# 1.2: Other Rules - Mathematics

Let’s play the dots and boxes game, but change the rule.

The 1←3 Rule

Whenever there are three dots in single box, they “explode,” disappear, and become one dot in the box to the left.

Example (PageIndex{1}): Fifteen dots in the 1←3 system

Here’s what happens with fifteen dots:

The 1←3 code for fifteen dots is: 120.

Problem 2

1. Show that the 1←3 code for twenty dots is 202.
2. What is the 1←3 code for thirteen dots?
3. What is the 1←3 code for twenty-five dots?
4. What number of dots has 1←3 code 1022?
5. Is it possible for a collection of dots to have 1←3 code 2031? Explain your answer.

Problem 3

1. Describe how the 1←4 rule would work.
2. What is the 1←4 code for thirteen dots?

Problem 4

1. What is the 1←5 code for the thirteen dots?
2. What is the 1←5 code for five dots?

Problem 5

1. What is the 1←9 code for thirteen dots?
2. What is the 1←9 code for thirty dots?

Problem 6

1. What is the 1←10 code for thirteen dots?
2. What is the 1←10 code for thirty-seven dots?
3. What is the 1←10 code for two hundred thirty-eight dots?
4. What is the 1←10 code for five thousand eight hundred and thirty-three dots?

Think / Pair / Share

After you have worked on the problems on your own, compare your ideas with a partner. Can you describe what’s going on in Problem 6 and why?

When rounding numbers, you must first understand the term "rounding digit." When working with whole numbers and rounding to the closest 10, the rounding digit is the second number from the right—or the 10's place. When rounding to the nearest hundred, the third place from the right is the rounding digit—or the 100's place.

First, determine what your rounding digit is and then look to the digit at the right side.

• If the digit is 0, 1, 2, 3, or 4, do not change the rounding digit. All digits that are on the righthand side of the requested rounding digit become 0.
• If the digit is 5, 6, 7, 8, or 9, the rounding digit rounds up by one number. All digits that are on the righthand side of the requested rounding digit will become 0.

## Sign Rules

Number is a mathematical object used to measure, label and other mathematical operations. Basic mathematical operations are Addition, Subtraction, multiplication and division.

The addition of two whole numbers is the total amount of those quantities combined.

Addition is written using the plus sign “+” between the terms that is, in infix notation. The result is expressed with an equals sign. For example,

(“one plus one equals two”) (“two plus two equals four”) and etc.,

If signs are the same, add and keep the sign same:

Case 1: If sign of both numbers are positive,then the result will have positive sign.

Case 2: If signs o both numbers are negative,then the result will have negative sign.

If signs are different, then subtract and keep the sign of larger value.

Case 1: If sign of the larger value us positive sign, then, the result will have positive sign.

Case 2: If sign of the larger value us negative sign, then, the result will have negative sign.

SUBTRACTION:

Subtraction is the operation of removing objects from a collection. It is written using the sign “-” between the terms. The result is expressed with an equals sign. For example,

2 – 1 = 1 (“two minus one equals 1”) 5 – 3 = 2 (“five minus three equals two”) and etc.

SUBTRACTION SIGN RULES:

For example,

= (-10) + (-8) = (-18) [Changed the sign from (+8) to (-8), then followed addition sign rule]

= (-10) + (+8) = ( -2) [Changed the sign from (-8) to (+8), then followed addition sign rule]

= (+10) + (+8) = (+18) [Changed the sign from (+8) to (-8), then followed addition sign rule]

= (+10) + (-8) = (+2) [Changed the sign from (+8) to (-8), then followed addition sign rule]

Multiplication:

The multiplication may be thought as a repeated addition that is, the multiplication of two numbers is equivalent to adding as many copies of one of them, the multiplicand, as the value of the other one, the multiplier.

“Normally, the multiplier is written first and multiplicand second.”

Multiplication is written using the sign “x” between the terms. The result is expressed with an equals sign. For example,

For example, 4 multiplied by 3 (often written as 3 x 4 and said as 𔄛 times 4”) can be calculated by adding 3 copies of 4 together:

IF THE SIGNS ARE SAME, MULTIPLY AND PUT POSITIVE SIGN.

Case 1: If the signs are positive then multiply and put positive sign.

Case 2: If the signs are negative then multiply and put positive sign.

IF THE SIGNS ARE DIFFERENT, MULTIPLY AND PUT NEGATIVE SIGN IRRESPECTIVE OF VALUE OF THE NUMBER.

Division is the opposite of multiplying. It is written using the sign “÷ or /” between the terms. The result is expressed with an equals sign.

When we know a multiplication fact we can find adivision fact:

DIVISION SIGN RULE:

DIVISION SIGN RULE IS SAME AS MULTIPLICATION, SO FOLLOW MULTIPLICATION SIGN RULES.

(a) (-15) /3 = (-5) [Multiplication sign rule: if signs are different then put negative sign]

(b) (15) /(-3) = (-5) [Multiplication sign rule: if signs are different then put negative sign]

(c) (-15) /(-3) = (+5) [Multiplication sign rule: if signs are same then put positive sign]

(d) (+15) /(+3) = (+5) [Multiplication sign rule: if signs are same then put positive sign]

## Other Rules

Let’s play the dots and boxes game, but change the rule.

### The 1←3 Rule

Whenever there are three dots in single box, they “explode,” disappear, and become one dot in the box to the left.

### Example: Fifteen dots in the 1←3 system

Here’s what happens with fifteen dots:

Solution: The 1←3 code for fifteen dots is: 120.

### Problem 2

1. Show that the 1←3 code for twenty dots is 202.
2. What is the 1←3 code for thirteen dots?
3. What is the 1←3 code for twenty-five dots?
4. What number of dots has 1←3 code 1022?
5. Is it possible for a collection of dots to have 1←3 code 2031? Explain your answer.

### Problem 6

1. What is the 1←10 code for thirteen dots?
2. What is the 1←10 code for thirty-seven dots?
3. What is the 1←10 code for two hundred thirty-eight dots?
4. What is the 1←10 code for five thousand eight hundred and thirty-three dots?

### Think / Pair / Share

After you have worked on the problems on your own, compare your ideas with a partner. Can you describe what’s going on in Problem 6 and why?

## Create a Rule to Generate a Number Pattern

Videos, solutions, worksheets, and examples to help grade 5 students learn how to create a rule to generate a number pattern, and plot the points.

### New York State Common Core Math Module 6, Grade 5, Lesson 12

1. Write a rule for the line that contains the points (0, 4) and (2 1/2, 2 3/4).
a. Identify 2 more points on this line, then draw it on the grid below.
b. Write a rule for a line that is parallel to BC and goes through point (1, 2 1/4)

2. Give the rule for the line that contains the points (1, 2 1/2), (2 1/2, 2 1/2)
a. Identify 2 more points on this line, then draw it on the grid above.
b. Write a rule for a line that is parallel to GH.

3. Give the rule for a line that contains the point (3/4, 1 1/2), using the operation or description below. Then, name 2 other points that would fall on each line.
b. A line parallel to the x-axis: _________
c. Multiplication: _________
d. A line parallel to the y-axis: _________

(Problem Set) 4. Mrs. Boyd asked her students to give a rule that could describe a line that contains the point (0.6, 1.8). Avi said the rule could be multiply by 3. Ezra claims this could be a vertical line, and the rule could be is always 0.6. Erik thinks the rule could be add 1.2 to Mrs. Boyd says that all the lines they are describing could describe a line that contains the point she gave. Explain how that is possible, and draw the lines on the coordinate plane to support your response.

Try the free Mathway calculator and problem solver below to practice various math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.

## 1.2: Other Rules - Mathematics

An Introduction to Logic

In debates, I often find that people are unwilling to accept the rules of logic, and they make foolish comments like, “well you’re entitled to your opinion.” In reality, the rules of logic are like the rules of mathematics. They are an inherent and immutable property of existence, not opinions. Just as 2+2 always equals four, the rules of logic are always true and must always be followed. To illustrate, the most basic rule upon which all other rules rely is known as the Law of Noncontradiction. It states that something cannot be A and not A simultaneously. In other words, two mutually exclusive things cannot exist simultaneously. For example, you cannot have a circular triangle, because a circle, by definition, has no straight lines and no corners, and a triangle, by definition, has three straight lines and three corners. An object cannot simultaneously have zero corners and zero lines and three corners and three lines. That’s not an opinion, it’s an immutable property. If you reject the rules of logic, then you have just acknowledged the possibility of a triangular circle, and, in fact, all rational thought disintegrates. You see, we all inherently and intuitively know that the rules of logic work, and we apply them in our daily lives, we just don’t often think about them in technical terms. For example, suppose that your fuel gauge shows that you are low on gas, and you know that your fuel gauge works, what do you conclude? Obviously, you will conclude that you are low on fuel, but why did you reach that conclusion? Without you even knowing it, your brain did the following:

1. My fuel gauge is designed to tell me how much fuel I have
2. I know that my fuel gauge works
3. My fuel gauge says that I am low on fuel
4. Therefore I am low on fuel.

That’s plain and simple deductive logic. If, however, you deny the laws of logic, and claim that they are just opinions, then you have just denied that syllogism. In other words, if the rules of logic don’t work, then the fact that your fuel gauge works and is currently showing that you’re low on fuel does not mean that you are low on fuel. Cause and effect relationships operate because of the rules of logic. So, if you deny the rules of logic, then you deny cause and effect.

I mentioned earlier that the rules of logic are like the rules of math. In fact, they aren’t just like math, math relies on them. For example, anyone who has taken geometry has probably been introduced to proofs. These are simple logical syllogisms. For example,

1. The sum of the angles of any triangle equal 180 degrees
2. For triangle ABC, angle A = 90
3. For triangle ABC, angle B= 45
4. Therefore, for triangle ABC, angle C = 45

Notice, the conclusion is made absolutely necessary by the premises. If 1–3 are true, then 4 absolutely must be true. Angle C cannot be anything other than 45. That is logic. It’s not an opinion, it is an inherent property of the universe that absolutely must be accepted. If you reject the rules of logic, then you must also reject the rules of mathematics.

Do Christians Have to Follow the Rules of Logic?

It may seem odd that I am singling Christians out in a blog about science, but on scientific issues like climate change and evolution, I often find that Christians are hesitant to accept logical arguments and often respond to them with statements like, “Logic is just man’s wisdom, but God is higher than man, therefore we shouldn’t trust man’s logic and should rely on God instead.” I want to address this argument, because I encounter it frequently, and it often seems to be an underlying reason for rejecting science. To be clear, I’m not going to enter into a debate about theism or atheism, rather I am simply going to address the issue of whether or not a belief in God somehow makes you exempt from the rules of logic.

First, this argument is obviously dependent on the belief that God is actually real. So the argument is predicated on faith, which is problematic to say the least (again, I’m not telling you what to believe, but you should be aware that this argument is based on a premise which cannot be proved, which means that it is going to be completely unconvincing to anyone who does not share your faith). Nevertheless, for the sake of argument, let’s assume for a second that Christians are right, and God does actually exist. If he is real, then he, like everything else, must be bound by the laws of logic. I can prove that via that law of non-contradiction. Consider the following hypothetical dialogue between two Christians:

1. (Christian 1) “Can God do anything evil?”
2. (Christian 2) “No” 3. (Christian 1) “Why not?
4. (Christian 2) “Because his inherent nature is perfectly good.”
5. (Christian 1) “Why does his inherent nature prevent him from doing anything evil?”
6. (Christian 2) “Because, it’s impossible to be perfectly good and do something evil.”

Does #6 look familiar? It’s an affirmation of the law of non-contradiction. If God was not bound by the laws of logic, then he could be evil and perfectly good simultaneously, but every Christian agrees that he cannot do anything evil, therefore if he exists, he must be bound by the laws of logic (note: this is also the appropriate response to creationists’ absurd and ad hoc allegation that logic would not exist without God, clearly it would since, if he exists, he must be bound by it).

What I have just argued often makes Christians irate because they see this as an assault on God’s omnipotence, but that is only because they misunderstand the concept of omnipotence. Philosophers universally agree that “the ability to do anything” is a terrible definition for omnipotence. The most widely accepted definition is, “The ability to do anything logically possible if one wanted to.” The reason for this definition becomes obvious if we return to the example of a triangular circle. No matter how powerful a being might be, he wouldn’t be able to make a triangular circle because it’s not logically possible for such an object to exist.

So, in short, even an omnipotent being would have to be bound by the laws of logic, and would not be capable of doing anything that is not logically possible. Therefore, claims that we should not follow the laws of logic because, “they are just opinions,” or “they are man’s wisdom,” or “all things are possible with God,” are silly and invalid. The laws of logic always hold true and must always be followed in all rational conversations and debates, regardless of your religious beleifs.

## Multiplication Facts - Tips, Rules and Tricks to Help You Learn

Memorizing the entire Multiplication Table can seem quite overwhelming at first. The key to learning your multiplication facts is to break the process down into manageable lessons. This is done through a series of rules or “tricks” that can be learned. Once these have been mastered, you will see that it is only necessary to memorize ten multiplication facts! First, however, there are several key concepts that must be understood. [caption align=“aligncenter” width=“640”] Multiplication can be performed with basic addition and subtraction[/caption]

• The first is that multiplication is simply a fast way of joining groups of equal size through repeated addition. Let’s look at a problem together:

Sarah has 4 boxes of crayons. There are 3 crayons in each box. How many crayons does Sarah have altogether? This problem can be solved through repeated addition:

A shortened version of this would be to use the multiplication sentence:

• The second concept that must be understood is what each number in the multiplication problem represents. Let’s look at that same problem again:

Sarah has 4 boxes of crayons. There are 3 crayons in each box. How many crayons does Sarah have altogether?

In this case, the (4) represents the number of groups in the problem. (There were 4 boxes.) The (3) represents how many objects/items were in each group. (There were 3 crayons in each box.)

• The third concept that will help you with learning your multiplication facts is the Commutative Property of multiplication. This states that when two numbers are multiplied together, the product (or answer) is the same regardless of the order of the numbers. For instance:

3 x 2 = 2 x 3

## Basic Divisibility Rules

Here are some example questions that can be solved using some of the divisibility rules above.

• Since the last digit of 65973390 is 0, it is divisible by 2.
• Since 6 + 5 + 9 + 7 + 3 + 3 + 9 + 0 = 42 6+5+9+7+3+3+9+0=42 6 + 5 + 9 + 7 + 3 + 3 + 9 + 0 = 4 2 , which is divisible by 3, it follows that 65973390 is divisible by 3.
• Since the last digit of 65973390 is 0, hence it is divisible by 5.
• To check divisibility by 7, as the initial step, we calculate 6597339 − 2 ( 0 ) = 6597339 6597339-2(0)=6597339 6 5 9 7 3 3 9 − 2 ( 0 ) = 6 5 9 7 3 3 9 . However, this number is still a little too big for us to tell whether it's divisible by 7. In such cases, we keep applying the divisibility rule again and again until we have a small enough number to work with: 659733 − 2 ( 9 ) = 659715 65971 − 2 ( 5 ) = 65961 6596 − 2 ( 1 ) = 6594 659 − 2 ( 4 ) = 651 65 − 2 ( 1 ) = 63. egin659733-2(9)&=65971565971-2(5)&=659616596-2(1)&=6594659-2(4)&=65165-2(1)&=63.end 6 5 9 7 3 3 − 2 ( 9 ) 6 5 9 7 1 − 2 ( 5 ) 6 5 9 6 − 2 ( 1 ) 6 5 9 − 2 ( 4 ) 6 5 − 2 ( 1 ) ​ = 6 5 9 7 1 5 = 6 5 9 6 1 = 6 5 9 4 = 6 5 1 = 6 3 . ​ Now we can see that we are left with 63 , 63, 6 3 , which we can easily identify as a multiple of 7. Hence 65973390 is a multiple of 7 also.

Try some problems for yourself to see if you understand this topic:

If we know an integer is a multiple of 5, how many possibilities are there for the last two digits of the integer?

## 1.2 Decimals and Real Numbers

We have a nice way to represent numbers including fractions, and that is as decimal expansions. Suppose we consider numbers like (frac<1><10>), (frac<2><10>), (which is the same as (frac<1><5>)), (frac<3><10>), and so on.

We write them as (.1 , .2, .3), and so on. The decimal point is a code that tells us that the digit just beyond it is to be divided by ten.

We can extend this to integers divided by one hundred, by adding a second digit after the decimal point. Thus (.24) means (frac<24><100>). And we can keep right on going and describe integers divided by a thousand or by a million and so on, by longer and longer strings of integers after the decimal point.

However we do not get all rational numbers this way if we stop. We will only get rational numbers whose denominators are powers of ten. A number like 1/3 will become (.33333. ), where the threes go on forever. (This is often written as (.3*), the star indicating that what immediately precedes it is to be repeated endlessly)

To get all rational numbers using this decimal notation you must therefore be willing to go on forever. If you do so, you get even more than the rational numbers. The set of all sequences of digits starting with a decimal point give you all the rational numbers between 0 and 1 and even more. What you get are called the real numbers between 0 and 1. The rational numbers turn out to be those that repeat endlessly, like (.33333. ), or (.1000. ), or (.14141414. ), (aka (.(14)*)).

Now neither you nor I nor any computer are really going to go on forever writing a number so there is a sense of unreality about this notion of real numbers, but so what? In your imagination you can visualize a stream of numbers going on forever. That will represent a real number.

If you stop a real number after a finite number of digits, you get a rational number (because all its entries after where you stopped are zeroes). As a result, the rules of addition, subtraction, multiplication and division that work for rational numbers can be used to do the same things for real numbers as well. Fortunately, the digits that are far to the right of the decimal point in a number have little effect on computations when there are non-zero digits much closer to the decimal point.

Since we cannot in real life go on forever to describe a non-rational real number, to do so we have to describe it some other way. Here is an example of different way to describe a number.
We define the number that has decimal expansion (.1101001000100001. ) between each consecutive pair of (1)'s there is a number of (0)'s that is one more than between the previous consecutive pair of 1's.This number is not rational it does not repeat itself.

We do not have to, but just for the fun of it, we will go one step further and extend our numbers once more, to complex numbers. This is required if you want to define inverses to the operation of squaring a number. (Complex numbers are entities of the form (a+bi) where (a) and (b) are real numbers and (i) squared is (-1).)

## Algebra Rules for Exponents

### The product of two powers with the same base is equal to that base raised to the sum of the two exponents.

As with many rules related to exponents, writing out the exponents as multiplications makes it obvious why the rule is true

### A number raised to a power raised to a power is equal to that number raised to the product of the two exponents.

Like the previous rule, this one can be demonstrated simply by expanding the exponents out into a series of multiplications

### Convert a multiplication with an exponent into the product of two factors each raised to the exponent.

Thanks to the commutative property of multiplication, any series of multiplications can be rearranged without changing its value. This means that we can take a multiplication raised to a power and rearrange the resulting series of multiplications to make two exponents

### The result of a negative exponent is the inverse of the same positive exponent.

It might seem odd to have a negative exponent (since you can't multiply something by itself a negative number of times). However, if we take a closer look at the rule a^na^m = a^ we can see that it implies that a^<-n> must equal <1 over a^n>, the multiplicative inverse or reciprocal of a^n.

This becomes clear looking at the a^ side of the equation from rule 11. What happens if m is negative? Obviously, this will reduce the combined value of the exponent (for example, 2^ <4-2>= 2^2). What does this mean for the left hand side of the a^na^m = a^ equation? It means that the value of, for example, 2^4 must be reduced to 2^2 when it is multiplied by 2^<-2>. If, as this rule states, a^ <-n>= <1 over a^n>, this works out perfectly: 2^4 * 2^ <-2>= 2^4 * <1 over 2^2>= 16 * <1 over 4>= 4 = 2^2 = 2^<4-2>

### A fraction raised to a negative exponent equals the inverse of the fraction raised to a positive exponent.

The reciprocal of a fraction is the fraction turned on its head: the reciprocal of <2 over 3> is <3 over 2>. We know from the previous rule that a^<-n> is the reciprocal of a^n, so we can simply convert the fraction to its reciprocal by exchanging the numerator and denominator, and then the exponent becomes positive. Positivity is such a nice thing!

### A fraction with an exponent is equal to the same fraction with the exponent on the numerator and denominator.

This looks weird at first, but the reasons behind it are pretty simple. If we were paying attention when someone told us how to multiply fractions (this is doubtful, but we'll continue anyhow) we will remember that to multiply two fractions you simply multiply the numerators with each other and multiply the denominators with each other to get the resulting fraction. This rule follows from that fact.

### If the top and bottom of a fraction are both exponents with the same base, the fraction is equal to the base raised to the numerator exponent minus the denominator exponent.

This one is very simple. Since division is the inverse of multiplication, multiplying a number by itself a few times and then dividing it by itself multiplied a few time is the same as just multiplying it by itself a few less times.

### Anything raised to the power of zero is equal to 1.

This rule may seem arbitrary, but it is necessary in order to maintain consistency with other properties of exponents. Consider the rule a^na^m = a^. What happens if m = 0? The right hand side of the equation will be a^, or a^n. This means that in the left hand side, a^n has to be multiplied by the value of a^0, but remain unchanged. The only way for this to be the case is if a^0 = 1. (For some discussion of the peculiar case of 0^0 and why it should (probably) equal 1, see this article.)