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Consider the equation x = 8 + 4 x 3. What do you get when you solve for x? Perhaps you got 36. Do you see how the answer could also be 20? The problem can be done in two ways. You can carry out the addition first, and obtain 36 as the answer. You could also carry out the multiplication first and arrive at 20 for your answer. This is why is it necessary to know the order of operations, the order in which mathematical operations are supposed to be carried out.

A common mnemonic (memory device) used for remembering the order of operations is the acronym PEMDAS. This stands for Parenthesis, Exponents (and roots), Multiplication/Division, Addition/Subtraction.

Parenthesis

Expressions enclosed by parenthesis or brackets are always analyzed first. Inner parenthesis/brackets are done before outer ones.

Example: (3 + x 4

• First analyze the parenthetical expression.

(11) x 4

• Complete the problem.

11 x 4 = 44

Example: [ 2 + ( 2 x 5)] x 4

• Analyze the inner parenthesis first.

[2 + (10)] x 4 = (2 + 10) x 4

• Analyze the outer parenthesis.

12 x 4 = 48

Exponents

After taking care of the parenthesis, move on to terms with exponents. If an expression with an exponent is inside of parenthesis, carry out the exponential operation before the parenthetical operation.

Example: (3 + 1) x 2³

• Take care of the parenthesis…

4 x 2³

• Simplify the exponential expression…

4 x 8 = 32

Multiplication/Division

When there are no remaining parenthesis or exponents, carry out any multiplication or division. The order in which multiplication and division are done does not matter at this point if they are not combined. However, most people choose to work from left to right for the sake of convenience. This is also the easiest way to finish a problem that included a combination of multiplication and division.

Example: 20 ÷ 2 x 5

• First carry out the division and then multiply…

10 x 5 = 50

Addition/Subtraction

Similar to multiplication and division, you should add and subtract from left to right to make things simpler. However, Repeated addition can be done in any order.

Example: 3 + 9 – 5

• Add 3 and 9, then subtract 5…

12 – 5 = 7

PEMDAS: Putting it all Together

It is possible to have a problem that includes many different operations. Use PEMDAS to solve them.

Example: (2 + 1)² x 3 + 10 ÷ 5

• Parenthesis first…

3² x 3 + 10 ÷ 5

• Simplify the exponential expression…

9 x 3 + 10 ÷ 5

• Carry out the multiplication and division from left to right…

27 + 10 ÷ 5

27 + 2

And the answer is 29. Solving complex problems becomes much simpler when you know what order to do them in. Eventually, this process will become very natural to you.  Recall the first equation  x = 8 + 4 x 3. The order of operations tells us to multiply first and then add. The correct answer is 20.

Calculating the area of a triangle is a very important part of geometry. It comes in especially handy when dealing with complex shapes. These can often be broken down into triangles, and calculating the area of these individual areas can aid you in finding the total.

A = ½bh

This is easily the most recognizable formula for the area of a triangle. It is based on the fact that, if you draw a rectangle of base b and height h, it will be exactly twice the area of the triangle. Remember that for the purposes of calculating the area of a triangle, b is the base and h is the altitude drawn to that base. An altitude is a line drawn from one vertex of the triangle to the side opposite of that vertex. Altitudes must be perpendicular to the line they are drawn to. In cases where the altitude is not given, you may need to use the Pythagorean theorem to calculate it or use a different formula.

A = ½ ab sinθ

This is one formula that can be used when the height of the triangle is not known. In this equation, a and b are two sides of a triangle while θ is their included angle. In geometry, this may be known as a side-angle-side (SAS) case. Notice that a sinθ is equivalent to h in the previous equation.

A = √s(s-a)(s-b)(s-c)

This is known as Heron’s formula. In this equation, a, b, and c are sides of a triangle. The semi-perimeter s, is equal to one half of the perimeter of the triangle. In other words, s = ½(a + b + c). This is obviously used in a side-side-side (SSS) situation.

Notice that calculating area becomes much simpler when right triangles are involved. The first formula applies when you are given the two legs of the right triangle, and it can be used without calculating an altitude. Provided with sufficient information, you can calculate the missing angles and side lengths of a right triangle and apply any other the formulas above.

Most work in algebra and essentially all work in geometry is done using the real number system. Because of this, it is important to understand the properties of real numbers in order to work with them in the future.

Commutative Property

Addition and multiplication are commutative. In order words, the order in which two numbers are multiplied or added does not affect the result. Mathematically, for any two numbers a and b

• a + b = b + a
• a • b = b • a

Associative Property

Closely related to the previous property is the associative property. This property states that the way one adds or multiples three (or more) numbers will not affect the outcome. For any three real number a, b, and c:

• (a + b) + c = a + (b + c) = a + b + c
• (a • b) • c = a • (b • c) = a • b • c

Distributive Property

This is one of the most important properties of real numbers, and is very commonly used in algebra when factoring or multiplying sums or differences. For any three real number a, b, and c:

• a • (b + c) = a • b + a • c

Additive Identity Property

• a + 0 = 0 + a = a

Multiplicative Identity Property

• a • 1 = 1 • a = a

Additive Inverse Property

For every real number a, there is another real number -a. This number is called the additive inverse or opposite of a. In many cases it make be called negative a, but this can be dangerous. Saying negative a implies that a is a negative number, which is not always the case. Consider a = -6. In this case, -a = 6, a positive number.

• a + (-a) = -a + a = 0

The difference between a and b can be rewritten as the sum of a and the additive inverse of b. That is to say,

• a – b = a + (-b)

Multiplicative Inverse Property

For every real number a (except 0), there is another real number 1/a called the multiplicative inverse of a. This is more commonly called the reciprocal of a. A real number and its reciprocal have the following property.

• a • (1/a) = (1/a) • a = 1

The quotient a/b can be rewritten as the product of a and the reciprocal of b.

• a/b = a • (1/b)

Sign Properties

Certain rules apply when performing multiplication or division with numbers that have different signs.

• a(-b) = (-a) b = -(ab)
• (-a)(-b) = ab
• -(-a) = a
• a/-b = -a/b = -(a/b)
• -a/-b = a/b

Cancellation Properties

• If ac = bc, then a = b (Divide both sides by c)
• ac/bc = (a/b) • (c/c) = (a/b) • 1 = a/b

Zero Product Property

• If ab = 0, then a = 0, b = 0, or both

In algebra, there is a lot of focus on using function and analyzing their different properties. Functions have exactly one unique value of x for every value of y. That is to say, no value of x will produce two or more values of y. A specific type of function is known as a one-to-one function. As the name implies, a one-to-one function has exactly one unique value of f(x) for every value of x in the domain.

If each x in the domain produces exactly one y in the range, and no value of y in the range is produced by more than one value of x in the domain, the function is one-to-one. In other words, a function f is a one-to-one function if for any two number a and b in the domain of the function, a ≠ b and f(a) ≠ f(b).

In determining whether or not a graph was a function, you may have used a method known as the vertical line test. In this method, you visualise a line x = k where k is any value in the domain. If, for any value of x, the graph is intersected at more than one point, it is not a function. If a vertical line is drawn at any value of x in the domain and intersects the graph only once then the graph is a function.

A similar test exists for determining if a given function is one-to-one. If the graph passes the vertical line test, then it is a function. The horizontal line test determines whether or not that function is one-to-one. If any horizontal line y = k where k in a value in the range of the graph intercepts the graph at more than one point then the graph is not a function. The combination of these two tests ensures that each value of y is produced by exactly one value of x, and also that each value of x produces exactly one value of y. This is a one-to-one function.

Example: Determine whether the graph y = x² is a function, and then if it is one-to-one.

First mentally perform a vertical line test. Notice that a vertical line at any point will not intercept the graph more than once. This graph is a function. Now perform a horizontal line test. This function is not one-to-one because a horizontal line can intercept the graph more than once. For example, the line y = 1 will intercept the graph at both (1, 1) and (-1, 1).

Example: Determine if the graph y = x³ is a function and then if it is one-to-one.

First perform a vertical line test. Once again, a vertical line drawn at any point on the graph will not intercept it more than once, so y = x³ is a function. Now perform a horizontal line test. Notice that a horizontal line drawn at any point on the graph will also intercept it no more than once. f(x) = x³ is a one-to-one function.

The Pythagorean theorem is an important part of geometry, as well as future math course. The Pythagorean Theorem applies to right triangles. A right triangle contains exactly one right (90°) angle. The side of the triangle opposite the right angle is called the hypotenuse. The remaining two sides are generally known as legs.

In the picture above, sides a and b are legs, while side c is the hypotenuse. The Pythagorean Theorem defines a special relationship between these sides that allows us to calculate the length of an unknown side if the other sides are known.

Verbally, the theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the square of the lengths of the legs. Mathematically, a² + b² = c² where a and b are the lengths of the legs and c is the length of the hypotenuse.

Solving for the Hypotenuse

Example: A right triangle has a leg of length 3 and another leg of length 4. Solve for the length of the hypotenuse.

Using the Pythagorean Theorem, we can immediately determine that c² = 3² + 4² where c is the length of the hypotenuse.

• Simplify the right side…

c² = 9 + 16 = 25

• Take the square root of both sides to solve for c…

c = √25 = 5

The length of the hypotenuse is 5.

Solving for a Leg

Example: A right triangle has a hypotenuse of length 13 and a leg of length 5. Solve for the length of the other leg.

Using the Pythagorean Theorem, we can conclude that 13² = 5² + b².

• Simplify the squared constants…

169 = 25 + b²

• Subtract 25 from both sides…

144 = b²

• Take the square root of both sides to solve for b…

b = √144 = 12

The other leg of this triangle has a length of 12.

Special Cases

There are some special cases and shortcuts that can be used when solving for the length of the sides of some right triangles. While it is essential to be able to understand and use the Pythagorean theorem, eventually you will be able to recognize certain patterns that eliminate the need for it.

Pythagorean Triples

Pythagorean triples are common combinations of side lengths in right triangles. The length of the sides in a Pythagorean triple are always integers. A common way to denote Pythagorean triples is using the ratio Leg 1 : Leg 2 : Hypotenuse, and the numbers are almost always written in ascending order. Some common Pythagorean triples are listed below.

• 3 : 4 : 5
• 5 : 12 : 13
• 8 : 15 : 17
• 7 : 24 : 25
• 9 : 40 : 41

Notice that for any Pythagorean triple a : b : c, ka : kb : kc is also a Pythagorean triple provided that k is a positive integer. For example, 3 :4 : 5 is a Pythagorean triple. 6 : 8 : 10 and 9 : 12 : 15 are multiples of 3 : 4 : 5 and also Pythagorean triples.

45° – 45° – 90° Triangles

For any triangle with one right angle and two 45° angles, the legs have the same length. The hypotenuse is √2 times the length of the legs.

Mathematically, for a 45-45-90 triangle with legs a and b, and hypotenuse c:

• a = b
• c = √2 a = √2 b

30°-60°-90° Triangles

For any triangle with one right angle, one 30 degree and and one 60 degree angle, the sides have the ratio 1: √3 : 2. That is to say, the hypotenuse is twice the length of the shortest leg. The longer leg is √3 times the length of the shorter leg.

Mathematically, in a 30-60-90 triangle where a is the shortest leg, b is the longer leg, and c is the hypotenuse:

• b = √3 a
• c = 2a

In previous courses like geometry, angles are usually measured in degrees (with or without minutes and seconds). For more advanced trigonometry and future courses like calculus, you will need to know how to use a unit called radians instead.

To understand radians, picture a circle with a radius of length r. If an angle is drawn in this circle with its vertex at the center of the circle, it is known as a central angle. If the rays that make up the angle form an arc of length r on the circle, the angle has a measure of 1 radian.

For a circle with a radius of length r, a central angle of θ radians creates and arc of the length rθ. To find the number of radians in a circle, consider an arc than encompasses the circumference of the circle. Recall from geometry that circumference is 2πr.

2πr = rθ

Dividing by r produces 2π = θ. The angle equal to one revolution or circumference of a circle has a measure of 2π radians. From geometry we know that an angle in degrees equal to one revolution of a circle has a measure of 360°. Therefore,

• 2π radians = 360°
• π radians = 180°

Converting between Degrees and Radians

From π radians = 180°, we can obtain two conversion factors.

• 1 degree = π/180 radians
• 1 radian = 180/π degrees

Example: Convert 60° to radians.

60 degrees

• Multiply by the conversion factor…

60π/180

• Reduce the fraction…

π/3

Example: Convert π/4 from radians to degrees.

π/4 radians

• Multiply by the conversion factor…

(π/4) (180/π)

• Remove the factor of π/π…

180/4 = 45°

Notice that angles in radians are (almost) always expressed in terms of π. This means that even fractional multiples of π rather than using decimal approximations. This allows for precise measurements rather than non-exact values that can result in rounding errors.

Here are some commonly used angles and their measurements in both degrees and radians.

• 0° = 0 radians
• 30° = π/6 radians
• 45° = π/4 radians
• 60° = π/3 radians
• 90° = π/2 radians
• 120° = 2π/3 radians
• 135° = 3π/4 radians
• 150° = 5π/6 radians
• 180° = π radians
• 210° = 7π/6 radians
• 225° = 5π/4 radians
• 240° = 4π/3 radians
• 270° = 3π/2 radians
• 300° = 5π/3 radians
• 315° = 7π/4 radians
• 330° = 11π/6 radians
• 360° = 2π radians

The absolute value of a number a is usually denoted as |a|. We can define absolute value in the following ways.

• |a| = a if a ≥ 0
• |a| = -a if a < 0

This means that the absolute value of any number is positive. In other words, the absolute value of any real number is the principal (positive) square root of that number squared.

• |a| ≥ 0
• |a| = √(a²)

Using the properties of absolute value given above, we can find another property regarding the absolute value of a product. Consider |ab|. Using the second property in the list above, we can rewrite this as √(ab)² = √(a²b²) = √(a²) √(b²) = |a| |b|. The absolute value of the product of two numbers is equal to the product of their absolute values.

• |ab| = |a| |b|

Geometrically, absolute value can be thought of as a length or distance from a point. Suppose a distance from 0 on the number line |x| = 5. There are two points that satisfy this condition. The distance can be measured in the positive direction, or in the negative direction. From this we can conclude that if the absolute value of a number is a, the number itself is either a or -a.

• If |x| = a, then x = a or x = -a

Solving Equations with Absolute Value

|x + 5| = 19

• This means that x + 5 is either 19 or -19, so we are left with two standard equations…

x + 5 = 19          OR          x + 5 = -19

• Subtract 5 from both sides of both equations….

x = 14          OR          x = -23

Solving Absolute Value Inequalities

Inequalities with absolute value are a little more complicated, but important to understand. Recall the distance concept once again. |x| < a would contain all numbers less than a units from 0 on the number line. |x| > a would contain all numbers more than a units away from 0.

Solving “Less Than” Equalities with Absolute Value

|x| < 5

• In order to be less than 5 units away from 0 in the negative direction, x must be more than -5. In order to be less than 5 units away in the positive direction, x must be less than 5. Therefore,

-5 < x  < 5

This is a consistent rule that can be generalized as follows:

• If |x| < a, then -a < x < a
• If |x| ≤ a, then -a ≤ x ≤ a

Solving “Greater Than” Inequalities with Absolute Value

|x| > 7

• In order for x to be more than 7 units away from 0 in the negative direction, x must be less than -7. In order for x to be more than 7 units away in the positive direction, x must be more than 7. Therefore,

x < -7          OR          x > 7

This rule can be generalized as well.

• If |x| > a, then x < -a OR x > a
• If |x| ≥ a, then x ≤ -a OR x ≥ a

The “or” in this case is very important. The solution should not be written as -5 > x > 5, since no value of x can be less than -5 and greater than 5 at the same time. In these cases, the solution is always two seperate inequalities and it is incorrect to combine them.

For most of algebra and trigonometry, the focus was on real numbers and the rectangular or Cartesian plane. “No solution” was an accepted answer for problems whose solutions were non-real, or complex. These solutions involve the “imaginary” numerical value of i, which is √-1. Algebra is rooted in the four fundamental mathematical operations (addition, subtraction, multiplication, division) as well as powers and root. Algebraic functions for complex numbers are limited, but with calculus we are able to apply more of these functions to complex values.

In the rectangular system, a complex number z has the form z = x + yi where (x, y) are the coordinates corresponding to the point on the complex plane. You will recall that the x-axis holds real number values on the complex plane, while the y-axis holds imaginary or non-real values. A point lying on the x-axis will have the form z = x + 0i = x, a real number. A point lying on the y-axis will have the form z = 0 + yi = yi, an imaginary number. As a result, the x-axis becomes the “real axis” and the y-axis becomes the “imaginary axis”.

In order to use DeMoivre’s Theorem, the complex number must be written in polar form. Recall that polar coordinates have the form (r, θ) where r is the radius (or distance from the pole) and θ is the angle formed between the polar axis and a line drawn from the pole to the point. The Cartesian point (x, y) can be converted into polar form through use of the Pythagorean Theorem. From this it is derived that x = r cos θ and y = r sin θ.

We will use the same properties to convert complex numbers between the rectangular and polar systems.

• z = x + yi = ( r cos θ) + (r sin θ)i =  r(cos θ + i sin θ)

Therefore, a complex number in the polar system has the form z = r(cos θ + i sin θ). The angle θ is called the argument of z.

DeMoivre’s Theorem provides a convenient way of raising a complex number z to a power of n. The theorem states that, if a complex number z = r(cos θ + i sin θ), then zⁿ= rⁿ[cos(nθ) + i sin(nθ)] as long as n is greater than or equal to 1.

Example:  Rewrite the complex number (1 + i)³ in standard form (a + bi) without exponents.

(1 + i)³

• In order to use DeMoivre’s theorem, we must rewrite the complex number in polar form.

[√2(cos π/4 + i sin π/4)]³

• Now we can begin using the theorem…

(√2)³ [(cos 3π/4 + i sin 3π/4)]

• Simplify (√2)³…

2(√2) [(cos 3π/4 + i sin 3π/4)]

• Solve for trig functions…

2(√2) [(-(√2)/2 + (√2)/2) i]

• Simplify…

-2 + i

As you should know by now, the domain of a function is the set of x-values for which the function is defined and the range is the set of y-values for which the function is defined. Domain and range are especially important when it comes to the graphs of trigonometric functions. Beginning with an angle θ drawn in standard position and a point (x, y) that corresponds to that angle on the unit circle, the trigonometric functions are defined as follows:

Definitions of the Trigonometric Functions

For an angle θ and its corresponding point on the unit circle (x, y),

• sin θ = y
• cos θ= x
• tan θ = y/x, x ≠ 0
• csc  θ = 1/y, y ≠ 0
• sec θ = 1/x, x ≠ 0
• cot θ = x/y, y ≠ 0

For sine and cosine, θ can be any angle so the domain of both functions contain the set of all real numbers. Remember that the unit circle is centered at the origin with a radius of length 1. Since (x, y) is the point on the unit circle corresponding to θ, it follows that both x and y exist on the interval [-1, 1]. Since sin θ = y and cos θ = x,  the range of these functions is the set of all real numbers between -1 and 1.

If x = 0 then the functions for secant and tangent are undefined. Since sin θ = x and x cannot equal 0,  θ cannot be π/2 (90°) or 3π/2 (270°). Since adding 2π to any angle will produce the same values for these functions, we can conclude that all odd multiples of π/2 (90°) must be excluded from the domains of tangent and secant. The range of the secant function consists of all real numbers less than or equal to -1, or all real numbers greater than or equal to 1. The range of the tangent function contains all real numbers.

if y = 0 then the cotangent and cosecant functions are undefined. On the unit circle, y = 0 at the angles 0 and π (180°). As in the previous two functions, the values are repeated every 2π or 360°. From this we can include that all even multiples of π/2 are excluded from the domain of cotangent and cosecant. Since all even numbers are multiples of two, this is the same set as all integral multiples of π. If θ is not one of these values then csc = 1/y = 1/sin θ. Since sin θ is defined between -1 and 1, csc θ must exclude this inner set. That is to say, the range of the co secant function contains all real numbers less than or equal to -1, or greater than or equal to 1. The range of cotangent contains the set of all real numbers.

Classifying different numbers is useful in noticing patterns and it becomes increasingly important in algebra when the properties of these numbers are very important. The numbers you are most familiar with are called counting numbers. These are the numbers you first learn to deal with when you learned to count (1, 2, 3, 4, 5 . . . etc.). These are also known as natural numbers, and are the numbers you use to count things.

Whole numbers are the numbers 0, 1, 2, 3, 4 . . .etc. They include the counting numbers and 0. The negative numbers are those numbers that are less than or before 0. They are preceded by a negative sign like this: -1, -2, -3, -4. . . etc. Integers include the whole numbers and the negative numbers. Each category expands to include more numbers, allowing us to do more complex problems. Integers allow us to solve problems involving both positive and negative numbers.

Another type of number is known as a rational number. This category includes any number that can be expressed as a fraction, or ratio, between two integers. If you have learned about fractions before, you will know that in the rational number a/b, a is the numerator and b is the denominator. The denominator cannot be 0. Examples of rational numbers include 1/2, 2/3, and 3/4. Since any integer a can be expressed as a/1, the integers are also rational numbers.

Though the rational numbers contain a large part of the number system, there is also a need for numbers than cannot be expressed as fractions. These include square roots like √2 as well as non-integer constants like π. These numbers cannot be expressed as a ratio between two numbers, and as a result they are known as irrational numbers. The irrational and rational numbers together form the real number system.

Numbers like 4.95 and 0.5 are known as decimals. Rational numbers can be expressed in decimal form by dividing the two numbers. 1/2, for example, becomes .5. The value .5 is called a terminating decimal because it terminates, or ends, after a certain number. The decimal form of 2/3 is .6666. . . This is known as a repeating decimal, because the series of sixes repeats forever.

Every rational number can be expressed as either a terminating or repeating decimal. However, some decimal values don’t terminate or repeat. Irrational numbers like √2 and π have decimal values that continue forever and don’t repeat. When expressing these as decimals, they are sometimes approximated for ease. The symbol ≈ means “is almost or approximately equal to”, and is used in these cases. For example, π ≈ 3.14.