CSET Math Subtest 1

A real number can be an integer (5), a rational number that is not an integer (3/5), rational number as a decimal (3.5), or an irrational number (pi=3.14...) along a continuum. Points on an infinitely long number line.
Let R denote the set of all real numbers. Then:

• The set R is a field, meaning that addition and multiplication are defined and have the usual properties.
• The field R is ordered, meaning that there is a total order ≥ such that, for all real numbers x, y and z:
  • if x ≥ y then x + z ≥ y + z;
  • if x ≥ 0 and y ≥ 0 then xy ≥ 0.

A number consisting of a real and imaginary part (5 + 6i, where i^2= -1). The complex numbers contain the ordinary real numbers, but extend them by adding in extra numbers and correspondingly expanding the understanding of addition and multiplication.

In abstract algebra, a field is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms. The most commonly used fields are the field of real numbers, the field of complex numbers, and the field of rational numbers. As an algebraic structure, every field is a ring, but not every ring is a field. The most important difference is that fields allow for division (though not division by zero), while a ring need not possess multiplicative inverses.The most common way to formalize this is by defining a field as a set together with two operations, usually called addition and multiplication, and denoted by + and ·, respectively, such that the following axioms hold;subtraction and division are defined implicitly in terms of the inverse operations of addition and multiplication:^{[note 1]}Closure of F under addition and multiplicationFor all a, b in F, both a + b and a · b are in F (or more formally, + and · are binary operations on F).Associativity of addition and multiplicationFor all a, b, and c in F, the following equalities hold: a + (b + c) = (a + b) + c and a · (b · c) = (a · b) · c.Commutativity of addition and multiplicationFor all a and b in F, the following equalities hold: a + b = b + a and a · b = b · a.Additive and multiplicative identityThere exists an element of F, called the additive identity element and denoted by 0, such that for all a in F,a + 0 = a. Likewise, there is an element, called the multiplicative identity element and denoted by 1, such that for all a in F, a · 1 = a. For technical reasons, the additive identity and the multiplicative identity are required to be distinct.Additive and multiplicative inversesFor every a in F, there exists an element −a in F, such that a + (−a) = 0. Similarly, for any a in F other than 0, there exists an element a⁻¹ in F, such that a · a⁻¹ = 1. (The elementsa + (−b) and a · b⁻¹ are also denoted a − b and a/b, respectively.) In other words, subtraction and division operations exist.Distributivity of multiplication over additionFor all a, b and c in F, the following equality holds: a · (b + c) = (a · b) + (a · c).Note that all but the last axiom are exactly the axioms for a commutative group, while the last axiom is a compatibility condition between the two operations.

Rational numbers and real numbers can be ordered and complex numbers cannot be ordered, but any polynomial equation with real coefficients can be solved in the complex field.

The commutative property of addition says that we can add numbers in any order. The commutative property of multiplication is very similar. It says that we can multiply numbers in any order we want without changing the result.addition
5a + 4 = 4 + 5amultiplication
3 x 8 x 5b = 5b x 3 x 8

Both addition and multiplication can actually be done with two numbers at a time. So if there are more numbers in the expression, how do we decide which two to "associate" first? The associative property of addition tells us that we can group numbers in a sum in any way we want and still get the same answer. The associative property of multiplication tells us that we can group numbers in a product in any way we want and still get the same answer.addition
(4x + 2x) + 7x = 4x + (2x + 7x)multiplication
2x²(3y) = 3y(2x²)

The distributive property comes into play when an expression involves both addition and multiplication. A longer name for it is, "the distributive property of multiplication over addition." It tells us that if a term is multiplied by terms in parenthesis, we need to "distribute" the multiplication over all the terms inside.2x(5 + y) = 10x + 2xyEven though order of operations says that you must add the terms inside the parenthesis first, the distributive property allows you to simplify the expression by multiplying every term inside the parenthesis by the multiplier. This simplifies the expression.

The density property tells us that we can always find another real number that lies between any two real numbers. For example, between 5.61 and 5.62, there is 5.611, 5.612, 5.613 and so forth.Between 5.612 and 5.613, there is 5.6121, 5.6122 ... and an endless list of other numbers!

The identity property for addition tells us that zero added to any number is the number itself. Zero is called the "additive identity." The identity property for multiplication tells us that the number 1 multiplied times any number gives the number itself. The number 1 is called the "multiplicative identity."Addition
5y + 0 = 5yMultiplication
2c × 1 = 2c

When you add two real numbers together, their sum will be a real number. Multiplication: when you multiply two real numbers together, you get a real number.

Take any real number and multiply it by -1 to get its additive inverse. When you add these numbers together, the result will always be zero.

For any real number that is not zero, its multiplicative inverse (or reciprocal) is one divided by that number. The multiplicative inverse says that when you multiply a real number by its reciprocal, the result will be one.

A complex number is one of the form a + bi, where a and b are real numbers. a is called the real part of the complex number, and b is called the imaginary part.Two complex numbers are equal if and only if their real parts are equal and their imaginary parts are equal. I.e., a+bi = c+di if and only if a = c, and b = d. every real number is a complex number (with imaginary part 0).

The formula for multiplying two complex numbers is(a + bi) * (c + di) = (ac - bd) + (ad + bc)i.

To add or subtract two complex numbers, you add or subtract the real parts and the imaginary parts.(a + bi) + (c + di) = (a + c) + (b + d)i.(a + bi) - (c + di) = (a - c) + (b - d)i.