The sixth article in an eight-part series, Dexter Booth tackles multiplication and irrational numbers.Multiplication, originally defined as the repetitive addition of integers and later modified to account for the multiplication of rational numbers that are not integers, makes our arithmetic faster and more efficient. A further efficiency can be gained by employing the idea of repetitive multiplication or, to give it its more common name, raising to a power. If, for example, 3 number 4s are multiplied together, we speak of the number 4 being raised to the power 3 and write: By using arithmetic arguments it can be shown that a number raised to the power zero is equal to unity and that raising a number to a negative power indicates a reciprocal. So that, for instance: A most important efficiency gain derived from raising to a power is the ability to write our numbers as a list of single numerals using the notion of place value where the value that a numeral represents is dictated by its place in the listing. For example, the numeral listing 7,347 has two numeral 7s and their positions in the listing dictate the numbers they represent; the first from the right represents 70 and the second represents 7000. Indeed, the entire listing stands for seven thousand (7 × 1000 = 7 × 10³), three hundred (3 × 100 = 3 × 10²) and seventy (7 × 10 = 7 × 10¹) four (4 × 1 = 40). That is: The list of numerals on the left of this equation and the sum of powers on the right are just alternative ways of describing the same number. The listing on the left of the coefficients of the respective powers of 10 is clearly the more efficient; the comma in the listing is there purely to make the numeral listing easier to read. This method of listing can easily be extended to the integers by judicious use of the minus sign but an extension to those rational numbers that are not integers requires more careful consideration. Look at the number represented by the mixed fraction:
has two numeral 7s and their positions in the listing dictate the numbers they represent; the first from the right represents 70 and the second represents 7000. Indeed, the entire listing stands for seven thousand (7 ´ 1000 = 7 ´ 103), three hundred (3 ´ 100 = 3 ´ 102) and seventy (7 ´ 10 = 7 ´ 101) four (4 ´ 1 = 4 ´ 100). That is
The list of numerals on the left of this equation and the sum of powers on the right are just alternative ways of describing the same number. The listing on the left of the coefficients of the respective powers of 10 is clearly the more efficient; the comma in the listing is there purely to make the numeral listing easier to read. This method of listing can easily be extended to the integers by judicious use of the minus sign but an extension to those rational numbers that are not integers requires more careful consideration. Look at the number represented by the mixed fraction: In terms of a sum of products of powers of 10 this is: Now, if we were to list the coefficients of the powers of 10 as we have just done we would obtain 53 which is clearly not right because it would be confused with fifty-three. To avoid this confusion the coefficients multiplying 10 to negative powers are separated from the rest by using a decimal point thus: 5.3 (five point three). This is called the decimal representation of the number: The decimal representation of a number can be best investigated with a calculator. For example a calculator will produce the result: Here the ellipsis (. . .) indicates that the listing of the 3s never ends. This is the problem with the decimal representation of our rational numbers - the numeral listing goes on forever, albeit with an infinitely repeated sequence of numerals after the decimal point. As examples we can use a calculator to show that: And: And a computer to show that: Each and every one of our rational numbers has a decimal representation that eventually involves an infinitely repeated sequence of numerals and each and every decimal listing that has an infinitely repeated sequence of numerals is a rational number. That’s neat and that’s tidy. But hold on a minute, what happens if we have an infinite numeral listing that does not have an infinitely repeated sequence of numerals such as: Where the numerals for the whole numbers are listed one after another. No pattern there but a perfectly reasonable listing of numerals. Oh dear, we have yet again a numeral structure that does not represent one of the only type of number we possess, namely the rational numbers. Do we deny this form of symbolic listing or do we dive in and call it a new kind of number? You’re right, we call it a new kind of number; we call it an irrational number. Irrational: now there’s an elephant trap if ever I saw one. No, it doesn’t mean that the number is incapable of behaving rationally. What it does mean is that the number cannot be represented as a fraction - as one integer divided by another integer; as a ratio. Consequently, the numeral representation of irrational numbers poses a real problem because of their never-ending, pattern-free decimal representation. Furthermore, as a result of this never-ending, pattern-free decimal representation they cannot be used to measure because all measurements are given as quantities of a repeated sub-unit that is numerically represented by a fraction. How weird is that? We have found some numbers that we can neither describe nor use! And, believe it or not there are infinitely more of them than there are rational numbers.
For certain special irrational numbers we have an alternative symbolism. The letter e, for instance, represents a number called the exponential number and the square-root of 2 (that number which when multiplied by itself equals 2) is an irrational number and for that we use the surd symbol √2. We even resort to the Greek alphabet where π (pronounced pi) is used for the irrational number that represents the value of the ratio of the circumference to the diameter of a circle (how odd - the ratio of the circumference to the diameter of a circle has a value that cannot be represented by the ratio of one whole number divided by another. But that is a subject for another blog!).
So now our numbers are complete. We have positive and negative whole numbers and zero called the integers and we have ratios of integers and all of these are called rational numbers. Complementing the rational numbers we have irrational numbers and collectively all these numbers can be plotted on a line thus:
To every number, rational or irrational, there corresponds a unique point on the line and to every point on the line there corresponds a unique number, rational or irrational. There are no gaps and no overlaps; the line is totally and uniquely occupied. Collectively the rational and irrational numbers form what are called the real numbers and the real numbers are closed under addition, subtraction, multiplication and division - how very neat and how very tidy. Hold on; I didn’t say they were closed under the operation of raising to a power and furthermore, why call them real?