This lesson summarize the first topic in Engineering Mathematics Review. It includes important notes and tips you have to remember and learn in order to have a better preparation in taking the Board Exam. The notes are properly organized and make easy to understand even for the undergraduate.
SYSTEM OF NUMBERS
Number is an item that describes a magnitude or a position Numbers are classified into two types, namely cardinal and ordinal numbers.
Cardinal numbers are numbers which allow us to count the objects or ideas in a given collection. Example, 1,2.3…
Ordinal numbers states the position of individual objects in a sequence.
Example, First, second, third…
Numerals are symbols, or combination of symbols which describe a number. The most widely used numerals are the Arabic numerals and the Roman numerals.
Arabic numerals were simply a modification of the Hindu-Arabic number signs and are written in Arabic digits. Taken singly, 0, 1, 2. 3, 4, 5, 6. 7. 8, 9 and in combination 30, 41, 62, 2014, …
The Roman numerals are numbers which are written in Latin alphabet.
The following are Roman numerals and their equivalent Arabic numbers:
|I = 1||C = 100|
|V = 5||D = 500|
|X = 10||M = 1000|
|L = 50|
To increase the number, the following are used:
- 1. Bracket to increase by 100 times.
|X| = 1000
- 2. Bar above the number – to increase by 1000 times.
- 3. A “doorframe” above the number – to increase by 1000000 times.
Digit is a specific symbol or symbols used alone or in combination to denote a number. For example, the number 24 has two digits, namely 2 and 4. In Roman numerals, the number 9 is denoted as IX. So the digits I and X were used together to denote one number and that is the number 9.
In mathematical computations or engineering applications, a system of numbers using cardinal numbers was established and widely used.
The number system is divided into two categories. Real numbers and Imaginary number.
A. Natural numbers – are numbers considered as ‘counting numbers’
Examples: 1, 2, 3, 4, 5, …
B. Integers – are all the natural numbers, the negative of the natural numbers
C. Rational numbers – are numbers which can be expressed as a quotient (ratio) of two integers. The term ‘rational’ comes from the word ‘ratio’
Examples: 0.4, ¼, -4, 0.333…
In the example, 0.4 can be expressed as 2/5 and -4 can be expressed as -8/2, hence rational numbers. The number 0.333…is a repeating and non-terminating decimal. As a rule. a non-terminating but repeating (or periodic) decimals is always a rational number. Also, all integers are rational numbers.
D. Irrational Numbers – are numbers which cannot be expressed as a quotient of two integers.
The numbers in the examples above can never be expressed exactly as a quotient of two integers. They are in fact, a non- terminating number with non-terminating decimal.
An imaginary number is denoted as i. In some other areas in mathematical computation, especially in electronics and electrical engineering it is denoted as j.
Imaginary number and its equivalent:
i2 = -1
i4 = 1
DIAGRAM SHOWING THE SYSTEM OF NUMBERS
Complex Number is an expression of both real and imaginary number combined. It takes the form of a + bi, where a and b are real numbers. If a = 0, then pure imaginary number is produced while real number is obtained when b = 0.
Absolute Value of a real number is the numerical value of the number neglecting the sign. For example, the absolute value of -5 is 5 while of -x is x. The absolute value |a| is either positive or zero but can never be negative.
Common Fractions are numbers which are in the form of a/b, where a is the numerator which may be any integer while b is the denominator which may be any integer greater than zero.
If the numerator is smaller than the denominator, it is called as proper fraction while improper fraction is when the numerator is greater than the denominator.
Unit Fractions are common fractions with unity for numerator and positive integer for the denominator.
Example: 1/5, 1/25
Composite Number is a number that can be written as product of two or more integers, each greater than 1. It is observed that most integers are composite numbers.
Example. 60 = 2 x 2 x3 x 5
Prime Number is an integer greater than 1 that is divisible only by 1 and itself. According to the fundamental theorem of arithmetic, “Every positive integer greater than 1 is a prime or can be expressed as a unique product of primes and powers of primes".
Example of prime numbers:
2. 3, 5, 7, 11, 13„17, 19. 23. 29. .. 1 000 000 009 649.
Example of unique product of power of primes:
360 = 23 • 32 •51
Twin primes are prime numbers that appear in pair and differ by 2.
Example: 3 and 5, 11 and 13, 17 and 19…
Perfect number is an integer number that is equal to the sum of all its possible divisors, except the number itself.
Example: 6, 28, 496..
In the case of 6, the factors or divisors are 1, 2 and 3.
1 + 2 + 3 = 6
Defective or deficient number is an integer number, the sum of all its possible divisor is less than the number itself. If the sum of the possible divisors is greater than the number, it is referred to as abundant number. There are around 30 numbers known today as perfect number and all of which are even numbers.
Amicable numbers or friendly numbers refers to two integer numbers where each is the sum of all the possible divisors of the other. The smallest known amicable numbers are 220 and 284.
The number 220 has the following factors/divisors 1, 2, 4. 5, 10, 11. 20. 22. 44, 55, an 110 which when added sums up to 284, while the number 284 has the following divisors 1, 2, 4, 71, and 142 which adds up to 220.
Factorial denoted as n!, represents the product of all positive integers from 1 to n, inclusive.
Example n! = n(n – 1). ..3, 2, 1
If n = 0, by definition:
(n!)(n + 1) = (n + 1)!
”This is known as recursion formula”
(0!)(0 + 1) = (0 + 1)!
0!(1) = 1!
0! = 1
The factorial symbol ( ! ) was introduced by Christian Kramp in 1808.
Significant figures or digits are digits that define the numerical value of a number. A digit is considered significant unless it is used to place a decimal point.
The significant digit of a number begins with the first non-zero digit and ends with the final digit, whether zero or non-zero.
1. 18.24 4 significant figures
2. 1.824 x 103 4 significant figures
3. 0.0018 2 significant figures
Example 2 is expressed in scientific notation and figures considered significant are 1, 8, 4 and 4 excluding 103.
Example 3 has 2 significant figures only because the 3 zeros are used only to place a decimal.
The number of significant digits is considered the place of accuracy. Hence, a number with 3 significant digits is said to have a three place accuracy and a number with 4 signi6cant figures is said to have a four place accuracy.
Rounding and Truncating:
The two forms of approximations are known as rounding and truncation.
Rounding of a number means replacing the number with another number having fewer significant decimal digits, or for integer number, fewer value-carrying (non- zero) digits.
1. 3.14159 shall be rounded up to 3.14.16
2. 3.12354 shall be rounded down to 3.1235
Truncation refers to the dropping of the next digits in order to obtain the degree of accuracy beyond the need of practical calculations. This is just the same as
rounding down and truncated values will always have values lower than the exact values.
3.14159 is truncated to 4 decimal as 3.1415
TEMPERATURE CONVERSION TIPS:
1. Revolution and its equivalent in units of angle.
1 revolution = 360 degrees
= 2π radians
= 400 grads
= 6400 mils
= 6400 centissimal degree
= 6400 gons
Relation between °Celsius and °Fahrenheit
°K = °C + 273
°R = °F + 460
Kelvin was named after British physicist, William Thompson (1824-1902) the First Baron, Kelvin.
Rankine was named after Scottish engineer and physicist. William John Macquom Rankine(1820-1872).
Fahrenheit was named after German physicist, Gabriel Daniel Fahrenheit (1686 – 1736).
Celsius (or Centigrade) was named after Swedish astronomer, Anders Celsius (1701 – 1744).
3. Density of water = 1000 kg / m3
= 62.4 lb/cu. ft.
= 9810 N/m3
= 1 gram / cc
The symbol π (pi), which is the ratio of the circumference of a circle to its diameter was introduced by William Jones In 1706 after the initial letter of the Creek word meaning “periphery”.