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Algebra and numbers: Numbers
We start this TEAS Study Guide mathematics section by focusing on numbers, the basic building blocks of the subject.
Let’s look at numbers in terms of some of their specific features first, and we will do this by looking at various terms that you should understand:
- Integer: This includes zero and any whole number that’s positive or negative. It won’t include any mixed numbers, fractions, or decimals.
- Prime number: This includes whole numbers that have only two factors (can only be divided by 1 and itself) and are greater than 1.
- Composite number: This includes whole numbers that have more than two different factors and are therefore not prime numbers. If we take 8, which is a composite number, the factors (by which it is divisible) are 1, 2, 4, and 8.
- Even number: When a number can be divided by 2 and no remainder is left, it is known as an even number. For example, 8 is an even number (divided by 2 = 4), while 7 is not (if divided by 2, it leaves a remainder).
- Odd number: When an integer can’t be divided evenly by 2, it is an odd number. For example, 3 cannot be divided evenly by 2, so it is, therefore, an odd number.
- Decimal number: When a number uses a decimal point and this shows a section of it less than one, it is a decimal number, for example, 1.452.
- Decimal point: A symbol used in numbers and currency where it will distinguish the ones place from the tens place.
- Decimal place: When using a decimal point, the decimal place is the position of the number on the right-hand side. The first number is in first place, the second from the decimal point is in second place, and so on.
- Decimal: This numbering system, also known as base 10, uses digits from 0 to 9. A base 2, or binary numbering system uses something other than 10 digits. Used mainly in computing, the only numbers in this system are 0 and 1.
- Rational numbers: This includes all fractions, integers, and decimals. Both repeating and terminating decimal numbers are considered rational numbers.
- Irrational numbers: Because they have an infinite number of decimal places, and, within the number, there are no recurring patterns of digits, they cannot be written as fractions. π is an example of an irrational number because it is recurring.
- Real numbers: All rational and irrational numbers are considered real numbers.
As a way to see the distance between numbers in mathematics, we make use of a number line.
Usually, it will have a point that signifies zero.
To the left of that,, you will find negative numbers, and to the right, positive ones.
The number line will also use dashes to demarcate points.
For example, there might be four dashes between the numbers 1 and 2.
If the first dash is marked with a dot, this would signify ¼, for example.
Let’s talk about place value and numbers in word form.
Understanding how the place value system works is necessary to properly understand how to write our numbers in word form.
Each digit of the number shows how many of the corresponding place values are included in the represented number in the decimal system.
So it’s by a comma that every three numbers to the left of the decimal place are preceded, and this makes reading numbers far easier.
Here’s a handy table that you can follow
- 103: Value = 1,000. Place = thousands
- 102: Value = 100. Place = hundreds
- 101: Value = 10. Place = tens
- 100: Value = 1. Place = ones
- 10-1: Value = 0.1. Place = tenths
- 10-2: Value = 0.01. Place = hundredths
- 10-3: Value = 0.01. Place = thousands
Let’s look at factors and the greatest common factor.
When numbers are multiplied together to get a product, they are known as factors.
So when we look at an equation like 2×2 =4, the product is 4, and the factors multiplied to arrive at the product are 2 and 2.
While most numbers can have many factors, a prime number can only have to – itself and 1.
When a number can be divided exactly into two or more other numbers, it is known as a common factor.
So if we take 12, the factors would be 1, 2, 3, 4, 6, and 12.
If we take 15, the factors would be 1, 3, 5, and 15.
The common factors of 12 and 15 would then be 1 and 3, as they feature as factors of both.
As for a prime factor, well, you can probably guess from the name that this is a prime number as well.
In the example above, 2 and 3 are the prime factors of 12, while for 15, the prime factors are 3 and 5.
Let’s just mention GCF, or the greatest common factor, which is a factor of two or more numbers, but specifically in terms of the largest number that meets those requirements.
So take the numbers 15 and 35.
Here, 1, 3, 5, and 15 are the factors of 15, while 1,5,7, and 35 are the factors of 35.
In this example, CGF is therefore 5.
Let’s move on to multiples.
Multiples are something that you will see in multiplication tables.
This takes a given factor, for example, 7, and provides integer increments thereof.
So you will get an integer if you multiply a multiple by a factor.
Here’s an example with 7: 1 x 7 = 7, 2 x 7 = 14, 3 x 7 = 21, and so forth.
Also, if you divide the numbers 7, 14, and 21 by 7, you get their respective integers, in this case, 1, 2, and 3.
The smallest number that is a multiple of two or more numbers is known as the least common multiple, or LCM.
Take the numbers 3 and 5, for example.
Multiples for 3 are 3, 6, 9, 12, 15, and so on, while multiples for 5 are 5, 10, 15, 20, etc.
In this example, 15 is the LCM for 3 and 5.
Algebra and numbers: Operations
When an output is produced from some values that are taken as input, then, in mathematical terms, an operation has taken place.
Value operation value is the way in which elementary operations are often written.
So in the expression 1+2, the operation is addition, while 1 and 2 are the values used in the operation.
When we carry out that operation, we get 3, which is the output.
In math, we say this in a way that you will be more familiar with – 1+2 = 3.
Let’s look at the various kinds of operations that can occur.
One quantity’s value is increased by another in the operation we know as addition, and the order in which this is carried out doesn’t make a difference.
These quantities are called addends, with their result being the sum.
When signed numbers are added, you can add the addend’s absolute values if the signs are the same.
The sum will have the original sign applied to it afterwards; for example, (+5) + (+8) = +13.
That changes, however, when the addends have different sides.
Now you look at the addends’ absolute value, and you will take the smaller value from that of the larger value.
The difference between the two will then have the original sign of the large value put in front of it.
So (+5) + (-8) = -3.
Next, we have subtraction.
Here, one quantity’s value is increased by another in the operation.
The quantity that has its value decreased is known as the minuend, while the value doing the decreasing is known as the subtrahend.
Here’s a simple example: 2-1 = 1, with 1 simply known as the difference.
Unlike with addition, order does matter when it comes to subtraction.
If signed numbers are subtracted, the subtrahend must have its sign changed.
Once this has been done, the same rules as those of addition can be followed.
Then there is multiplication.
Repeated addition is what this is all about.
Here we have one number, which shows us how many times we should add another number.
The first number is called the multiplier, and the second, is the multiplicand.
So in the equation 2×2 or (two x two), it really just says 2+2, which we know equals 4 (which is known as the product).
As with addition, the order in which this is done doesn’t matter at all.
When multiplying signed numbers, if the signs are the same, the product will be positive; for example, (+5) x (+8) = +40.
The product is negative, however, if the signs are opposite, so (+5) x (-8) = -40.
The sign of the product depends on how many negative factors there are when multiplication involves more than two factors.
The product is negative if there are an odd number of negative factors and positive if there are an even number of negative factors.
So (+5) x (-8) x (-2) = +80, and (-5) x (-8) x (-2) = -80.
The last mathematical operation in this section is division.
Division is the opposite of multiplication in that one number indicates how many parts the other number is to be divided into.
The first number is called the divisor, and the second is the dividend, with the quotient being the result.
Unlike multiplication, however, the order of the numbers is something to take note of when dividing, so 3/2 is not the same as 2/3.
When it comes to signed numbers, division follows much the same rules as we learned with multiplication.
The quotient will be positive if the dividend and divisor have the same sign, while it will be negative if they have opposite signs.
Parentheses play a role in operations in that they dictate which should be carried out first if there are more than one.
So in the example, 4 – (2+1), 2+1 should be carried out first because they are in parentheses.
A superscript number that’s placed next to another in the top right position is known as an exponent.
They show us that the base number must be multiplied by itself, and the number of times this has to happen depends on the number of the exponent.
For example, 22 means 2×2, or in this example, it’s 2 squared.
If the exponent was 3, then it would be 3 cubed.
We can also call the exponent a power, so 8 to the 4th power is 84.
The inverse of raising a number to an exponent is known as a root.
Roots make use of the symbol √ as a way to show this particular operation.
So √5 is simply the square root of 5.
If a number has an integer for its square root, it is known as a perfect square.
In the number sequence 1 to 100, there are 10 perfect squares.
These are: 1, 4, 9, 16, 25, 49, 64, 81, and 100.
When dealing with operations, there is a certain order in which they should be carried out.
For this, you should remember PEMDAS.
- Parentheses first (inside of parentheses, apply PEMDAS as well).
- Exponents (roots, powers)
- Multiplication and division (from left to right)
- Addition and subtraction (from left to right)
Here’s an example: 13 – (3 – 4 x 6) + 24
= 13 – (3 – 24) + 24
= 13 – ( -21) = 24
= 13 + 21 + 16
Algebra and numbers: Percentages, decimals, and fractions
We start this section with fractions.
When a number is expressed as one integer above another, it’s known as a fraction.
These numbers are separated by a dividing line, for example, ½, which indicates 1 is divided by 2, with the line representing the quotient of these numbers.
The numerator is the name given to the number on top of the fraction, while the one on the bottom is called the denominator.
A fraction won’t have a denominator of 0.
By multiplying or dividing the numerator and the denominator, a fraction can be manipulated (but this does not apply to addition or subtraction).
The fraction will be simplified (reduced) when the numerator and denominator are divided by a common factor.
In equivalent fractions, they will have the same values, but they will be expressed differently.
For example, ²⁄₁₀ and as ³⁄₁₅ are equivalent fractions.
Finding the common denominator in fractions sees them being manipulated so that their denominator is then the same.
It’s the least common of the two original denominators that the chosen common denominator should be.
Let’s look at an example.
Take ¾ and ⅚.
12 is the least common multiple of 4 and 6, so let’s get to the common denominator by manipulating them.
That leaves us with ¾ = ¾ and ⅚ = ¹⁰⁄₁₂.
Let’s look at mixed numbers and proper fractions too.
A proper fraction is one where the denominator is bigger than the numerator.
If it’s the other way around, then you have an improper fraction.
Another thing to note is the value of the fractions.
With a proper fraction, that’s less than one, while with an improper fraction, it is more than one.
What about mixed numbers?
Well, that’s where you will find both a fraction and an integer.
It’s as a mixed number that any improper fraction can be rewritten, and vice versa.
Let’s look a little into adding and subtracting fractions.
This can happen if two fractions have the same denominator.
To do this, the same denominator is kept while the numerator is either added or subtracted, depending on what the operation requires.
One or both of the fractions will need to be manipulated should they not have the same denominator.
This is done so they can both then have the same denominator, allowing the operation to be carried out.
Here’s an example: ½ + ¼ = ²⁄₄ + ¼ = ¾.
What about multiplying fractions?
It’s possible, yes, and to do so, the numerators are multiplied to find the new numerator.
The same is carried out with the denominator as well.
As an example, ⅓ x ⅔ = ¹ˣ²⁄₃ₓ₃ = ²⁄₉.
And dividing fractions?
Well, when this operation is carried out, the denominator and the numerator are flipped around in the second fraction.
The operation is then treated like when you multiply fractions.
Here is an example: ⅔ ÷ ¾ = ⅔ x ⁴⁄₃ = ⁸⁄₉.
While we won’t go into it in any detail here, look a little deeper into your coursework with regards to multiplying fractions, specifically with regards to mixed numbers, whole numbers, and decimals.
We move on to decimals in our next section.
They are used to show parts of a whole.
From the decimal point, each number placed to the right of it shows the number of units of a corresponding negative power in the place value system.
But how do you write a decimal as a fraction?
Well, to start, make the denominator equal to 1 and the numerator equal to the value of the decimal.
Now to remove the decimal places, you will multiply both the numerator and the denominator by 10.
Now you will need to get the fraction to its lowest terms, and this can be done by simplifying it.
Here’s an example where we convert the decimal 0.24 to a fraction: 0.24 = 0.24/1 = 0.24×100/1×100 = ²⁴⁄₁₀₀ = ⁶⁄₂₅.
Let’s look at decimal operations too.
It’s critical to align the decimal point when adding and subtracting decimals, and as long as you do, it’s just like adding whole numbers.
Here’s an example: 4.5 + 2.5 = 7.0
To make sure you align the decimal points properly, use a vertical column to arrange them and add from right to left.
If the column adds up to more than 9, always remember to carry the number from the left.
You can apply these exact rules to the subtraction of decimals as well.
Next, we move on to percentages.
Based on the whole of 100, when you think of percentages, you can equate them to fractions.
In other words, when you think of one whole, it’s the equivalent of 100%.
In the TEAS exam, you will find three main ways in which these problems are presented:
- You may be asked to find the percentage one number is of another for example, what percentage is 10 of 100.
- You may be given a number and then asked to find what number is some percentage of that, for example, what number is 20% of 100.
- You may be asked to find out what number is another percentage that you are given, for example, what number is 8 20% of.
In each of these cases, there are three things you need to take note of – the whole (W), the part (P), and the percentage (%).
These three components play a part in the equation: P = W x %.
The great thing about this equation is that, depending on how the question is structured, it can be changed to suit it for example, % = P/W and W = P/%.
A big part of working out percentage problems is working out the quantities properly, and this is because they often take the form of word problems.
Algebra and numbers: Percentage, fractions, and decimal conversion
By moving the decimal point, it is possible to easily convert percentages to decimals as well as decimals to percentages.
It just matters which way you move the decimal point.
For example, you would move it two places to the right when you convert from a decimal to a percentage and two places to the left when you convert from a percentage to a decimal.
Here’s a helpful hint – the decimal number is always smaller than the equivalent percentage number.
When converting a fraction to a decimal, just take the denominator and divide it into the numerator.
For a conversion from a decimal to a fraction, place the decimal in the numerator and then put 1 in the denominator, following which the numerator and denominator are multiplied by tens until no decimal places remain.
From here, it’s to the lowest form of the fraction that you should simplify it.
Here’s an example: 0.24 = 0.24/1 = 0.24×100/1×100 = 24/100 = ⁶⁄₂₅.
By finding the equivalent fraction that has a denominator, you can convert fractions to a percentage; for example, 7/10 = 70/100 = 70% or ¼ = ²⁵⁄₁₀₀ = 25%.
By dividing the percentage number by 100, you can convert a percentage to a fraction but remember to simplify as far as possible, for example, 60% = 60/100 = ⅗.
Next, we move to rational numbers.
When a number can be expressed as a fraction or a ratio, it’s known as a rational number.
This means that r (which signifies the number) will be considered a rational number if it can also be shown as a fraction – a/b.
There are other provisions in play here, as b may not equal 0 and both a and b must be integers.
Both integers and decimals are included in a set of rational numbers, and a number is irrational if there is no possible way in which a value can be expressed with a fraction of an integer.
π is an example of an irrational number.
Algebra and numbers: Ratios and proportions
A link between two variables that defines how each changes as the other does is dubbed a proportion.
When a quantity grows or decreases by a certain amount based on changes in another quantity, this relationship is called a direct proportion.
For example, given a set speed, a trip’s duration goes up if the distance thereof increases.
The time it takes to reach a certain distance is directly proportional to it.
An inverse proportion sees a gain in one quantity cause a decline in the other, for example.
So if a car is driving on a long-distance trip, if it increases its speed, the time to reach the destination drops, while when the speed decreases, the time to reach the destination increases.
The comparison of two values in a specific order is known as a ratio.
For example, if a science lab has 7 microscopes but 10 students, the ratio of microscopes to students is 10:7.
If you are able, you should always reduce the ratio to its smallest number representation.
Let’s talk about proportionality constants.
There is a constant of proportionality across two quantities if they have a proportional relationship.
One quantity is equal to the product of this constant and the other value.
So if one sweet costs $0.50, two cost $1.00, and three cost $1.50, between the number of sweets bought and their cost, a proportional relationship exists.
Here, the unit price is the proportionality constant, and their price (t) can be worked out by multiplying their number by their unit price (p).
Next up is the work/unit rate.
This is when an amount of one item is expressed as a unit of another.
Here’s an example.
Let’s say that you travel on a train for 20 miles every two hours.
The unit rate would then show this comparison based on one hour, which is then 10 miles.
So 10 miles is considered the unit rate.
When using a unit rate, note that the value associated with the denominator thereof is always one.
Working out the unit rate can help you compare products, for example.
Take two butchers who are selling a cut of meat.
One sells it for a pound for $1.50, while the other sells it for two pounds for $2.75.
The unit rate on the second one is $1.37, so it’s cheaper per pound than the first example.
Unit rates can be used in many ways.
We move on to slope.
If you have a graph with two points, they will be plotted as (x1, y1) and (x2, y2).
The formula m=y2-y1/x2-x1 is then used to determine the slope.
M stands for the slope, while x1 ≠ x2.
The line will move from left to right and in an upward direction when the slope is positive, but it will move from left to right and in a downward direction when the slope is negative.
Algebra and numbers: Inequalities, expressions, and equations
Expressions, coefficients, and terms
It’s as a combination of one or more values, which are then added together after they are first arranged in terms that mathematical expressions are made up of.
A single variable term plus a constant (that could be equal to zero) creates a single variable linear expression.
So in the following example, 3w+8, 8 is the constant term, while 3w is the variable term.
Because of this, single numbers, of which zero is part of the group, can be an expression.
Constants, also known as constant terms, can also be included in expressions; these are numbers that don’t have a variable.
So let’s take 8s, for example.
Here, s is the variable, while the real number coefficient is 8.
If you see a term that does not have a real number, for example, just s, then you know that 1 is that term’s coefficient.
One variable linear equations are those that may be stated as a+b=0, where a = 0.
A root is the term given to the answer for these equations types.
Let’s look at a quick example.
So if we try to find a solution for x in this equation, – 5x+10=0 – its value is equal to -2 (the root).
Solving this is relatively simple if you take 10 away from both sides, which leaves you with 5 = -10.
You get the answer of -2 by dividing the coefficient of the variable (5) into both sides.
You can check if you are right as well in a simple manner, and that’s by placing the root back into the original equation to see if it works.
In other words, (5) (-2)+10=-10+10=0.
All the possible solutions to an equation are called the solution set.
In our above example, there is one solution (-2), so that is the solution set for that equation.
In multivariable equations, there is more than one answer, however, and these will then form part of that equation’s solution set.
You get an empty set too, and this is when an equation doesn’t have any true solution to it.
If equations have the same set of solutions, they are termed equivalent equations.
When a term has a determinant/value of 1, it is called an identity.
Here are some examples:
- Standard form: Ax + By = C. Here, -A/B is the slope, while C/B is the y intercept.
- Slope intercept form: y = mx + b. Here, the slope is m and the y intercept is b.
- Point slope form: y – y1 = m (x – x1). Here, the slope is m while a point on the line is shown by (x1, y1).
- Two point form: y – y1/x – x1 = y2 – y1/x2 – x1. Here, two points on the line are signified by (x1, y1) and (x2, y2).
- Intercept form: x/x1 + y/y1 = 1. Here, the point where the x axis is intersected by the line is shown by (x1, 0). (0, y1) is where the y axis is intersected by the same line.
How to solve one-variable linear equations
Start by eliminating all fractions, and to do this, multiply all terms by the lowest common denominator.
Now isolate all variables on one side of the equal signs by finding any addition or subtraction that you will need to undo.
The next step involves the coefficient of the variable which is divided into both sides of the equation.
By doing so, you will then have the value of the variable, and then this can be plugged into the equation, which will reveal the true equation.
Your coursework will include examples of how this can be done, so we are not going to go into any detail here.
Manipulating equations: Rules to remember
We start with the most important rule of all, and that’s remember to carry out the same operations on both sides of the equation.
The usual method for solving an equation is to do a number of steps on both sides thereof, and by doing so, the equation is simplified as much as possible.
Because it is important that the process preserves the equation’s meaning and gives a result that is equivalent to the original equation, the same operation must be performed on both sides thereof.
If we performed an operation on one side of an equation but not the other, this wouldn’t be true. Remember, an equation is a statement that two values/expressions are equal.
The two sides of the equation are changed in the same way and stay equal if we do so, and in that way, it’s easier to solve.
The next rule is combining like terms and the advantages thereof.
Adding or deleting like terms, which are those with the same variable, is often called combining like terms.
By doing so, sets of like terms are simplified into a single term, and through this, the benefit achieved is a simplified equation.
More often than not, when solving an equation, this is the very first step that should be carried out.
In some cases, like after distributing terms in a product, it can be carried out at a later point, however.
Let’s look at this example with an equation to consider: 2(x+3)+3(2+x+3) =-4.
Because they are like terms, 2 and 3 in the second set of parentheses can be combined, which results in the following: 2 (+3) + 3 (x + 5) = -4.
The parenthesis implies multiplication, so that’s the next step, which sees the outer 2 and 3 distributed, leaving: 2x+6+3x+15=-4.
In that, we have like terms in the form of 2x and 3, so these can be added together, leaving: 5 + 6 + 15 = 4.
Constants 6, 15, and -4 are also like terms, so they can be combined, which sees both sides of the equation having 6 and 15 subtracted from it.
That leaves 5x=-4-6-15 and that is 5-25, so if we simplify for x, the answer is -5.
What about terms on the opposite side of the equation and their cancellation?
When two terms on different sides of an equation are an exact match, then they can be canceled.
The same variable must be used when doing so, increased to the same power, and using the same coefficient.
For instance, consider this equation: 3x+2x2+6 = 2x2-6.
2x2 stands on both sides, and therefore, you can cancel it.
You then have 3x+6=-6.
Because 6 is added on one side of the equation and subtracted on the other, you may not cancel it.
The 6 and -6, on the other hand, are like terms and therefore can be combined.
This results in 3x = 12, which, when simplified, means x = -4.
It’s also crucial to remember that the terms that must be canceled cannot be a subset of a longer term; rather, they must be independent.
Consider the equation 2(x+6) = 3(x+4)+1 as an example.
Even though the x’s match, because they are part of the larger terms 2(x + 6) and 3(x + 4), they cannot be canceled.
Distributing 2 and 3 is the first course of action, and that results in 2x + 12 = 3x + 12 + 1.
That leaves terms with x’s that do not match, but we can cancel the 12s because they do.
This leaves 2x = 3x+1, and when simplified, -1 remains.
Manipulating equations: The process
In math, manipulating equations is such a necessary skill, but there’s a process that you need to follow.
To start, you must isolate variables.
What does this mean?
Well, it ensures that a variable, via manipulation of the equation, will appear by itself only on one side of the equation.
Remember, equations are usually solved when they are simplified as much as possible, but that can only happen when the variable has been isolated on one side.
Note, however, that it’s necessary to isolate only one variable in a two-variable (inequality) equation.
That’s because, for the most part, isolating both variables at the same time isn’t possible.
When an equation sees a variable appear and it’s only raised to the first power, this is known as a linear equation.
Here, you can move all the terms that appear with the variable to one side while the other terms are placed on the other side, and that will help isolate it.
On each side, you will then combine the like terms and, if necessary, take the coefficient of the variable and divide it into both sides.
When we use the term “moving,” what we mean is that the inverse of the term is added to both sides of the equation.
Remember, its sign is flipped when a term moves across to the equation’s other side.
How do you deal with inequalities then?
Often in algebra, mathematical expressions simply do not equal each other.
In that case, you will often see the symbol for less than or greater than used (<; >).
Statements in which these appear are called an inequality and here is an example: 7x > 5.
Divide both sides by 7 to solve for x so that leaves x > ⁵⁄₇.
It’s on a number line that inequality solution sets will be represented.
To show when expressions are approaching a number but never equaling it, open circles can be used.
Let’s look at how you would determine solutions to inequalities.
You can start by substituting the values of the coordinate into the inequality when trying to determine if it is a solution.
Your next step is to simplify and then see if the statement that results is true or not.
So let’s take the inequality, y ≥ -2x = 3, and the coordinates (-2, 4).
To begin, substitute (-2, 4) into the quality, leaving us with: 4 ≥ -2(-2) + 3.
4 ≥ 7 results when you carry out the next step, which is simplifying the inequality.
Is this statement true or false?
Well, it’s false, so that means that it’s not possible for the coordinates of (-2,4) to be a solution to the inequality y ≥ -2x = 3.
The part of the graph of an inequality that is shaded can also be determined through this method.
There is a solid line, y = -2x + 3 in the graph y ≥ -2x = 3.
We know it doesn’t include the point (-2,4) from solving the equation, and because of this, the area to the right of the line will be shaded, while the area to the left isn’t.
It’s possible to turn an entire inequality around, which means swapping its two sides and then changing its sign.
So x + 2 > 3x – 3 is the same as 2x – 3 < x + 2.
Other than that, usually the inequality won’t change when operations are carried out on both sides of it.
We did say usually, because there is one time to take note of when it will.
The inequality will flip when a negative number is multiplied or divided into both sides thereof.
Here’s an example with the following inequality: -2x < 6
By dividing both sides by – 2, we are left with x > -3 when both sides are flipped.
Note that this is only with negative numbers and only when multiplying and dividing.
When reciprocals are used, the inequality sign is flipped as well.
So while 3 > 2, the reciprocal relation is ½ < ⅓.
Graphical equation solutions
Usually, it’s on a cartesian coordinate plane that equations are shown graphically.
This is made up of two number lines that run perpendicular to each other, and it’s at point zero, or the origin, that they will intersect, dividing the plane into four quadrants – I, II, III, and IV.
The x-axis runs horizontally, while the y-axis will run vertically.
On the x-axis, positive values run to the right of the origin and negative values to the left.
On the y-axis, positive values run upwards from the origin and negative values downward.
Points on the plan are tracked in coordinates, designated in the form (x,y).
The name abscissa is given to the x-coordinate and ordinate to the y-coordinate.
- Quadrant 1: x>0 and y>0
- Quadrant 2: x<0 and y<0
- Quadrant 3: x<0 and y<0
- Quadrant 4: x>0 and y>0
The slope line will go upwards from left to right when the value is positive and downwards from left to right if it is negative.
The line is horizontal if the y-coordinates are the same at two points on it, but if that’s the case for the x-coordinates, then the line is vertical.
When two or more lines have the same slope, they are parallel, while negative reciprocals will produce slopes forming perpendicular lines.
When graphing simple inequalities, start with the value that shows the endpoint of the inequality and mark it on the number line.
Use a hollow circle to mark when the inequality is strict (which means it involves a greater or less than).
Use a solid circle if the inequality is not strict (which means greater than or equal to or less than or equal to).
Now, to satisfy the inequality, fill in the part of the number line that does so.
For less than (or less than or equal to), this is to the left of the marked point, while for greater than (or greater than or equal to), this is to the right.
Data and measurement: Principles of measurement
Accuracy, precision, and error
Let’s start this section by looking at what we mean when we say precision.
Well, in a nutshell, this looks at a measurement with two factors in mind – how reliable it is and whether it is repeatable or not.
Think of a dartboard.
If you throw a dart in the same spot over and over again, but that’s not the bullseye you were actually aiming for, is that not precise?
Well, it is, but that’s not what you are looking for, right?
What about accuracy?
This looks at the data collected from the standpoint of just how close it is to the data you know to be correct.
If you hit the bullseye each time but not in the exact same spot, that’s considered accuracy.
Data can be precise but still not be accurate.
Consider a scale that’s not set up properly, and adds two pounds to someone’s weight.
When you take a weight reading on it, it will be precise, but it’s not accurate because it’s off by two pounds due to a faulty setup.
Data that is repeatable and correct can be considered to be both precise and accurate.
The next thing to look at is the approximate error.
This deals with a physical measurement, specifically from the standpoint of the amount of error in it.
When reported, it is shown as a measurement with a ± next to it as well as the approximate error amount.
What about maximum possible error?
This is half the size of the smallest measurement unit.
Here’s an example.
Let’s say that 1 centimeter is the unit of measurement used; the maximum possible error for this would then be ½ centimeters.
When written, it will be in the following way: ± 0.5 cm.
When reporting the maximum possible error, it’s critical that vital figures are used.
In other words, when reporting the answer, compared to the least accurate measurements taken, it should not be made to appear more accurate.
Estimation and rounding
When you try to keep the value similar, but want to reduce the number of digits in a number, this is known as rounding.
While the result leaves the value in a form that’s easier to use thanks to it being simpler, it’s obviously not as accurate.
That said, we use rounding in math all the time, with whole numbers rounded either to the nearest ten, hundred, or thousand.
In some cases, estimation is used to come up with a potential solution to a problem in the form of an approximate figure.
It’s vital that all numbers must be rounded to the same level when estimating a sum.
So one cannot be rounded to the nearest thousand, another to the nearest ten, and yet another to the nearest hundred.
They must all be rounded to the nearest hundred, for example.
Here are the metric measurement prefixes you must know:
- Giga: one billion (for example, 1 gigawatt)
- Mega: one million (for example, 1 megahertz)
- Kilo: one thousand (for example, 1000 grams in a kilogram)
- Deci: one-tenth (for example, 1 decimeter makes up one-tenth of a meter)
- Centi: one-hundredth (for example, 1 centimeter makes up one-hundredth of a meter)
- Milli: one-thousandth (for example, 1 milliliter makes up one-thousandth of a liter)
- Micro: one-millionth (for example, 1 microgram makes up one-millionth of a gram)
Let’s talk a little about measurement conversion.
While you want to change the way it is displayed, ultimately, when converting between units, you still have to keep the same meaning.
Multiply the number of the known amount by the equivalent amount when converting from a larger unit to a smaller one.
You will divide the number of the known unit by the equivalent amount when converting from a smaller unit to a larger one.
Conversion fractions are the way to go when dealing with very complicated conversions.
Here, one will be the conversion factor, and the numerator in the other fraction will be unknown.
It’s in the denominator that the known value is then placed.
In some cases, the problem’s known value is in the numerator, and the unknown in the denominator in the second fraction.
To get the converted measurement, you would multiply the two fractions, and the value of the fraction is 1 because the numerator and the denominator of the factor are equivalent.
Because of this, even though they have different numbers, the result in the new units is equal to that in the old units.
Look through your coursework for measurement conversion examples to further improve your knowledge thereof.
We move on to common units and equivalents.
First, metric equivalents.
- 1000 μg (micrograms): 1 mg
- 1000 mg (milligram): 1 g
- 1000 g (gram): 1 kg
- 1000 kg (kilogram): 1 metric ton
- 1000 ml (milliliter): 1L
- 1000 μm (micrometer): 1 mm
- 1000 mm (millimeter): 1 m
- 100 cm (centimeter): 1 m
- 1000 m (meter): 1 km
We need to discuss distance and area measurement too.
- Inch (in): US equivalent = 1 inch; metric equivalent = 2.54 centimeters
- Foot (ft): US equivalent = 12 inch; metric equivalent = 0.305 meters
- Yard (yd): US equivalent = 3 feet; metric equivalent = 0.914 meters
- Mile (mi): US equivalent = 5280 feet; metric equivalent = 1.609 kilometers
- Acre (ac): US equivalent = 4840 square yards; metric equivalent = 0.405 hectares
- Square mile (sq. mi or mi2): US equivalent = 650 acres; metric equivalent = 2.590 kilometers
And then there are capacity measurements.
- Fluid ounce (fl oz): US equivalent = 8 fluid drams; metric equivalent = 29.573 milliliters
- Cup (c): US equivalent = 8 fluid ounces; metric equivalent = 0.237 liters
- Pint (pt): US equivalent = 16 fluid drams; metric equivalent = 0.473 liters
- Quart (qt): US equivalent = 2 pints; metric equivalent = 0.946 liters
- Gallon (gal): US equivalent = 4 quarts; metric equivalent = 3.785 liters
- Teaspoon (t or tsp): US equivalent = 1 fluid dram; metric equivalent = 5 milliliters
- Tablespoon (T or tbsp): US equivalent = 4 fluid drams; metric equivalent = 15 or 16 milliliters
- Cubic centimeter (cc or cm3): US equivalent = 0.271 drams; metric equivalent = 1 milliliter
Next up, weight measurements.
- Ounce (oz): US equivalent = 16 drams; metric equivalent = 28.35 grams
- Pound (lb): US equivalent = 16 ounces; metric equivalent = 453.6 grams
- Ton (tn): US equivalent = 2,000 pounds; metric equivalent = 907.2 kilograms
Data and measurement: Geometric quantities
Volume, perimeter, and area
Having the measurements of the square, cube, or other two or three dimensional figures can help you determine the volume, perimeter, or area of these figures.
There are well-defined formulas that will help you do that, once you know the measurements of these figures.
In a two-dimensional shape, if you are looking to establish its perimeter, you need to add together the edges’ total lengths.
While that’s easy in some cases – like with a square, for example – doing so in a two-dimensional shape like a circle isn’t that simple.
That said, the same principle applies when measuring perimeter in both of these figures.
What about the area of a two-dimensional shape?
This looks at the flat space within a shape and helps determine just how much that is.
With a rectangle, for example, this isn’t difficult, as you would just need to multiply the length by the width.
But it gets more complicated than that with a compound figure, for example, where the various parts thereof must be added to each other to find out the full extent of the area it takes up.
Volume deals with solids, or, as they are more commonly known, three-dimensional figures, and this determines the space within them.
Again, much like we covered with perimeter and area, some of these calculations are easier than others.
These are closed figures that have three (or higher) sides (straight line segments).
They are two-dimensional.
The vertex is the point where two sides of a polygon intersect.
If you look at a polygon, you can always be sure that the number of sides it has is exactly the same as the number of vertices it has.
A regular polygon is one that has equal angles and congruent sides.
Let’s look at a list of common polygons you should know:
- 3-sides: Triangle
- 4-sides: Quadrilateral
- 5-sides: Pentagon
- 6-sides: Hexagon
- 7-sides: Heptagon
- 8-sides: Octagon
- 9-sides: Nonagon
- 10-sides: Decagon
- 11-sides: Dodecagon
By adding together the lengths of its three sides, you can easily determine the perimeter of a triangle.
The equation for this is P = a + b + c.
That changes in an equilateral triangle, however.
So because the sides are all equal in length, the equation can now be written as P = 3a, where a is the length of one of the sides of the triangle.
When a geometric figure has four straight sides and is closed, it is known as a quadrilateral.
Diagonals are lines that connect together in a quadrilateral, but specifically in opposite corners.
Now let’s look at various types of quadrilaterals, starting with a kite.
With this shape, there is congruence between two pairs of adjacent sides, which results in perpendicular diagonals.
Kites have one line of symmetry and can be both convex and concave.
Our second quadrilateral shape is the trapezoid.
These will always have one pair of sides that run parallel to each other.
When it comes to the second pair of sides, however, anything goes, and because of this, in a trapezoid, there are no rules to consider regarding diagonals.
Lines of symmetry have no rules either.
If a trapezoid has equal base angles, it is known as an isosceles trapezoid.
Because of the equal base angles:
- The non-parallel sides are exactly the same length
- The non-base angles are equal
- A line of symmetry exists between the parallel sides’ midpoints
The following equation is used to work out the area of a trapezoid: ½h (b1 + b2).
- The height is defined by h (this is the segment that connects the parallel bases)
- b1 + b2 relate to the two parallel sides
Note that unless one of the sides is perpendicular to the parallel bases, it cannot be used as the height measurement in this equation.
To work out a trapezoid’s perimeter, you would use the following equation: P = a + b1 + c + b2.
Here, the four sides of the trapezoid are represented by a, b1, c, and b2.
We move on to the parallelogram.
When there are two pairs of opposite parallel sides in a figure that are also congruent, it is known as a parallelogram, which is a type of trapezoid.
When comparing the consecutive interior angles, you will find that they are supplementary, while the opposite interior angles are congruent.
The parallelogram is divided by the diagonals, and this forms two congruent triangles.
At its midpoint, a parallelogram does have 180-degree rotational symmetry, but it does not have a line of symmetry.
When working out the area of a parallelogram, the following formula is used: A = bh, where b is the base’s length and h is its height.
When working out the perimeter of a parallelogram, the following formula is used: P = 2a + 2b or P = 2 (a + b).
Here, the lengths of the two sides of the parallelogram are signified by a and b.
When a quadrilateral has four right angles, it’s known as a rectangle.
While not all trapezoids or parallelograms are considered rectangles, all rectangles are indeed trapezoids and parallelograms.
Congruence exists between the diagonals of a rectangle, while this shape has 180-degree rotational symmetry at the midpoint as well as two lines of symmetry that are through the opposing midpoint pairs.
When working out the area of a rectangle, the following formula is used: P = 2l + 2w or P = 2(l + w) with w being the width and l being the length.
When working out the perimeter of a rectangle, the following formula is used: A = lw.
Next, we look at the rhombus.
When a quadrilateral has four congruent sides, it is a rhombus.
A rhombus is also a kite and a parallelogram, and because of this, it has the properties of both.
In a rhombus, the diagonals run perpendicular to each other, while there are two lines of symmetry along them as well as 180-degree rotational symmetry.
When working out the area of a rhombus, the formula to use is A = d1d2/2.
Then we have the square.
This has four congruent sides, as well as four right angles, and is from the quadrilateral family.
In a square, all diagonals are perpendicular and congruent to each other.
In a square, you will find four lines of symmetry, which run along the diagonals as well as through each pair of opposing midpoints.
At the midpoint, it also has 90-degree rotational symmetry.
To find the area of a square, use the formula A =s2, with s being the length of one side.
For the perimeter of the square, you would use the formula P = 4s.
What about the circle?
Every point on a circle is equally distant from its center.
When we mention the radius, this joins the center of the circle from any point and is a line segment.
In a circle, all radii are equal.
A concentric circle will not have the same length of radii but will have the same center.
A line segment that has both endpoints on the circle and moves through the center is known as the diameter.
When compared to the radius, the diameter is twice as long.
To work out the area of a circle, use the formula A = πr2.
To work out the circumference of a circle, use the formula C = 2πr.
Next up are solids.
To work out the volume of any prism, use the formula V = Bh, with B the base’s area and h the height.
To work out the volume of a rectangular prism, use the formula V = lwh, where h is the height, w is the width, and l is the length.
To work out a cube’s volume, use the formula V = s3.
Here, the length of the sides is signified by s.
For a sphere’s volume, the formula is a little more complicated.
It’s V = ⁴⁄₃πr3 with r as the radius.
While we won’t go into any great detail here, check your coursework to make sure you understand how to work out the volume of a cone, a pyramid, and a cylinder as well.
Data and measurement: Statistics
Measures of central tendency
A reasonable estimate of the center of a group of data is a statistical value that is known as a measure of central tendency.
It can be described in various ways, but each way is calculated in its own manner while looking at the dataset in its own unique way.
One of the most critical things is using the same units when giving a measure of central tendency.
This means converting the data to the same units, where necessary.
For example, if the data set is made up of hours, minutes, and seconds, decide on the best unit and convert the others to that.
Do not give units for the measure of central tendency if none are given in the data.
Let’s move on to the statistical mean.
This is when the arithmetic average of a group, and the group of data itself are the same.
A set of data’s mean can be found by changing all values to the same units, where necessary.
Next, add all the data values together and count their total number together too.
When doing so, make sure you have each and every data value that’s available, so count it more than once if it appears more than once.
Now, the sum of the values is divided by their total number, and then the units are applied.
The mean may not divide evenly and doesn’t have to be one of the set’s data values.
So the equation for calculating the mean is: sum of the data values/quantity of the data values.
If used as the only measure of central tendency, the mean can be pretty imprecise.
In some cases, the mean is distorted when there are some values that are far higher or lower than most of the others.
These data values that are significantly different from the majority are called outliers.
We now move on to the median.
The value in the middle of a data set is known as the statistical median.
It’s relatively straightforward to work out, and you begin by first listing all data values from the smallest to the largest.
If a value is repeated, it must be listed each time.
The median is then the value in the middle of the list of data values, which is an odd number.
If an even number, then take the two middle values and work out their arithmetic mean.
This is a very accurate tool when there is even dispersion in the data set.
When the distribution of values is skewed because of a large group of values, using a median may not be that accurate.
Next up is mode.
The data value that occurs the most often in a data set is called the mode.
In data sets, you could have multiple modes in some cases, one mode, or, at times, even no mode.
Finding the mode starts with arranging the data in much the same way as you would when trying to establish the median, as we covered above.
Now look at each value and note the number of times it appears in the data set.
This is important because all values appear the same number of times; for example, if they appear just once, there is no mode.
The value that appears more times than any other will be the mode.
All values are the mode if two or more are present the same number of times, but others are present fewer times, and none are present more times.
As a way to understand the measure of central tendency by looking at how the set’s data values are distributed, we can use a single value known as a measure of dispersion.
In doing so, we can reduce the disadvantages that single measures of central tendency, such as mean, median, and mode, have.
This provides a clearer picture of the dataset on an all-encompassing level.
Calculating the range, standard deviation, or variance of a data set will help work out the measure of dispersion.
Let’s start with how to calculate range.
The difference between the biggest and smallest values in a data set is their range, so it’s simple enough to work out.
To start, all the units for these values need to be the same.
Then, note the largest and smallest values.
In the formula, you only need to use one of the largest and smallest values if there are multiples of them.
We move on to standard deviation.
This dispersion contrasts a set’s data values with its mean as a way to provide a clearer picture.
There is a larger variance in the data values from the mean when a larger standard deviation occurs.
The data graph takes on a bell curve shape when there is a normal distribution of the data around the mean.
A larger standard deviation is an indication of graphs that don’t follow this shape.
This also indicates that the mean is farther away from the data.
A smaller standard of deviation will show that the mean and most of the data set’s values are closer together.
Data and measurement: Probability
A particular outcome likely to occur for a certain event (a scenario that yields a result) is known as probability.
Events can be both simple, like playing heads or tails, or more complex, like launching a space shuttle.
Because events can be simple or complex, so can their probability outcome.
When looking at probability, there are a range of terms that you should know.
- Compound event: This has two (or more) independent events that take place (for example, picking two cards from a pack and then adding their value).
- Desired outcome: When a certain set of criteria are met by an outcome (for example, rolling 6 on a dice to roll again).
- Independent events: When the outcomes of two or more events do not affect one another (two coins tossed simultaneously).
- Dependent events: When the outcomes of two or more events do affect one another (drawing two cards from one deck at the same time)
- Certain outcome: 100% probability of the outcome happening.
- Impossible outcome: 0% probability of the outcome happening.
- Mutually exclusive outcomes: When a single event cannot satisfy the criteria of two or more outcomes
- Random variable: This is all of a single event’s possible outcomes, which can be continuous or discrete.
Let’s move on to theoretical and experimental probability.
Without even carrying out the event, one can determine theoretical probability.
The following formula provides the likelihood or probability of an outcome occurring: P (A) = number of acceptable outcomes/number of possible outcomes.
The probability is denoted by P (A), and that’s of the outcome (denoted by A) occurring.
Every outcome has an equal chance of happening.
Therefore, the acceptable outcome total must be equal to or less than the possible outcome total.
When these two are equal, the probability is said to be 1, because the outcome will happen.
If there are zero acceptable outcomes, then the probability is said to be 0, because the outcome won’t happen.
Here’s an example.
Let’s say there are 20 cupcakes under a cloth, and children are asked to put their hands under and pick one.
Of the 20 cupcakes, 5 have red decorations and 15 have white decorations.
The probability of a child choosing a cupcake with red decorations is therefore 5 in 20.
If we plug that into our equation, we have ⁵⁄₂₀ which, when simplified, is ¼.
That’s 0.25, which, made into a percentage, becomes a 25% chance.
That’s as much as we cover in this guide regarding probability, but be sure to read up in your coursework on sample space and the addition and multiplication rules.
Data and measurement: Displaying information
Dependent and independent variables
You must know which variables are dependent and which are independent when displaying information.
- An independent variable is one that helps establish the order of information or a grouping, for example, categories, or time.
- A dependent variable, when compared across independent variables, is of particular interest, for example, frequency or growth.
The frequency of each value in a set can be shown using a frequency table.
The proportions of each unique value contrasted against the entire site can be seen in a relative frequency table.
Here, the relative frequency values are expressed as percentages, but note that due to rounding, a frequency table’s total percentage might not equal 100%.
With a line plot, there are points plotted, but line segments won’t connect them.
It is also sometimes known as a dot plot.
Different data values are plotted along the horizontal axis of this type of graph, while on the vertical axis, you will plot each value according to the number of times it occurs.
To show this, a single dot will be placed on the graph for each of the values.
While the word line is used to explain these plots, it’s not that closely related to a line graph but is similar to a bar graph.
Remember that data is misconstrued when you connect line segments to the dots, so this is something that should never be done when using line plots.
It is for discrete items that occur in fairly small numbers that line plots are best suited.
For example, in a hospital setting, you can use this to plot a patient’s fruit consumption over a specific time period.
To show the different values of data sets, a line graph, which consists of one or more lines that can be broken or solid, is used.
As on a Cartesian plane, it is in ordered pairs that individual data is represented.
The units used defined the x and y axes on the graph.
Here, line segments join together points plotted on the graph, and this will indicate an increase or decrease.
If the data indicates an increase, the line will slope upward, while a decrease is characterized by a downward-sloping line.
A horizontal line indicates that the data is not changing from point to point (until there is a change, of course).
As a way to compare data sets easily, they can be graphed at the same time.
These graph types can be drawn both horizontally and vertically and are one of the few that can be displayed in this manner.
The way data is placed on a bar graph is very similar to how it is done on a line plot.
For the graph to work correctly, both axes will have to have clear definitions as to what their categories are.
A thick line or bar is used instead of the single dot of a line plot to mark data value points on the graph.
This line runs from zero to the data’s exact value.
This value could be a percentage, a number, or other types of numerical values.
The longer the bar is represented by a data set on the graph, the bigger the value.
Reading a bar graph is simple.
Start by establishing the units that it is reporting by reading the axes where they will be indicated.
From there, it’s easy to determine the value associated with each bar in the graph.
As a way to show small or large counts across various categories, bar graphs excel.
It’s easy to mistake a histogram for a vertical bar graph when you first look at it.
There is a major difference in the fact that a histogram has one long bar for each data range, while a bar graph has a separate bar for each data range.
A histogram shows numerical values on both axes, while a bar graph only shows those values on one axis.
The ranges on a histogram occur in order from left to right and from lowest to highest.
The number of data values within a given range is shown by the height of each column.