“Mathematics is the queen of the sciences—and number theory is the queen of mathematics” – Carl Friedrich Gauss.
Numbers
- The number is the basic unit of arithmetic that symbolizes a mathematical object that can be used for counting, labeling, and measuring.
- A mathematical object is any abstract concept that can be defined and proved to exist using philosophical logic. Numbers can be proved to exist using both deductive and inductive arguments.
- Numbers are classified into the following categories:
- Counting Numbers: This is the set of numbers used for determining the numerical quantity of objects or elements in a finite set. Numbers were invented for counting in Ancient Sumeria. To be able to count, one must understand what a number is. Some primitive tribes such as the Pirahã who do not understand the concept of numbers have difficulty counting objects. Counting numbers are always positive numbers, and they start with 1.
- Natural Numbers: The set of counting numbers that can be used for ordering objects in the set. Thus, all counting numbers are natural numbers. When the natural number is used to order an item in a set, it assigns a numerical label to that item, (e.g item 6) and this number becomes a nominal number. Therefore, natural numbers are the entire set of counting numbers and nominal numbers. Nominal numbers assign a qualitative value to items in the set while counting numbers assign a quantitative value to the set.
- Whole Numbers: This is the set that includes all natural numbers and the value zero (0). It has no fractional part. This means that 0 is not a natural number.
- Integers: This is the set of whole numbers that includes negative (whole) numbers. A negative number is less than zero and is thus prefixed by a negative (-) sign. In the number line, all negative numbers are located to the left of zero (0) while all positive numbers are located to its right.
- Rational Numbers: This is the set of numbers that are written as an integer divided by another integer. For example, 5/1 is a rational number that is equal to 5 or -5/-1. Rational numbers exclude division by zero because this mathematical operation is undefined. All integers and whole numbers are rational numbers.
- Irrational Number: Any number that cannot be written as an integer divided by another integer. An example of an irrational number is the square root of 2 i.e √2.
- Real Numbers: This is the entire set of rational numbers and irrational numbers.
- The philosophical study of numbers is covered in the number theory which forms the basis of pure mathematics.
- A fractional number is less than 1 but greater than 0. It is normally expressed as a fraction, which is written as an integer divided by another integer (that is not zero). If this fraction represents a value less than 1 but greater than zero which is symbolized as 1>value>0, then it is called a proper fraction. If the fraction represents a value greater than 1, then it is called an improper fraction because it can be resolved to a whole number and a proper fraction e.g 7/3 is an improper fraction that resolves to 2 + 1/3 or 2 1/3. The combination of a whole number and a proper fraction is called a mixed fraction. Therefore, an improper fraction resolves into a mixed fraction.
- In division, a real number known as the dividend (i.e the number to be divided) is divided into or broken down into a number of equal parts. This number of equal parts is determined by a number known as the divisor, and the value of each equal part is known as the quotient. For example, 8 ÷ 2 = 4, with 8 being the dividend, 2 being the divisor, and 4 being the quotient. This shows that 8 has been divided into 2 equal parts, and the value of each part is 4. During some divisions, the size of the last part is not equal to the size of the other parts, and this last part is called the remainder. For example, 7 ÷ 2 results in 3 parts that have whole number values i.e 3, 3, and 1. The last part is the remainder and its value is 1. The remainder can be written as a proper fraction as follows: remainder/divisor. Therefore, 7 ÷ 2 = 3 1/2.
- A fraction is a notation for a division where the division symbol is substituted with the slash (/). For example, 8 ÷ 2 can be written as a fraction as follows: 8/2. The fraction has two parts – the numerator and the denominator. The numerator is the dividend and the denominator is the divisor. In the proper fraction, the numerator is less than the denominator (numerator < denominator), which results in a quotient whose value is less than 1. In an improper fraction, the numerator is greater than the denominator (numerator > denominator). If the numerator is equal to the denominator (numerator = denominator), the quotient is described as unity which is represented numerically as 1.
- A proper fraction can be expressed as a decimal value by dividing the quotient by a power of 10. For example, ½ can be written as 1/2 × 10/10 = ((1/2) × 10) ÷ 10 = (10/2) ÷ 10 = 5 ÷ 10 = 0.5.
- If the decimal value has a finite (i.e fixed) number of digits after the decimal point, then it is called a terminating decimal. All terminating decimals are rational numbers.
- A decimal value that has an infinite number of digits to the right of the decimal point is called an infinite decimal. If these digits contain a number sequence that is repeated, then the decimal value is known as a recurring decimal or repeating decimal. The repeating number sequence is called the reptend or repetend. For example, 9 ÷ 7 = 1.285714285714285714, and the reptend can be identified to be the number sequence 285714. This reptend is written with a bar above the reptend or as bar (reptend) i.e 1. bar (285714). The repetend can be used to define the terminating decimal as a decimal value whose reptend is zero. Therefore, a recurring decimal does not have a reptend whose value is 0. A recurring decimal is a rational number.
- An infinite decimal that is not a recurring decimal represents an irrational number.
Introduction to Algebra
- Addition and subtraction were the original mathematical operations developed for manipulating numbers.
- Multiplication was developed from addition. This means that multiplication is a special type of addition where a fixed number of copies of the original number are added to the original number. For example, 2 + 2 + 2 + 2 + 2 = 10 means that 2 is the original number, and four copies of 2 are added to 2 so that the operation has 2 repeated five times. The original number is called the multiplicand and the multiplier is the number of times that the multiplicand is repeated in the addition operation. In this example, 2 is the multiplicand and 5 is the multiplier (because 2 is repeated five times). This operation can be described as adding five copies of 2 together. Therefore, multiplication can be described as adding a finite number of copies of the multiplicand together, and the number of copies is set by the multiplier.
- The result of a multiplication operation is known as a product.
- The multiplication operation is written as follows:
Multiplicand × Multiplier = Product
- The above mathematical expression is called an equation because the value of numbers on either side of the equality symbol (=) is the same. In this equation, the multiplicand and the multiplier are known as operands because they are the values that are manipulated according to the law set by a symbol called the operator, which in this case is the multiplication symbol that can be denoted by the cross (×) or asterisk (*) or midline dot (•) symbol. Multiplication is described as a binary arithmetic operation because it requires at least 2 operands in order to produce a product.
