Introduction to Abstract Mathematics

What are Rational and Irrational Numbers?

• The counting numbers 1,2, 3,… are called positive integers or natural numbers. This forms the set of positive integers, here represented as Z⁺.
• To create the set of all integers, here represented as Z, there is need to include 0, and negative numbers. The entire set of Z can be represented on a number line whose middle value is 0, as it bisects the line into a positive half (Z⁺) and a negative half (Z^–).
• The mathematical theory of natural numbers which governs the combinations of all integers is called arithmetic.
• Any number expressed as a/b where both a and b are integers whose value is not 0 is called a fraction.
• A rational number is any integer expressed as a/b, including a/1.
• If Z is extended to include all rational numbers, then a full set of rational numbers is created, hereby represented as Q.
• Any number not included in set Q is called an irrational number because it cannot be represented as a/b (where b≠0). For example (e.g), √2 is an irrational number and this can be proved using the proof of contradiction as explained below:

Start by writing √2 as a rational number as √2=a/b,

Therefore, 2=a²/b²; and a²= 2b².

This means that a² is an even number (because it is equal to 2 x b²).

Because square roots of even numbers are even numbers, therefore a is an even number that can be written as a=2c.

Thus (2c)²= 2b² = 4c².

This means that b²=2c², which means that b is an even number.

Therefore, if both a and b are even numbers, then a/b is not written in its simplest form because both a and b can be divided by 2.

This shows that √2 cannot be written as √2=a/b, that is, √2 cannot be written as a rational number which means that it is an irrational number.

• The set of numbers formed by both rational numbers (set Q) and irrational numbers is called the set of real numbers, hereby represented as R. The number line represents set R.
• There is no limit to the sequence of natural numbers. This is an example of a mathematical infinite. Likewise, in any set of numbers, there is an infinite number of objects, which is another example of a mathematical infinite. Therefore, a set of numbers can be described as an infinite set.
• If a rational number is expressed as decimals, it either recurs or has a finite number of decimal places. If an irrational number is expressed as decimals, it has an infinite number of decimal places.
• The notations a>b and a<c shows an inequality which means that a is greater than b, and a is less than c respectively. The notations a ≥ b and a ≤ c also show an inequality as they show that a has the potential of being greater than b, as well as and being lesser than c. In the number line, this notation can be rewritten as:

a ε R: b≤a≤c

which means that the set of real number values of a is equal to either b or c, or any of the numbers in the range of real (R) numbers between b and c.

The R: means that R obeys the aforementioned rule. If one wants to exclude a number, e.g 3 from this b≤a≤c, then the notation is written as a ε R: a ≠3, b≤a≤c.

Modulus

• When multiplying or dividing an inequality with a negative number, the inequality must be revised (i.e reversed) in the answer e.g (-4<4) x -1 results in 4>-4.
• If one is only interested in the magnitude of a number, and not whether it is a positive or negative number, then the modulus sign |number| is used, e.g modulus of -7 is |-7| which is equal to |7|. If -3<a<3 is written using the modulus sign to show its magnitude, it is written as |a|<3, which means that a lies between -3 and +3. If a>3, then |a|>3 means that a is always greater than +3.

Laws of Arithmetic

• Arithmetic has general laws which can be represented symbolically as follows:
  1. a + b = b+ a
  2. ab = ba
  3. a + (b + c) = (a + b) + c
  4. a (bc) = (ab) c
  5. a (b + c) = ab + ac
• LAWS 1 and 2 are called commutative laws that govern the way numbers are added or multiplied together. They allow for the order of numbers to be interchanged without altering the sum and product respectively.
• LAW 3 and 4 are collectively called the associative law. LAW 3 is the associative law of addition that allows one to add either the first two elements, or the latter two elements, to the third element, and still get the same result. LAW 4 is the associative law of multiplication.
• LAW 5 is called the distributive law.
• Addition and subtraction are inverse operations as explained symbolically as follows: (a + b) – b = a.
• In the operation of subtraction, b – a where (b>a), then the difference it yields is a positive integer, but if (b<a), then the difference is a negative integer. However, if b ≤ a or b ≥ a, then the difference can be zero (0).

Decimal System and Positional Notation

• The ten-digit symbols are 0 and the first 9 natural numbers of the decimal system. This means that the decimal system has ten digits which are 0,1,2,3,4,5,6,7,8,9.
• If the symbol 1 is written as a single digit, then it is placed in the unit position in the positional notation system, i.e 1. This 1=01=001. In 01, it is in the tens position, and in 001, it is in the hundreds position, with all numbers to the left of 1 being occupied by zero. In 101, the hundreds position is occupied by 1, which makes it 100 times greater than 001. The general rule of positional notation is expressed as:

Z = (a x 10²) + (b x 10) + c = abc, e.g if Z = 101 = (1 x 10²) + (0 x 10) + 1

• In the positional system, the base must be identified first, and in the decimal system, the base is 10. Therefore, Z can be represented as:

Z = (a x 10ⁿ) + (b x 10^n-1) + (c x 10^n-2) = abc. If a, b, and c are replaced with their positional notations based on base 10; then the results are

Z = (a_n x 10ⁿ) + (a_n-1 x 10^n-1) + (a_n-2 x 10^n-2) = a_n b_n-1 c_n-2.

For example, 111 = (1₂ x 10²) + (1_2-1 x 10^2-1) + (1_2-2 x 10^2-2) = (1₂ x 10²) + (1₁ x 10¹) + (1₀ x 10⁰) = (1₂ x 100) + (1₁ x 10) + (1₀ x 1) = 100 + 10 + 1 = 111.

This shows that Z = a_n a_n-1 a_n-2…a₁a₀^.

• The Base 10 system is called the decimal system, the Base 2 system is called the dyadic system, and the Base 12 system is called the duodecimal system. In computer science, the dyadic system is called the binary system, and there exists Base 16 which is called the hexadecimal system, as well as a Base 8 system which is called the octal system. The Babylonians used the Base 60 system, which is now called the sexagesimal system.

Principle of Mathematical Induction

• The principle of mathematical induction affirms that a general theorem A which has an infinite set of objects is proved if:
  1. The first statement, symbolized as A₁ is shown to be true.
  2. The next statement A_r+1 is true.

• Consider this example:

If the general theorem A states that the number of sides, n, of a polygon relates to the total sum of all its internal angles in this polygon per this mathematical expression 180(n-2);

then the principle of matematical induction can be applied as follows:

For a triangle; n = 3, therefore total sum of all its internal angles is 180(3-2) = 180 X 1 = 180°.

In an equilateral tringle, each of the 3 internal angles is 60°, thus the sum of its internal angles is: 60° + 60° + 60° = 180°.

For a pentagon; n = 5, therefore sum of angles is 180(5-2) = 180 X 3 = 540°.

In an equilateral pentagon, each of its 5 internal angles is 108°, thus the sum of its internal angles is: 108° + 108° + 108° + 108° + 108° = 540°.

• Mathematical induction proves the truism of the general theorem of arithmetical progression which states for any value of n integers whose objects are sequential with the first object being 1, then the sum of all the objects can be calculated as n (n + 1)/2. For example, 1,2,3,4,5,6,7; their sum is 7 (7 + 1)/2 = 7 x (8 /2) = 7 x 4 = 28.