System of Numbers and Conversion
Systems of numbers and conversion
Key ideas
- Numbers form the foundation of engineering calculations and are classified into real and imaginary categories.
- Different numeral systems exist, such as Arabic numerals and Roman numerals.
- Understanding significant figures ensures precision in physical measurements and calculations.
- Mastering basic unit conversions for angles, temperature, and density is critical for board exams.
Types of numbers
Number is a concept that describes a magnitude or a position.
Cardinal number is a number used simply for counting objects, like 1, 2, and 3.
Ordinal number is a number that indicates the position of an object in a sequence, such as first, second, or third.
Numeral is a symbol or a combination of symbols used to express a number. We commonly use Arabic numerals (0 to 9) and Roman numerals (I, V, X, L, C, D, M).
Digit is a single symbol used to build a numeral.
To write larger Roman numerals, specific markings modify the base symbol:
- A bracket increases the value 100 times.
- A bar above the numeral increases the value 1,000 times.
- A doorframe symbol above the numeral increases the value 100,000 times.
Real number is any number that falls into the rational or irrational categories.
Natural number is a basic counting number starting from 1.
Integer is any natural number, its negative version, or zero.
Rational number is a number that can be written as a ratio or fraction of two integers. If a decimal is non-terminating but repeats in a pattern, it is always a rational number.
Irrational number is a number that cannot be expressed exactly as a fraction of two integers. Examples include \(\pi\), \(e\), and \(\sqrt{2}\).
Imaginary number is the square root of negative one. Engineers use \(i\) (or \(j\) in electrical engineering) to represent it.
Complex number is a combination of a real and an imaginary number, written in the standard format \(a + bi\).
Absolute value is the positive numerical magnitude of a number, completely ignoring its positive or negative sign.
Fractions and Prime Numbers
Common fraction is written as a numerator over a denominator. A proper fraction has a smaller numerator, while an improper fraction has a larger numerator.
Unit fraction is a fraction where the numerator is exactly 1.
Composite number is an integer greater than 1 that is the product of two or more smaller integers.
Prime number is an integer greater than 1 that can only be evenly divided by 1 and itself.
Twin primes are pairs of prime numbers that have a difference of exactly 2, like 11 and 13.
Perfect number is an integer that exactly equals the sum of its divisors, excluding the number itself. For example, 6 is perfect because \(1 + 2 + 3 = 6\).
Defective number is an integer where the sum of its divisors is less than the number itself. Conversely, an abundant number has divisors that sum to more than the number.
Amicable numbers are pairs of numbers where the divisors of each number sum up to the other number. The smallest pair is 220 and 284.
Factorial is the continuous product of all positive integers from 1 up to a given number \(n\), written as \(n!\). By mathematical definition, \(0! = 1\).
Significant Figures and Rounding
Significant figures are the digits that actually define the precise value of a number. You start counting significant digits from the first non-zero digit.
Rounding is replacing a number with a shorter, simpler value that is numerically closest to the original.
Truncation is chopping off the extra decimal digits without adjusting the last remaining digit. This always results in a value lower than or equal to the exact number.
Formulas
| Quantity | Conversion / Formula | Use it when… |
|---|---|---|
| Angle | \(1 \text{ rev} = 360^\circ = 2\pi \text{ rad} = 400 \text{ grads} = 6400 \text{ mils}\) | Converting between common angular units. |
| Celsius | \(^\circ\text{C} = \dfrac{5}{9}(^\circ\text{F} - 32)\) | Converting Fahrenheit to Celsius. |
| Fahrenheit | \(^\circ\text{F} = \dfrac{9}{5}^\circ\text{C} + 32\) | Converting Celsius to Fahrenheit. |
| Kelvin | \(^\circ\text{K} = ^\circ\text{C} + 273\) | Finding the absolute temperature from Celsius. |
| Rankine | \(^\circ\text{R} = ^\circ\text{F} + 460\) | Finding the absolute temperature from Fahrenheit. |
| Water Density | \(1000 \text{ kg/m}^3 = 62.4 \text{ lb/ft}^3 = 1 \text{ g/cc}\) | Calculating fluid weight or mass. |
Common mistakes
- Counting leading zeros as significant figures. For example, \(0.0016\) only has 2 significant figures, because the zeros just locate the decimal point.
- Confusing scientific notation significant figures. In \(1.672 \times 10^3\), the power of ten is ignored; there are 4 significant figures.
Board exam focus
- Identifying properties of numbers, such as distinguishing rational from irrational numbers.
- Unit conversions for angles (especially converting mils to degrees) and absolute temperature scales.
Quick check
Is the fraction \(22/7\) a rational or irrational number?
Show answer
It is a rational number. By definition, any number that can be expressed as a quotient of two integers is rational.
If a calculation requires truncating \(3.14159\) to four decimal places, what is the result?
Show answer
The result is \(3.1415\). Truncation drops the remaining digits without rounding up.
Additional facts. The famous symbol \(\pi\) was introduced in 1706 by William Jones, taking the first letter of the Greek word for “periphery”. Also, the factorial symbol (!) was first introduced by Christian Kramp in 1808.