amounts which makes any analysis easier to understand.



US public debt as a percentage of GDP

The first chart shows the US public debt as a percentage of GDP. While the debt continually increases (and is therefore larger than it ever has been), GDP also increases and is currently $17 trillion. The current US public debt as a percentage of GDP is 73% and the 1945 high was 113%.

For comparison, the UK public debt stands at 76% of GDP with a 1947 figure of 238%. Japan's public debt is currently 220% of GDP.



US federal deficit as a percentage of GDP

The second chart shows the US federal deficit as a percentage of GDP. The current US deficit is $680 million - about 4.1% of GDP. It was $1.4 trillion in 2009 or about 10% of GDP.

The US deficit was 30% of GDP in 1945.

 

A brief guide to US monetary policy

When demand for goods is falling (such as during a recession), a central bank can respond by lowering the short-term interest rate. This results in more borrowing and spending and higher employment. Stocks go up, investment increases, the dollar goes down and exports go up. Good times!

However a special problem with the 2008 global economic crisis was that interest rates in areas like the US and Europe were close to 0%. Unable to reduce short-term rates further, the Federal Reserve has resorted to unconventional monetary policy to reduce long term interest rates. The Fed achieves this by declaring that it will keep short-term rates low, which helps lower bond yields. And it also buys bonds on the open market which further lowers bond yields.

For example, with Quantitative Easing, the Fed credits its own account (the digital equivalent of printing money) and then uses that money to purchase government bonds on the open market from financial firms such as banks and insurance companies. This procedure increases the price of the bonds which makes them a less attractive investment. The money that firms make from the bond sales can then be invested in other companies or lent to individuals, usually at a lower interest rate to attract borrowers.

Once the US economy is in good shape, the Fed will tighten (decrease) the money supply to prevent high inflation. It will do this by selling the originally purchased bonds and then debiting its account (thus destroying the money it originally created).