Definitions/Notations

• Number Line

  A number line is a visual representation of the real number system, where numbers are represented as points or markers on a straight line. The number line extends infinitely in both directions, with zero $(0)$ located at the center. Numbers to the right of zero are positive, and numbers to the left of zero are negative.

  The number line is divided into equal intervals or units, which can be labeled with integers, fractions, decimals, or other types of numbers. The distance between any two points on the number line represents the difference between the corresponding numbers. For example, the distance between $2$ and $5$ on the number line is $3$, since $5 - 2 = 3$.

• Rational Number A rational number is a number that can be expressed as a ratio of two integers, where the denominator is not equal to zero. That is, a rational number can be written in the form $\frac pq$, where $p$ and $q$ are integers and $q$ is not equal to zero. For example,$\frac 23, \frac45$ and $-\frac72$ are all rational numbers. However, numbers like $\sqrt2$ and $π (pi)$ are not rational, because they cannot be expressed as a ratio of two integers.

• Real Number

  Real numbers are a set of numbers that include all rational and irrational numbers. Real numbers can be represented on a number line, and they are used to measure continuous quantities, such as distance, time, temperature, and weight. Real numbers can be positive, negative, or zero, and they can be expressed as either a terminating or a non-terminating decimal or as a fraction.

  Examples of real numbers include:

  • $2, 1/2, 0, 3.14,$ $\sqrt2$ (which is an irrational number), $π (pi)$ (which is also an irrational number)

• Irrational Number

  An irrational number is a number that cannot be expressed as a ratio of two integers, meaning it cannot be written in the form $\frac pq$, where $p$ and $q$ are integers and $q$ is not equal to zero.

  Irrational numbers are characterized by an infinite, non-repeating decimal expansion. For example, $\sqrt2$ is an irrational number because its decimal expansion is $1.41421...$, which goes on forever without repeating. Similarly, $π (pi)$ is an irrational number because its decimal expansion is $3.14159...$, which also goes on forever without repeating.

  Some other examples of irrational numbers include $\sqrt3, \sqrt5$, and $e$ (the base of the natural logarithm).

• Terminating Decimal Fraction

  A terminating decimal fraction is a decimal number that ends after a finite number of digits, that is, it has a finite number of decimal places. In other words, it can be expressed as a fraction where the denominator is a power of $10$.

  For example:

  • $0.25$ is a terminating decimal fraction because it ends after $2$ decimal places and can be written as $\frac14$.

  • $1.375$ is a terminating decimal fraction because it ends after $3$ decimal places and can be written as $\frac{11}8$.

  • $3.0$ is a terminating decimal fraction because it ends after $0$ decimal places and can be written as $\frac31$ or simply $3$.

• Non-terminating Recurring Decimal Fraction

  A decimal number that continues indefinitely with a repeating pattern of digits after a certain point. This repeating pattern is called a repeating block or a repeating decimal.

  For example:

  • $0.666...$ is a non-terminating recurring decimal fraction, where the digit $6$ repeats infinitely. It can be written as $\frac23$.

  • $0.142857142857...$ is a non-terminating recurring decimal fraction, where the block of digits $142857$ repeats infinitely. It can be written as $\frac17$.

• Non-terminating Non-recurring Decimal Fraction

  A decimal number that continues indefinitely without a repeating pattern of digits. This type of decimal cannot be expressed as a simple fraction with a finite number of digits, and its decimal representation goes on forever without any repeating pattern.

  Examples of non-terminating non-repeating decimal fractions include:

  • $Pi (π) = 3.1415926...$

  • e (Euler's number) $= 2.718281...$

  • The square root of $2 = 1.4142...$

  • The golden ratio (φ) $= 1.6180...$

  These numbers are irrational, which means they cannot be expressed as a ratio of two integers.

• Radical It is a mathematical symbol that represents the operation of taking the nth root of a number. The radical symbol is denoted by $\sqrt[n]{\ }$, where $n$ is the index of the radical.

• Radicand

  The number under the radical symbol is called the radicand. For example, in the expression $\sqrt{25}$, 25 is the radicand, and the index is $2$ since we are taking the square root.

  Similarly, in the expression $\sqrt[3]{8}$, $8$ is the radicand, and the index is $3$ since we are taking the cube root.

  Here are some more examples:

  • $\sqrt{9}$ $= 3,$ since $3 × 3 = 9$

  • $\sqrt{16}$ $= 4,$ since $4 × 4 = 16$

• Base The base refers to the number that is being multiplied by itself a certain number of times. For example, in the expression $2³$, the base is $2$, since we are multiplying $2$ by itself $3$ times.

• Exponent An exponent (or power) is a number or variable that represents the number of times a base is multiplied by itself. An exponent is written as a superscript number or variable to the right of the base, and indicates the number of times the base is multiplied by itself. For example, in the expression $2³$, the base is $2$ and the exponent is $3$. This means that $2$ is multiplied by itself 3 times: $2³ = 2 × 2 × 2 = 8$.

• Complex Number

  A complex number is the sum of a real and an imaginary number. It is usually represented by $z=a+ib$, where $a$ and $b$ are both real numbers. The real part is represented by $R_e(z)=a$ and the imaginary part by $I_m(z)=b$

• Imaginary Number:

  An imaginary number is a number that can be written in the form $a+ib$, where $a$ and $b$ are real numbers and $i$ is the imaginary unit, which is defined as the square root of $-1$ or $i=\sqrt{-1}$.

  Note that $i$ is not a real number because the square of any real number is always positive, whereas $i²$ is equal to $-1.$ Thus, $i$ is considered to be an imaginary number.

Notations:

• Iota: $i=\sqrt{-1}$

• Complex number: $z=a+ib$

• Real part of a complex number: $R_e(z)=a$

• Imaginary part of a complex number: $I_m(z)=ib$

• Conjugate of a complex number: $\bar{z}$

• Modulus of a complex number: $|z|$