Numbers: the Early Days

As a child, certain numbers burnt their way into my memory. One of them was the number of the house that I grew up in from age two onwards. It remained the family home until the late 1980s. The house is still there in an area that is now zoned light industrial but was once residential. All the surrounding houses have long gone but the one at 21 Mayneview Street remains. See Figure 1. As it turned out, I would leave home to join the army shortly after my 21st birthday. I didn't sign up, instead I was conscripted.

Figure 1: the ancestral home as shown
on Google Earth

This was the time of the United State's intervention in Vietnam and Australia's obsequious Liberal government, eager to demonstrate its unconditional support for the Americans, was happy to supply its young males as cannon fodder. And so 21 turned out to be a turning point in my life. Thereafter, it would always be a case of life before the army and life after it. I was discharged before my 22nd birthday, so the whole "adventure" took place while I was 21 years old. 

Figure 2: my maternal grandparents's house

There are other house numbers that I recall. One was that of my maternal grandparent's house where I spent the first two years of my life. The address was 69 Hale Street, Petrie Terrace, Brisbane. See Figure 2. Around the corner lived my first friend, George Marczyk, at 56 Sheriff Street. For an overview of these locations, follow this link.

Area codes are always important and I remember that for Milton, the suburb where the house was located, the code was 4064. However, this four digit system was only introduced in 1967.

The telephone number in the family home in those early days was FM4676 because of the rotary dialer. This translates to 364676 as can be seen from the image below that also compares the system used in Australia with that in the UK. See Figure 3.

Figure 3

Clearly, it was an easy number to remember. I don't remember any other telephone numbers from that era or any telephone numbers from any era for that matter but I remember that one. Here is a little more about the system:

Older Australian rotary dial telephones also had letters, but the combinations were often printed in the centre plate adjacent to the number. The Australian letter-to-number mapping was A=1, B=2, F=3, J=4, L=5, M=6, U=7, W=8, X=9, Y=0, so the phone number BX 3701 was in fact 29 3701. When Australia changed to all-numeric telephone numbers, a mnemonic to help people associate letters with numbers was the sentence, "All Big Fish Jump Like Mad Under Water eXcept Yabbies." However, such letter codes were not used in all countries. Wikipedia.

Later, as the population grew, an extra digit was added sometime in the late 1960s or early 1970s. I'm certain the extra digit was a 9 and I'm fairly sure that it was placed after the initial 36 so that the number became 3694676.

Of course, a combination of letters and numbers is still used today with car number plates. I can remember the number plate of the first car I owned: PAE 615. It was a red, Ford Falcon station wagon of an early 60s vintage. It looked something like the car shown in Figure 4.

Figure 4

on December 29th 2020

Novel Integration Technique

How does one integrate the integral shown below? Well, the standard approaches don't work here and so a little trickery is called for. Here's the integral in question:  \[ \int_0^\infty \! \frac{\sin x}{x} \mathrm{d}x \]Firstly, let's define a function \( I(b) \) as follows:\[ I(b)=\int_0^\infty \! \frac{\sin x}{x}\mathrm{e}^{-bx} \mathrm{d}x \; \text{where } b>=0 \]Now let's differentiate both sides with respect to \(b \), not \( x \):\[ I'(b)=-\int_0^\infty  \! \sin x \, \mathrm{e}^{-bx} \mathrm{d}x \; \]Using integration by parts, \(I(b) \) can be expressed as follows: \[ I'(b)=\left. \frac{\mathrm{e}^{-bx}(\cos x + b \sin x)}{b^2+1} \right| _{x=0} ^{x=\infty} =-\frac{1}{b^2+1} \]The demonstration of this integration by parts can be found here. Now integrating both sides with respect to b, we get:\[I(b)=-\int \!\frac{1}{b^2+1} \mathrm{d}b =-\arctan b+\mathrm{C} \]Comparing this result for \( I(b) \) with our earlier result, we can write:\[ -\arctan b +\mathrm{C}=\int_0^\infty \! \frac{\sin x}{x}\mathrm{e}^{-bx} \mathrm{d}x \]Now as \( b \rightarrow \infty \), the equation reduces to: \[ -\frac{\pi}{2} + \mathrm{C}=0 \text{ and so } \mathrm{C}=\frac{\pi}{2} \]Finally, we can write:\[I(b)=-\arctan(b)+\frac{\pi}{2}=\int_0^\infty \! \frac{\sin x}{x}\mathrm{e}^{-bx} \mathrm{d}x \]Now setting \( b \)=0, we achieve our desired result, namely:\[ I(0)=\int_0^\infty \! \frac{\sin x}{x}\mathrm{d}x=\frac{\pi}{2} \]A video demonstration of this technique (known as Feynman's Technique, presumably after the famous physicist) can be found here: