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Essential Concepts
Real Numbers
A real number is any value that can represent a distance along a continuous line, which includes both rational and irrational numbers.
- The set of natural numbers includes the numbers used for counting: [latex]\{1,2,3,\dots\}[/latex].
- The set of whole numbers is the set of natural numbers plus zero: [latex]\{0,1,2,3,\dots\}[/latex].
- The set of integers adds the negative natural numbers to the set of whole numbers: [latex]\{\dots,-3,-2,-1,0,1,2,3,\dots\}[/latex].
- The set of rational numbers includes fractions written as [latex]\{\frac{m}{n}|m\text{ and }n\text{ are integers and }n\ne 0\}[/latex]. Rational numbers may be written as fractions or terminating or repeating decimals.
- The set of irrational numbers is the set of numbers that are not rational, are nonrepeating, and are nonterminating: [latex]\{h|h\text{ is not a rational number}\}[/latex].
The order of operations is a set of rules used in mathematics to determine the sequence in which operations should be performed to correctly solve an expression. The standard order in which these operations must be carried out is often remembered by the acronym PEMDAS: Parenthesis, Exponents, Multiplications, Divisions, Additions, and Subtractions.
The real numbers under the operations of addition and multiplication obey basic rules, known as the properties of real numbers. These are the commutative properties, the associative properties, the distributive property, the identity properties, and the inverse properties.
The following properties hold for real numbers a, b, and c.
| Addition | Multiplication | |
|---|---|---|
| Commutative Property | [latex]a+b=b+a[/latex] | [latex]a\cdot b=b\cdot a[/latex] |
| Associative Property | [latex]a+\left(b+c\right)=\left(a+b\right)+c[/latex] | [latex]a\left(bc\right)=\left(ab\right)c[/latex] |
| Distributive Property | [latex]a\cdot \left(b+c\right)=a\cdot b+a\cdot c[/latex] | |
| Identity Property | There exists a unique real number called the additive identity, 0, such that, for any real number a
[latex]a+0=a[/latex]
|
There exists a unique real number called the multiplicative identity, 1, such that, for any real number a
[latex]a\cdot 1=a[/latex]
|
| Inverse Property | Every real number a has an additive inverse, or opposite, denoted –a, such that
[latex]a+\left(-a\right)=0[/latex]
|
Every nonzero real number a has a multiplicative inverse, or reciprocal, denoted [latex]\frac{1}{a}[/latex], such that
[latex]a\cdot \left(\dfrac{1}{a}\right)=1[/latex]
|
An algebraic expression is a collection of constants and variables joined together by the algebraic operations of addition, subtraction, multiplication, and division. They take on a numerical value when evaluated by replacing variables with constants.
- An equation is a mathematical statement indicating that two expressions are equal.
- A formula is an equation expressing a relationship between constant and variable quantities.
Exponents and Scientific Notation
- Exponential form is a way of expressing a number or function using a base raised to a power, which indicates how many times the base is multiplied by itself, such as [latex]a^n[/latex] where [latex]a[/latex] is the base and [latex]n[/latex] is the exponent.
- Products of exponential expressions with the same base can be simplified by adding exponents.
- Quotients of exponential expressions with the same base can be simplified by subtracting exponents.
- Powers of exponential expressions with the same base can be simplified by multiplying exponents.
- An expression with exponent zero is defined as 1.
- An expression with a negative exponent is defined as a reciprocal.
- The power of a product of factors is the same as the product of the powers of the same factors.
- The power of a quotient of factors is the same as the quotient of the powers of the same factors.
- The rules for exponential expressions can be combined to simplify more complicated expressions.
- Scientific notation is a shorthand method used to write very large or very small numbers by expressing them as a product of a coefficient and a power of [latex]10[/latex].
[latex]a\times {10}^{n}[/latex]
-
- The coefficient [latex]a[/latex] is typically a number between [latex]1[/latex] and [latex]10[/latex].
- The power of [latex]10[/latex], [latex]n[/latex], represents how many times the coefficient should be multiplied or divided by [latex]10[/latex]. So, [latex]n[/latex] is an integer.
Radicals and Rational Exponents
- The square root of a number [latex]a[/latex] refers to any number [latex]x[/latex] such that [latex]x^2 = a[/latex]. For positive number [latex]a[/latex], there are always two square roots: one positive and one negative.
- The principal square root of [latex]a[/latex] is the nonnegative number that when multiplied by itself equals [latex]a[/latex].
- The Product Rule for Simplifying Square Roots: If [latex]a[/latex] and [latex]b[/latex] are nonnegative, the square root of the product [latex]ab[/latex] is equal to the product of the square roots of [latex]a[/latex] and [latex]b[/latex].
[latex]\sqrt{ab}=\sqrt{a}\cdot \sqrt{b}[/latex]
- The Quotient Rule for Simplifying Square Roots: The square root of the quotient [latex]\dfrac{a}{b}[/latex] is equal to the quotient of the square roots of [latex]a[/latex] and [latex]b[/latex], where [latex]b\ne 0[/latex].
[latex]\sqrt{\dfrac{a}{b}}=\dfrac{\sqrt{a}}{\sqrt{b}}[/latex]
- We can add and subtract radical expressions if they have the same radicand and the same index.
- Radical expressions written in simplest form do not contain a radical in the denominator. To eliminate the square root radical from the denominator, multiply both the numerator.
- The [latex]\text{n}^{\text{th}}[/latex] root of [latex]a[/latex], denoted as [latex]\sqrt[n]{a}[/latex], is a number that, when raised to the nth power, equals [latex]a[/latex].
- If [latex]a[/latex] is a real number with at least one [latex]\text{n}^{\text{th}}[/latex] root, then the principal [latex]\text{n}^{\text{th}}[/latex] root of [latex]a[/latex], written as [latex]\sqrt[n]{a}[/latex], is the number with the same sign as [latex]a[/latex] that, when raised to the [latex]\text{n}^{\text{th}}[/latex] power, equals [latex]a[/latex]. Here, [latex]n[/latex] is what we call the index of the radical, which tells us the degree of the root.
- If a and b are nonnegative, the square root of the product ab is equal to the product of the square roots of a and b.
- If a and b are nonnegative, the square root of the quotient (a/b) is equal to the quotient of the square roots of a and b.
- We can add and subtract radical expressions if they have the same radicand and the same index.
- Radical expressions written in simplest form do not contain a radical in the denominator. To eliminate the square root radical from the denominator, multiply both the numerator and the denominator by the conjugate of the denominator.
- The principal nth root of a is the number with the same sign as a that when raised to the nth power equals a. These roots have the same properties as square roots.
- Rational exponents are another way to express principal [latex]\text{n}^{\text{th}}[/latex] roots. The general form for converting between a radical expression with a radical symbol and one with a rational exponent is
[latex]\begin{align}{a}^{\frac{m}{n}}={\left(\sqrt[n]{a}\right)}^{m}=\sqrt[n]{{a}^{m}}\end{align}[/latex]
- The properties of exponents apply to rational exponents.
Key Equations
| Rules of Exponents For nonzero real numbers [latex]a[/latex] and [latex]b[/latex] and integers [latex]m[/latex] and [latex]n[/latex] |
|
| Product rule | [latex]{a}^{m}\cdot {a}^{n}={a}^{m+n}[/latex] |
| Quotient rule | [latex]\dfrac{{a}^{m}}{{a}^{n}}={a}^{m-n}[/latex] |
| Power rule | [latex]{\left({a}^{m}\right)}^{n}={a}^{m\cdot n}[/latex] |
| Zero exponent rule | [latex]{a}^{0}=1[/latex] |
| Negative rule | [latex]{a}^{-n}=\dfrac{1}{{a}^{n}}[/latex] |
| Power of a product rule | [latex]{\left(a\cdot b\right)}^{n}={a}^{n}\cdot {b}^{n}[/latex] |
| Power of a quotient rule | [latex]{\left(\dfrac{a}{b}\right)}^{n}=\dfrac{{a}^{n}}{{b}^{n}}[/latex] |
Glossary
- algebraic expression
- constants and variables combined using addition, subtraction, multiplication, and division
- associative property of addition
- the sum of three numbers may be grouped differently without affecting the result; in symbols, [latex]a+\left(b+c\right)=\left(a+b\right)+c[/latex]
- associative property of multiplication
- the product of three numbers may be grouped differently without affecting the result; in symbols, [latex]a\cdot \left(b\cdot c\right)=\left(a\cdot b\right)\cdot c[/latex]
- base
- in exponential notation, the expression that is being multiplied
- commutative property of addition
- two numbers may be added in either order without affecting the result; in symbols, [latex]a+b=b+a[/latex]
- commutative property of multiplication
- two numbers may be multiplied in any order without affecting the result; in symbols, [latex]a\cdot b=b\cdot a[/latex]
- constant
- a quantity that does not change value
- distributive property
- the product of a factor times a sum is the sum of the factor times each term in the sum; in symbols, [latex]a\cdot \left(b+c\right)=a\cdot b+a\cdot c[/latex]
- equation
- a mathematical statement indicating that two expressions are equal
- exponent
- in exponential notation, the raised number or variable that indicates how many times the base is being multiplied
- exponential notation
- a shorthand method of writing products of the same factor
- formula
- an equation expressing a relationship between constant and variable quantities
- identity property of addition
- there is a unique number, called the additive identity, 0, which, when added to a number, results in the original number; in symbols, [latex]a+0=a[/latex]
- identity property of multiplication
- there is a unique number, called the multiplicative identity, 1, which, when multiplied by a number, results in the original number; in symbols, [latex]a\cdot 1=a[/latex]
- index
- the number above the radical sign indicating the nth root
- integers
- the set consisting of the natural numbers, their opposites, and 0: [latex]\{\dots ,-3,-2,-1,0,1,2,3,\dots \}[/latex]
- inverse property of addition
- for every real number [latex]a[/latex], there is a unique number, called the additive inverse (or opposite), denoted [latex]-a[/latex], which, when added to the original number, results in the additive identity, 0; in symbols, [latex]a+\left(-a\right)=0[/latex]
- inverse property of multiplication
- for every non-zero real number [latex]a[/latex], there is a unique number, called the multiplicative inverse (or reciprocal), denoted [latex]\dfrac{1}{a}[/latex], which, when multiplied by the original number, results in the multiplicative identity, 1; in symbols, [latex]a\cdot \dfrac{1}{a}=1[/latex]
- irrational numbers
- the set of all numbers that are not rational; they cannot be written as either a terminating or repeating decimal; they cannot be expressed as a fraction of two integers
- natural numbers
- the set of counting numbers: [latex]\{1,2,3,\dots \}[/latex]
- order of operations
- a set of rules governing how mathematical expressions are to be evaluated, assigning priorities to operations
- principal nth root
- the number with the same sign as [latex]a[/latex] that when raised to the nth power equals [latex]a[/latex]
- principal square root
- the nonnegative square root of a number [latex]a[/latex] that, when multiplied by itself, equals [latex]a[/latex]
- radical
- the symbol used to indicate a root
- radical expression
- an expression containing a radical symbol
- radicand
- the number under the radical symbol
- rational numbers
- the set of all numbers of the form [latex]\dfrac{m}{n}[/latex], where [latex]m[/latex] and [latex]n[/latex] are integers and [latex]n\ne 0[/latex]. Any rational number may be written as a fraction or a terminating or repeating decimal.
- real number line
- a horizontal line used to represent the real numbers. An arbitrary fixed point is chosen to represent 0; positive numbers lie to the right of 0 and negative numbers to the left.
- real numbers
- the sets of rational numbers and irrational numbers taken together
- scientific notation
- a shorthand notation for writing very large or very small numbers in the form [latex]a\times {10}^{n}[/latex] where [latex]1\le |a|<10[/latex] and [latex]n[/latex] is an integer
- variable
- a quantity that may change value
- whole numbers
- the set consisting of 0 plus the natural numbers: [latex]\{0,1,2,3,\dots \}[/latex]