The Properties of Real Numbers

Now that we have an alphabet, let's establish the rules of grammar for the language of mathematics. All of these rules work for any real numbers, so we'll use our new variables to express numbers in general. You can use the rules for substituting for variables with any of these rules, just be sure to use different numbers for different variables.

There are several kinds of rules. There are axioms which are so basic of a rule they are simply taken for granted. There are laws that are the consequence of axioms and are true everywhere all the time. And there are theorems which are rules that have been accepted as true because they have been built from axioms and laws and other theorems. Some of them we've already seen.

Closure Property
Adding any two real numbers results in another real number.
Multiplying any two real numbers results in another real number.
Transitive Property
If two numbers are equal to the same number, then they are equal to each other.	If a=c and b=c then a=b
Equality Axioms
If two numbers are equal and you add the same number to each the resulting numbers are equal.	If a=b then a+c=b+c.
If two numbers are equal and you mutliply each by the same number the resulting numbers are equal.	If a=b then ac=bc.
Commutative Property
When adding real numbers, the order doesn't matter.	a+b=b+a
When multiplying real numbers, the order doesn't matter.	ab=ba
Associative Property
When adding real numbers, the grouping doesn't matter.	a+(b+c)=(a+b)+c
When multiplying real numbers, the grouping doesn't matter.	a(bc)=(ab)c
Distributive Property
Multiplying a sum of real numbers by a real number is the same as multiplying each real number of the sum first then adding.	a(b+c)=ab+ac
This is a powerful property. It allows us to pull a common factor out of the parts of an equation and deal with it separately.
Identities
Adding zero to a real number doesn't change the number.	a+0=a
Multiplying a real number by 1 doesn't change the number.	a×1=a
Inverses
Adding the additive inverse of a real number to itself results in the identity.	a+(-a)=0=(-a)+a
Multiplying the reciprocal of a real number to itself results in the identity.	a(1/a)=1 if a≠0
Zero Principle
Multplying by zero results in zero.	a×0=0
If the product of two real numbers is zero, then one of them must have been zero.	If ab=0 then a=0 or b=0
Trichotomy Property
If two real numbers are compared, the second number can have only one of three outcomes, either greater than, less than or equal to.	a<b, a>b or a=b
Definition a^-1	a^-1=1/a
Properties of Negatives
Multiplying a real number by -1 changes the number's sign.	(-1)a=-a
	-(-a)=a
For any negative product of two real numbers, the negative sign can be assigned to either one the numbers.	(-a)b=-ab=a(-b)
The product of two negative real numbers is equal to the product of the additive inverses of the two numbers.	(-a)(-b)=ab
Negating the difference of two real numbers changes to signs of each number.	-(a-b)=b-a
Properties of Quotients
For any two rational numbers, the first rational number is equal to the second rational number if the product of the numerator of the first and the denominator of the second is equal to the product of the numerator of the second and the denominator of the first	a/b=c/d if and only if ad=bc
Mutliplying a rational number by a rational number equal to 1 changes the appearance of the rational number but not its value.	a/b=ad/bd
a/b+c/b=(a+c)/b
a/b+c/d=(ad+bc)/bd
a/b × c/d = ac/bd
a/b ÷ c/d = a/b × d/c = ad/bc
Definition
The principle square root of a number √a is the nonnegative real number b such that b squared is equal to a.	b²=a
Laws of Exponents
Definition
The absolute value of a real number is the magnitude of the number without a sign.	|a|=a, |-a|=a
Absolute Value Properties
The absolute value of a real number has the following properties
Positive Definite
For all real numbers, the absolute value of the number a is greater than or equal to 0 with |a|=0 only if a=0
Symmetric
For any two real numbers, |a-b|=|b-a|
Triangle Inequalities
For any two real numbers, the absolute value of the sum of the numbers is less than or equal to the sum of the absolute values of the numbers.	|a+b|≤|a|+|b|
For any two real numbers, the absolute value of the difference of the numbers is greater than or equal to the difference of the absolute values of the numbers.	|a-b|≥|a|-|b|
For any two real numbers, the absolute value of the difference of the absolute values of the numbers is less than or equal to the absolute value of the difference of the numbers.	||a|-|b||≤|a-b|

Variables

Mathematics often uses letters in place of numbers to illustrate a general principle. In a mathematical equation, the letters are referred to as variables because they can represent any number or numbers that can change depending on the situation. There are two types of variables; independent and dependent. An independent variable can change its value, usually without any restrictions. A dependent variable's value usually depends on the value of the independent variables in an equation. If a number alone is present in an equation, it represents a constant or a value that doesn't change.

Letters represent numbers sort of like letters can represent other letters in a code. If the letter A represents F in a code, the letter A represents F everywhere it appears. Similarly, if the letter a represents the number 5 in an equation, it represents 5 everywhere in the equation. The letter b would indicate another number besides 5.

Variables always have a combination of letters and numbers. Numbers in front of variables like 3a means there are three a's in the equation. If no number is in front of a variable it is understood that it is preceded by 1. Exponents are also always present with variables. If they're not explicitly shown they're undertood to be 1.

Variables can behave like numbers. Any operation you can perform on a number can be theoretically performed on a variable.

Multiplying a variable by itself changes the variables's dimension. If the variable a represents a line, then a² represents a square and a³ represents a cube and so forth. You can't add a, a² and a³ because they're not the same shape. You can only add lines to other lines, squares to other squares and cubes to other cubes. You can't add a² and b² because a² is a square and b² is a circle. To add variables together, they have to be the same letter and the same exponent. If the letters are the same and the exponents are the same, you add the numbers that come in front.

3a + 2a = 5a

When you multiply variables of the same kind together, you add the exponents. When dividing variables of the same kind, you subtract the exponent of the second variable from the first.

a²b × ab = a³b²

In a rational expression, a fraction involving variables and numbers, to add or subtract, the numbers and letters and exponents on the letters must be the same. If you have a² in one denominator and a³ in the other denominator, you multiply the first rational number by a/a. Remember that a fraction where the numerator and the denominator are the same is the same as 1, and multiplying a number by 1 doesn't change the value of the number. In this case, we only change the way the number looks.

1/a²b + 2/a³ =
(a/a × 1/a²b) + (b/b × 2/a³) =
a/a³b + 2b/a³b

Now that the denominators are the same we can add the two rational expressions. But notice that the numerators have different numbers and letters.

a/a³b + 2b/a³b = (a + 2b)/a³b

How would this work if we replaced the letters with numbers? Let's let a=3 and b=5.

1/3²5 + 2/3³ =
(3/3 × 1/3²5) + (5/5 × 2/3³) =
3/3³5 + 10/3³5 =
3/135 + 10/135 = 13/135

We can let a and b represent any two different numbers and we'll get a different result each time. Using the numbers only works for a specific case, but using variables allows us to talk about what happens in a general case. This is what variables are used for. We build mathematical sentences to describe what happens in a general sense using variables and then use specific numbers to explore a specific case. We can use variables to describe a general case of how long it takes to get to a destination at any speed, then put in numbers to see the results at specific speeds. To define how long it takes to get to a destination at any given speed we would write an equation using variables for distance and speed. If we let d represent the distance we want to travel and m represent the miles we travel in an hour, our equation would be d × 1/m.

Let's see what happens if we want to travel 100 miles at 55 mph. We put 100 in place of d and 55 in place of m.
100 × 1/55 = 1.812 hrs

To see what happens if we want to travel the same 100 miles at 65 mph we put 100 in place of d and 65 in place of m.
100 × 1/65 = 1.538 hrs

Using variables, we've set up a general case for how long it takes, then we can look at specific cases by substituting numbers for the variables. This is called modeling and is a very important topic in mathematics. All kinds of things can be modeled from the growth of tiny bacteria to landing a man on the moon. There is a famous equation developed by Isaac Newton to determine the gravity exerted by a body in space. All you have to do is substitute the size of the bodies and the distance between them. The same equation works everywhere in the universe, on Earth, on the Moon even on Jupiter or Mars. We have one equation and we can take it anywhere we want to go and substitute numbers for the variables depending on where we are.

Building Simple Words: Exponents and Roots

Multiplication is multiple additions. Exponentiation is multiple multiplications. Exponents are written as superscript numbers above and to the right of the number to be exponentiated. 2×2×2 is written as 2³ and equals 8. This is read as 2 raised to the 3rd power. Not all exponents can be shown in this format, so I will use a carat (^) to indicate the number that follows is the exponent.

Numbers raised to the second power are said to be squared because the area of a square is the product of the two equal dimensions (length, width) multiplied together. Numbers raised to the third power are said to be cubed because the volume of a cube is the product of the three equal dimensions (length, width, height) multiplied together. All other exponents use the ordinal form of the exponent (to the fourth, to the fifth, to the sixth, ect.) Any number raised to the 0 power equals 1.

Squares are so common that the squares from 1 to 10 should be memorized. They are taken from the multiplication table.

1² = 1×1 = 1
2² = 2×2 = 4
3² = 3×3 = 9
4² = 4×4 = 16
5² = 5×5 = 25
6² = 6×6 = 36
7² = 7×7 = 49
8² = 8×8 = 64
9² = 9×9 = 81
10² = 10×10 = 100

You can't directly add numbers with exponents. Each must be exponentiated first, then the results are added together. With multiplication, if the numbers are the same the exponents are added together. 2^3×2^4=2^(3+4)=2^7. If you exponentiate an exponent, you multiply the exponents. (2^3)^4=2^(3×4)=2^12. With a rational number, when the number in the numerator and the number in the denominator are the same, you subtract the exponents. 2^4/2^3=2^(4-3)=2^1=2. If the exponent in the numerator is larger, you subtract the exponent in the denominator from the exponent in the numerator and replace the denominator with 1. 2^6/2^3=2^(6-3)/1=2^3/1=2^3. If the exponent in the denominator is larger, you subtract the exponent in the numerator from the exponent in the denominator and replace the numerator with 1. 2^3/2^6=1/2^(6-3)=1/2^3.

Just as division is the opposite of multiplication, there is an opposite operation for exponentiation. The root of a number is the number that when multiplied by itself gives the orignal number. Roots are indicated with a symbol called a surd √ with an index over the little bar. The index shows the root and the surd separates the number to be broken down. The index and surd together with a number are called a radical. If the root is the square root the index is not shown. ³√27 = 3 because 3³=27.

Like with addition of exponents, you can't add different radicals. You have to get the roots first, then add the resulting numbers. We can mix numbers and radicals. A number can be added to a radical or a radical can be multiplied by a number. 3+√5 is a valid number as is 3√5. We can add two radicals together if they are exactly the same. In that case, you add the numbers on the outside and keep the radical unchanged. 4√5 + 3√5 = 7√5. I can't add 3√25 and 4√36 because the radicals are different. I would have to convert the radicals to numbers then add the result. 3√25+4√36=3(5)+4(6)=15+24=39.

Roots can be expressed as rational exponents. A square root is a ½ exponent. √4=4^½. If the number inside the radical is raised to a power, the power inside the radical becomes the numerator of the rational exponent and the index of the radical becomes the denominator of the rational exponent. Converting roots to exponents means we can use our rules for multiplying and dividing exponents to multiply two roots together. If the numbers inside the radical are the same we simply add the exponents together using the rules for adding and subtracting rational numbers. If the exponent on the number under to surd is the same as the index of the root, the index and the exponent cancel each other out and the surd disappears. ³√27³=27^(3/3)=27^1=27. If the indexes are the same, multiply the numbers under the surd and keep the index. ³√9׳√8=³√(9×8)=³√72.

To simplify a radical you remove groups of common factors based on the index. First you factor the number under the surd into its prime factors. Then you remove groups of factors based on the index. When all of the groups have been removed, you multiply the numbers on the outside together and multiply the numbers remaining under the radical. Let's simplify ³√72. The prime factors of 72 are 3×3×2×2×2. The index is 3 so we want to remove factos in groups of 3. There are 3 2's so we remove them from inside the surd and place a single 2 on the outside;2³√(3×3). There are only 2 3's under the surd and we need to pull out in groups of three, so the 3's stay under the surd. Multiply the 3's back together again and ³√72=2 ³√9. How about ³√1296. First we factor 1296 into its prime factors; ³√(3×3×3×3×2×2×2×2). We need to remove factors in groups of 3. There are 4 3's, so we can remove 3 of them and there are 4 2's so we can remove 3 of them as well; (3)(2)³√(3×2). There are no more groups of 3 left so we multiply the numbers on the outside and multiply the numbers on the inside; 6 ³√6.

Roots of prime numbers don't have an even representation because there are no factors of prime numbers, so you can't get a prime number by multiplying any number to itself. All roots of prime numbers are irrational numbers. We'll discuss them next.

Even roots of negative numbers are also a problem. If you think about the rules for multiplying negative numbers, you can see where the problem lies. Remember that if there are an even number of negative numbers being multiplied together, you get a positive number. If there are an odd number of negative numbers, you get a negative number. A negative number exponentiated an even number of times gives a positive number. So there is no way to get a negative number if the exponent is even, thus there can't be an even root of a negative number. Finding an even root of a negative number requires us to use a very special number called the imaginary number. We won't worry about the imaginary number for now.

Modern calculators usually have a key for finding any root of any number, but for a method for finding any root of any number see this article in Wikipedia Nth Root Algorithm. Now that we know about roots, let's talk about the last group of numbers that make up our alphabet.

The Alphabet of Math: The irrational numbers

The last group of numbers that we are going to discuss are irrational numbers. Remember that rational numbers are numbers that can be represented as a ration of two numbers. Irrational numbers are numbers that can't be represented as a ration of two numbers. Irrational numbers fill in the spaces between rational numbers on the number line.

The Greek Pythagorean Society studied numbers and their effect in the universe. They believed all numbers were rational. When one of them asked what the square root of 2 was, they were so upset by the resulting irrational number that they took out in a boat, tied him up and threw him into the sea! Mathematicians are more receptive to new ideas these days.

Rational numbers are numbers that are either terminating or repeating and nonterminating. Irrational numbers are nonrepeating and nonterminating. Many numbers fall in this category. The roots of prime numbers are all nonrepeating and nonterminating. Roots of prime numbers are all irrational. An ancient special number is irrational. Pi is the ratio of the diameter of a circle to its circumference. This ratio has been known for thousands of years, but everyone in antiquity used a rational approximation to pi. Today, we recognize that pi can't be represented as a a rational number so we simply use the Greek letter π to represent it. Another special number is used to find an exponential growth like with rabbits or bacteria. It's called e and it's irrational also.

With the natural numbers, the integers, the rational numbers and the irrational numbers our alphabet is essentially complete. These numbers together are called the real numbers.

With our alphabet complete, we can begin talking about all topics in mathematics.

The Alphabet of Math: Decimal numbers

Rational numbers are actually the division of whole numbers into smaller parts. A fraction represents a division problem. The numerator is the dividend and the denominator is the divisor. So 7/4 could also be written as 7÷4. 4 divides 7 once with a remainder of 3. We can represent this as a mixed number, a whole number for the whole number of divisions and a rational number for the remaining parts of the whole left over. So 7/4 is the same as 1 ¾. But, we won't represent rational numbers with mixed numbers. We are going to use a new type of number that looks more like whole number called a decimal number.

A decimal number has two parts, a whole number part and a fractional part separated by a decimal. To get a decimal representation of a rational number we use long division and instead of stopping when we get to the remainder, we keep dividing until we get to a zero remainder. 7/4 would be divided like this

		1
4	(	7
	-	4
		3

This is where we have been stopping. But we can continue the division process to get a decimal representation by placing a decimal after the 1 and using zeros to continue dividing until we get a zero remainder.

		1.	7	5
4	(	7
	-	4
		3	0
	-	2	8
			2	0
		-	2	0
				0

So 7/4 = 1.75

This is a special decimal number called a non-repeating, terminating decimal. This decimal number doesn;t repeat and it comes out to a zero remainder. What happens if the number doesn't come out to a zero remainder?

		2.	3	3
3	(	7
	-	6
		1	0
	-		9
			1	0
		-		9
				1

This decimal number repeats 3 over and over again. This is called a repeating decimal number. We usually only show the repating part once and draw a bar opver the top of the repeating part.

All rational numbers can be represented as either non-repeating, terminating decimals or repeating decimals.

Adding decimal numbers is done just like whole numbers. Line up the decimals then add the numbers just like whole numbers.

	1	.	5	4	3
+	2	.	3	4
	3	.	8	8	3

Multipying decimal numbers is similar to multiplying whole numbers. Multiply the numbers just like whole numbers disregarding the positions of the decimals. After you've finished multiplying, count the number of digits to the right of the decimal in each number. This is the total number of decimals that will be in the product. Starting from the right, you count positions to the left for as many decimals as you have in the multiplicand and multiplier and place the deicmal in that position.

	1	.	5	4	3
×		2	.	3	4
		6	1	7	2
	4	6	2	9
3	0	8	6
3	6	1	0	6	2

There are 3 decimals in the multiplicand and 2 in the multiplier so our answer needs 5 decimals. Starting from the right we count back 5 positions and place a decimal between 3 and 6. So 1.543×2.34=3.61062

Dividing decimals works just like dividing we have seen above but we start by making the divisor into a whole number by miving the decimal to the right until there are no more numbers. We also have to move the deicmal in the dividend the same number of places to the right. The decimal in the quotient will be at the same position as the new position in the dividend.

2.

3

4

(

3.

6

1

0

6

2

There are 2 decimal positions in the divisor. So move the decimal 2 places to the right in the divisor and the dividend. Now divide as usual.

						1.	5	4	3
2	3	4	(	3	6	1.	0	6	2
			-	2	3	4
				1	2	7	0
			-	1	1	7	0
					1	0	0	6
					-	9	3	6
							7	0	2
						-	7	0	2
									0

What happens if the quotient doesn't repeat and doesn't terminate? This is the last of our number types for our alphabet. They are called irrational numbers because they can't be represented as rational numbers. There are lots of irrational numbers and we'll discuss them in another post.

The Alphabet of Math: The rational numbers

So far, we've only looked at whole numbers. We've had no way to represent parts of numbers or numbers in between the points on a numberline. Rational numbers give us the ability to represent points in between the whole numbers. To understand the points between whole numbers we need to look at the concewpt of infinity.

Infinity is a number that is so large or so small that it can't be represented, but we know it exists. Something can be infinitly large meaning it's so big there isn't a number that can be written to represent it. There are an infinite number of whole numbers because for any whole number you can add 1 to it and come up with another whole number.

Everyone has heard the joke routine of a person counting down to an ultimatum. I'm going to count to 5 and then I'm going to ... Then the person counts; one, two ,three, four, four and one half, four and three quarters... This joke works because there are an infinite number of parts bewteen 4 and 5. Have you ever wondered why the football never crosses the goalline when a team is third and goal and the defense gets charged with a penalty that results in half the distance to the goal? That's because you can always cut the distacne to the goal in half and get closer to the goal, but you never actually cross the goal.

Cutting something in half can;t be represented by whole numbers because half isn't a whole anymore. We use something called rational numbers to represent halves. Rational numbers are also called fractions because they represent fractions of a whole. Rational numbers are ratios of a number to a whole. One half is a ratio of 1 to 2. It is also a fraction of 1 part of 2 parts of a whole.

A rational number is made up of two parts. The number is represented as a number over another number. The number on top, called the numerator, is the part of the whole that we're talking about. The number on the bottom, called the denominator, is the number of parts that the whole is divided into. In the number 1/2, we are talking about 1 part of 2. Two halves make a whole.

The number 1/2 is between 0 and 1 on the numberline.
<——0——½——1——>

If we divide the distance between 0 and 1 into fourths we get this numberline
<——0—¼—½—¾—1——>

Notice how we count up the fourths from 1 to 3. At 1, we would be at 4 fourths. When the numerator equals the denominator, we are at 1 or a whole. If we continued to count fourths passed 1 we would get 5/4, 6/4, 7/4, 8/4.

Notice that 8 is a multiple of 4. 8/4 is the same as 2. Also notice that the gcd of 6 and 4 is 2. If we divide 6 and 4 by 2 we get 3 and 2. 6/4 is the same as 3/2. If the numerator and denominator of a rational number have a gcd, we have to use this gcd to divide the rational number to its reduced form. We never represent rational numbers with a gcd between the numerator and denominator. Rational numbers must always be represented in their reduced form.

Anytime the denominator is 1, the reduced number is a whole number. 4/1 would be 4 wholes or just 4. Anytime you see a rational number where the denomintaor is 1, simply drop the denominator.

The numberline divided into fourths would look like this
<——0—1/4—1/2—3/4—1—5/4—3/2—7/4—2—9/4—5/2—11/4—3——>

Multiplying ratinoal numbers is easy. You simply multiply the numerators and multiply the denominators. Then reduce the rational number by removing the gcd of the numeratoar and denominator, if it exists.

3/2×4/3 = (3×4)/(2×3) = 12/6

The gcd of 12 and 6 is 6, so divide the numerator by 6 and divide the denominator by 6
(12÷6)/(6÷6) = 2/1 = 2

Adding rational numbers requires us to add numbers with common denominators. To get a common denominators, we have to have the lcm of the denominators. To refresh the concept of the lcm look at the blog Division Revisited.Then we change each rational number so it looks different, but isn't really changed. We can do this by multiplying each number by 1. Remember that in multiplication, 1 is the identity meaning that when we multiply by 1 we don't change the value of the number. From above, if the numerator and denominator are the same, the rational number is actually 1. So we are going to multiply each of the rational numbers by a rational number where the numerator and the denominator are the same. This changes the way the rational numbers look but not their actual value. This is confusing, so let's work one out. Let's add 1/2 to 1/3.

First, we need the lcm of 2 and 3. The lcm of 2 and 3 is 6. So we need to convert 1/2 and 1/3 to sixths. If we multiply 2 by 3 we get 6, so we multiply 1/2 by 3/3 to get 3/6. If we multiply 3 by 2 we get 6, so we multiply 1/3 by 2/2 to get 2/6. Now that we have a common denominator, we simply add the numerators.

1/2+1/3 = 3/6+2/6 = (3+2)/6 = 5/6

So to add two rational numbers, we first have to find the lcm of the denominators, multiply both numbers by 1 to get a common denominator, then add the numerators and reduce if needed.

Subtraction works like addition. Get a common denominator then subtract the second numerator from the first.

5/6-1/3 = 5/6 - 2/6 = (5-2)/6 = 3/6 = 1/2

Division works similar to multiplication. But we flip the divisor then we multiply the numerators and the denominators and reduce if necessary.

Rational numbers give us the ability to represent the inverse for multiplication. Remember from addition that the inverse is the number that we add to a number to get the identity which is 0 in addition. In multiplication, the identity is 1, so we want the number that when we multiply we get 1. In multiplication, the inverse is called the reciprocal and we get it by flipping the rational number. The reciprocal of 1/2 is 2/1, and if we multiply 1/2 by 2/1 we get (1×2)/(2×1) = 2/2 = 1. When we multiplied by the reciprocal, we got the identity. So the reciprocal is the inverse.

Rational numbers actually represent a division problem and we'll investigate this further in the next blog.

Division Revisited

Just as subtraction is the opposite of addition, division is the opposite of multiplication. We learned that multiplication is adding a number a multiple of times. Division, likewise, is subtracting multiple times. We start with the dividend and we subtract the divisor until we get to zero or we can't subtract any more.

6-2=4-2=2-2=0
6÷2=3

9-2=7-2=5-2=3-2=1
Subtracting by 2 again would go past zero.
9÷2=4 with a remainder of 1 or 9=2×4+1

What happens when we divide by negative numbers? We divide just as we would with positive numbers, but we have to remember our sign rules for multiplication. Remember that an odd number of negative signs in multiplication gives a negative number. So if we multiply two negative numbers, we get a positive product. If we multiply two positive numbers we get a positive product. So if we divide a positive number, we must have gotten it by multiplying two numbers with the same sign. That means, in division, the divisor and the quotient must have the same sign. If the dividend starts negative, then the divisor and quotient must have different signs.

8÷-2=-4
The dividend is positive, the divisor gives the sign of the quotient.

-8÷-2=4
-8÷2=-4
The dividend is negative, the quotient has the opposite sign of the divisor.

9÷-2=-4 with a remainder of 1 because 9=-2×-4 + 1
-9÷-2=4 with a remainder of -1 because -9=-2×4 - 1

Let's look at division of large numbers.
12345÷567
Just start subtracting by 567 and keep count.
12345-567=11778 (1st division)
11778-567=11211 (2nd division)
11211-567=10644 (3rd division)
10644-567=10077 (4th division)
10077-567=9510 (5th division)
9510-567=8943 (6th division)
8943-567=8376 (7th division)
8376-567=7809 (8th division)
7809-567=7242 (9th division)
7242-567=6675 (10th division)
6675-567=6108 (11th division)
6108-567=5541 (12th division)
5541-567=4974 (13th division)
4974-567=4407 (14th division)
4407-567=3840 (15th division)
3840-567=3273 (16th division)
3273-567=2706 (17th division)
2706-567=2139 (18th division)
2139-567=1572 (19th division)
1572-567=1005 (20th division)
1005-567=438 (21st division)

So 12345÷567=21 with a remainder of 438

Doing division by multiple subtractions can be tedious and presents lots of opportunites for errors. We start by looking only at the number in the highest positions. So we want to divide 10000 by 500. Since there are two zeros on the right of 5, we can eliminate two zeros from the right in both numbers. So now we are looking at 100÷5. From our multiplication table, we don't have a multiple of 5 that equals 100. But we do have a multiple of 10 that equals 100 and a multiple of 5 that equals 10. 5×2×10=100. Using the associativity of the integers, we have 5×(2×10)=5×20=100. Putting the zeros back in, we have that 10000÷500=20.

Start by multiplying 567 by 20

		5	6	7
		×	2	0
1	1	3	4	0

12345-11340=1005
1005 > 567 so multiply by 21 instead.

		5	6	7
		×	2	1
1	1	9	0	7

12345-11907=438
438 < 567
So 12345÷567=21 with a remainder of 438

This is essentially how the Indians developed long division. Their notation is different, though. We're going to use the ) notation of long division with the divisor to the left and the dividend to the right. We are going to write our quotient above the dividend, carfeully observing positioning. The result at the bottom will be the remainder. We start like this:

				—	—	—	—	—
5	6	7	)	1	2	3	4	5

Now we count off positions in the dividend until we get a number that is larger than the divisor. 1234 is larger than 567. How many times does 5 go into 12? 2 times. We place 2 above the 4 in 1234. We will multiply 567 by 2 and place that product underneath 1234. Then we subtract that from 1234 and place the difference underneath. It looks like this.

							2
				—	—	—	—	—
5	6	7	)	1	2	3	4	5
			-	1	1	3	4
				—	—	—	—
					1	0	0

Now we bring down the 5 in the next column of 12345 to make a number greater than 567. How many times does 5 go into 10? 2 times. Place the 2 above the 5. Multiply 567 by 2 and place the product underneath 1005.

							2	2
				—	—	—	—	—
5	6	7	)	1	2	3	4	5
			-	1	1	3	4
				—	—	—	—
					1	0	0	5
				-	1	1	3	4
					—	—	—	—

1134 is greater than 1005, so we reduce the multiplier by 1. Place 1 above the 5 and multiply 567 by 1. Place 567 underneath 1005 and subtract.

							2	1
				—	—	—	—	—
5	6	7	)	1	2	3	4	5
			-	1	1	3	4
				—	—	—	—
					1	0	0	5
					-	5	6	7
					—	—	—	—
						4	3	8

438 < 567 so we're done.
12345=567×21+438

Division is neither associative nor commutative. The grouping and the order of the numbers matters.
(18÷6)÷3 = 3÷3 = 1
18÷(6÷3) = 18÷2 = 9

Finally, there is a special number that we look at regarding division. It will become very important in the next blog. It is the greatest common divisor. The greatest common divisor, or gcd, is the largest number that evenly divides two numbers. We look at the prime number divisors of a two numbers and collect the divisors they have in common.

You get the prime factors of a number by dividing it by prime numbers until you get to 1. If the number is even then you can divide by 2, for example, 246 is even and 246÷2=123. If a number ends in 5 then you can divide dy 5, for example, 85 ends in 5 and 85÷5=17. If the sum of the digits of a number is a multiple of 3 then the number can be divided by 3, for example, in 123 the digits add up to 6 which is a multiple of 3 and 123÷3=41. Here's the url in Wikipedia that lists the divisibility rules. http://en.wikipedia.org/wiki/Divisibility_rule

The prime factors of 8 are 2,2 and 2. The prime factors of 12 are 2,2 and 3. Both 8 and 12 have a pair of 2's in common. So the greatest common divisor of 8 and 12 is 2×2=4. Another way to find the gcd is to multiply the two numbers together and divide by the lcm. You can similarly find the lcm by multiplying the two numbers together and divide by the gcd. The product of 8 and 12 is 96. The lcm of 8 and 12 is 24. 96 divided by 24 is 4.

The Alphabet of Math: The integers and subtraction

We can do some simple forms of math using only the natural numbers and zero. The natural numbers and zero are called the whole numbers. Right now we haven't developed the notion of pieces of numbers, but since we call the natural numbers and zero the whole numbers, you can guess that we are going to eventually talk about numbers that aren't whole. We will, but one step at a time.

All of our numbers, so far, have fit on a number line that starts at zero and moves to the right. But what happens if you want to start at zero and move to the left? The number line can be extended to the left from zero using the concept of negative numbers. We simply number from zero, counting up just like we do on the right, but we preface each number with a negative sign to indicate it is to the left of zero.

-6···-5···-4···-3···-2···-1···0···1···2···3···4···5···6

This extended set of numbers (the natural numbers, zero and the negative numbers) are called the integers. The integers with a negative sign are called negative numbers and those without a negative sign are called positive. We can do addition and multiplication on positive numbers just as before. But, since negative numbers are the opposite of positive numbers, when we encounter them in addition or multiplication, we change our direction and move to the left. If an addend is positive, move to the right. If an addend is negative, move to the left.

Let's look at some examples of addition using negative numbers.

6+ -2. Start at 6 and move to the left by 2.
-6···-5···-4···-3···-2···-1···0···1···2···3···4···5···6
6+ -2 =4

-6+ 2. Start at -6 and move to the right by 2.
-6···-5···-4···-3···-2···-1···0···1···2···3···4···5···6
-6+ 2 =-4

6+ -8. Start at 6 and move to the left by 8.
-6···-5···-4···-3···-2···-1···0···1···2···3···4···5···6
6+ -8=-2

8+ -6. Start at 8 and move to the left by 6.
-6···-5···-4···-3···-2···-1···0···1···2···3···4···5···6···7···8
8+ -6=2

Notice the similarity between 8+ -6 and 6+ -8. They both equal 2 but the signs are different. Notice how in 6+ -8 the answer has the sign of the number with the higher magnitude. The same thing happened with 6+ -2 and -6+2. The answer is still 4, but with the sign is the same as the number with the higher magnitude.

This process of adding by moving to the left instead of the right is called subtraction. Subtraction is the opposite of addition. In addition, you start at a number and move to the right. In subtraction, you start at a number and move to the left. Negative signs tend to have more power than a positive sign, so when you see them side by side, the negative sign takes over. Because of this, 6+ -8 is written as 6-8 and 6+ -2 is written as 6-2.

What happens if we start with a negative number and subtract?
-4-2 = -6
-6···-5···-4···-3···-2···-1···0···1···2···3···4···5···6

When the signs are the same, you add the numbers and keep the sign. When the signs are different, you subtract the numbers and keep the sign of the larger number.

The integers inherit the commutativity and associativity of the natural numbers. Zero is still the identity of addition. With the integers, addition gets another special number called an inverse. The inverse is a number that you add by to get the identity. (Multiplication also has an inverse, but we need another type of number that will come later.) So the inverse of 4 is -4 because 4-4=0. The inverse of -4 is 4 because -4+4=0.

How about multiplication of negative numbers? We need to add a step to our multiplication process to account for negative multiplicands. If the multiplicand (the number on the left) is positive, we face to the right. If the multiplicand is negative, we face to the left. If the multiplier is positive we jump forward. If the multiplier is negative we jump backward.

3×-3
Start facing to the right and jump backwards.
························································>
-9···-8···-7···-6···-5···-4···-3···-2···-1···0···1···2···3···4···5···6···7···8···9
3×-3=-9

-3×3
Start facing to the left and jump forwards.
························································<
-9···-8···-7···-6···-5···-4···-3···-2···-1···0···1···2···3···4···5···6···7···8···9
-3×3=-9

-3×-3
Start facing to the left and jump backwards.
························································<
-9···-8···-7···-6···-5···-4···-3···-2···-1···0···1···2···3···4···5···6···7···8···9
-3×-3=9

Notice when the signs are different, the answer is negative and when the signs are the same the answer is positive. If you are multiplying a string of numbers and the number of negative signs is odd the answer is always negative.

So far we have been looking at what happens when we add or subtract two numbers. We can also compare two numbers. We look at the relative position of the two numbers on the number line and determine which one is further to the left or right. If a number is further to the right than the other, it is said to be greater than the other. If a number is further to the left than the other, it is said to be less than the other. If two numbers are at the same place on the number line, then they are said to be equal to each other. There is a law in mathematics called the trichotomy law. It says that a number can compare to another number in only one of three ways (tri is Greek for three). It is either greater than, less than or equal to the other number. Greater than is denoted by < and less than is denoted by >. The arrow always points to the smaller number.

3 < 6 and -3 > -6.

It gets confusing with negative numbers so you have to mentally place the numbers on the number line and look at which number is further to the right. That number is the greater number. On a number line, -3 is further to the right than -6, so -3 is greater than -6.

Comparisons are not usually done on individual numbers. You are more likely to see them in reference to two equations. You simplify each equation separately, then compare the resulting numbers.
3+6 < 3×4
9 < 12

4×4 = 12+4
16 = 16

3×1-2 > 3+4×-2
3-2 > 3-8
1 > -5

Now that we have the operation of subtraction, let's revisit division in the next blog.

Deconstruction: Division

Division is the opposite of multiplication. In multiplication, we start at zero and jump to the right a given number of times to the answer. In division we start on a number and jump to the left to zero counting the number of jumps. In division, the number of jumps to zero is the answer.

We start at 6 and start jumping by 2's to the left towards zero. We count each jump.
0-----1-----2-----3-----4-----5-----6
It took three jumps to get to zero, so 6÷2=3.

The numbers involved in division are called the dividend (the number we start with), the divisor (the number we subtract by or the size of the jumps we make) and the answer is called the qoutient. Let's divide 6 by 2. The equation is written 6÷2. 6 is the dividend and 2 is the divisor.

You can learn the result of division for some common small numbers by using the multiplication table in reverse. Find the dividend in the middle of the chart with the proper divisor on the left, then follow the column to the top to find the quotient.

What happens if the jumps don't exactly land on zero. The left over is called the remainder. Let's try 9÷2 and see what happens.

0-----1-----2-----3-----4-----5-----6-----7-----8-----9

We made 4 jumps to the left and landed on one. That means 9÷2 is 4 with a remainder of 1. What we are actually saying is that 4×2+1=9. (Remember from our order of operations that multiplication is carried out before addition.) This illustrates the division algorithm. Any whole number is the product of a whole number and a quotient plus a remainder.

To divide larger numbers, we need another method of deconstructing numbers. And for that method we need to extend our alphabet to include a new type of number. We need the operation of subtraction and we need the concept of negative numbers to do subtraction. We will introduce negative numbers in the next blog and use the operation of subtraction to extend division to large numbers.