1. Fundamentals

Real Numbers

Different types of real numbers(denoted by the symbol ) were invented to meet specific needs. Rational numbers are constructed by taking ratios of integers, thus any rational number is expressed as

Real numbers such as cannot be expressed this way and are therefore called irrational numbers. Rational numbers’ corresponding decimal representations are repeating. For example,

To convert a repeating decimal to a ratio of two integers, we can eliminate the repeating part by multiplying it to appropriate powers of 10 then subtract. For ,

Thus .

For any real number , we have , therefore the number is called the additive identity(加法单位元) and the number is called the multiplicative identity(乘法单位元). is the inverse of any real number that satisfies , similar to how division undoes multiplication:

Recall that we refer to as the quotient(商) of and or as the fraction over ; is the numerator and is the denominator or divisor.

We rewrite fractions so that they have the Least(smallest possible) Common Denominator(LCD) when adding fractions with different denominators. For example, evaluate: .

First, factor each denominator into prime factors(质因数):

Then, find the LCD by forming the product of the highest power occurred of each of the prime factors, in this case, . So

To express positive number in scientific notation:

The exponent indicates how many places the decimal points should be moved. For instance,

The symbol means the positive(principle) square root of, thus

The same applies to the principal th root, and if is even, and . For instance, is not defined.

For any rational exponent (), we define

It is often useful to rationalize the denominator, making the fractional expression to be in standard form, by multiplying both the numerator and the denominator by an appropriate expression. In general, with and the denominator being of the form , multiple by rationalizes the denominator:

With , take as the quotient of and as the remainder, recall that

Then

So naturally,

In this case, the expression to multiply by is .

When comparing roots, multiply the rational exponents by their LCD. For instance,

Expressions

An algebraic expression combines real numbers and variables. A monomial is an expression of the form , where the coefficient is a real number and is a nonnegative integer, while a polynomial is a sum of monomials called terms of the form

The degree of the polynomial is the highest power of the variable appears.

Use the Distributive Property to expand algebraic expressions, and reverse the process by factoring them. To factor a trinomial of the form , note that

so find the numbers that satisfy , and .

A trinomial is a perfect square if it is of the form , and can be recognized by checking if the middle term is twice the product of the square roots of the outer terms. For instance, notice that . The technique of completing the square is used to make a perfect square. To make a perfect square, add :

Factor the coefficient of first if it isn’t .

These special factoring formulas need to be memorized:

When factoring expressions with fractional exponents, try to factor out the smallest exponent, for instance

Rational expressions are fractional expressions, which can be simplified by cancelling common factors. To simplify a compound fraction in which the numerator or the denominator is a fractional expression, multiply by the LCD of all the fractions. For instance,

The form can be rationalized by multiplying by the conjugate radical :

Equations

An equation is a statement of equality of two mathematical expressions satisfied by its solutions. Equivalent equations share the same solutions, and solving an equation is to find the equivalent equation in which the variable stands alone on one side.

Linear equations or first-degree equations has each term being either a constant or a nonzero multiple of variable. With one variable, it is of the form . Quadratic equations are second-degree equations of the form . Complete the square to find a formula for its solution:

The quantity under the square root is the discriminant.

Using LCD to factor equations can be efficient. For instance, to solve , multiply by LCD and it’s immediately factored to a general quadratic equation.

Operations such as multiplying by an expression with variables or squaring (a radical) may turn a false equation into a true one and therefore introduce extraneous solutions, so always check the answers to make sure each satisfies the original equation.

Complex Numbers

The complex number system is an expanded number system. We define

And for any negative number , similar to how positive real number has two square roots , its square roots are , and the principle square root is .

A complex number is an expression of the form . The real part of the complex number is and the imaginary part is which are both real numbers. A pure imaginary number has a real part of . Notice that

For , we define its complex conjugate to be , so

which is always a nonnegative. This property is used to divide complex numbers:

In the complex number system, a quadratic equation always have solutions, even when the discriminant , in which case the solutions are complex conjugates of each other: