• 1. Natural Numbers and Whole Numbers
• 2. How Big Can Numbers Get?
• 3. Integers
• 4. Rational Numbers
• 5. Real Numbers
• 6. Properties of Numbers

High School Modules > Algebra by Gregory A. Moore

The Nature of Numbers

An exploration of the various subsets and properties of the real numbers, and interesting features of each.

[Directions : Execute the Code Resource section first. Although there will be no output immediately, these definitions are used later in this worksheet.]

1. Natural Numbers and Whole Numbers

Mankind began as most children do...counting objects they see. Numbers are the first level of abstraction in mathematics because there is nothing in common with five apples, five rocks, and five days... except the number five!

> restart;

> `\nSome Natural Numbers ` = {k $ k = 1..100};

It took many centuries before mankind introduced a new number ...0... a number which signifies no objects at all. This was a huge step.

> `\nSome Whole Numbers ` = {k $ k = 0..100};

>

2. How Big Can Numbers Get?

Pretty big!

> 3^2; `\nThis has this many digits :` ,length(%);

> 3^20; `\nThis has this many digits :` ,length(%);

> 3^200; `\nThis has this many digits :` ,length(%);

> 3^2000; `\nThis has this many digits :` ,length(%);

> 3^20000; `\nThis has this many digits :` ,length(%);

Lets try to roughly compute some common numbers we may encounter. Would you guess that there are more blades of grass on a football field, or printed characters in a library?


Little Question 2.1 : A movie theatre has 40 rows with 26 seat per row. How many people are there when the theatre is full?

> 40*26;

Little Question 2.2 : How many characters are there on the page of a book?
(count the number of lines per page and the average number of characters per line)

> 52*80;

Little Question 2.3 : How many characters are there in a 350 page book?

> 350*52*80;

Little Question 2.4 : How many books are there in the library? (42 bookcases, with 6 rows of book per shelf, with 32 books per shelf).

> 42*6*32;

Little Question 2.5 : How many printed characters are there in a library?

> (350*52*80) *(42*6*32) ;

Little Question 2.6 : If there are 19 blades of grass in a square inch of lawn, how many blades of grass are there on a football field 350 feet by 125 feet?

> 19 * (12 * 350) * (12 * 125);

Little Question 2.7 : If there are 400 grains of sand in a teaspoon of sand that would cover a square inch if spreadout, how many grains of sand are there on the top surface of a beach 200 feet wide, by 3 mile?

> 400 * (12*12) * 200 * 3* 5280;

3. Integers

The integers, Z , consist of three components : natural numbers, zero, and negatives of naturals.

> `\nSome Integers ` = {k $ k = -50..50};

> `\nPositive Natural Numbers ` = {k $ k = 1..50};
`\nZero ` = {k $ k = 1..50};
`\nNegative Natural Numbers ` = {k $ k = -50..-1};

Some people like to think of a bank statement. Deposits are positive numbers, and checks or withdrawals are negative numbers. Two deposits add to a greater balance, while two withdrawals add to be equivalent to a greater withdrawal. This principle is seen when adding two numbers of the same sign.

> (123) + (456);

> (-123) + (-456);

However adding numbers of opposite signs causes a net transaction that depends on which is larger. A deposit and withdrawal at the same time can result in a net deposit if the deposit is bigger, or a net withdrawal if the withdrawal is bigger. We compute the result by subtracting the two numbers and taking the sign of the larger.

> (123) + (-456);

> (-123) + (456);

Note that negative numbers have this reversal property when it comes to greater than. 3 is less than 4, but -3 is greater than -4. Thats because its better to have $4 in the bank rather than $3, and its also better to be only $3 in debt rather than $4 in debt.

> 3 < 4 ; -3 > -4;

4. Rational Numbers

Rational numbers are defined as the set of all fractions of integers where the denominator is non-zero.

> Q := { a/b };

> `\nSome Rational Numbers ` = [ seq( seq( i/j, i = 1..j-1), j = 1..10)];

Converting from Rational to Repeating Decimal

ALL rational numbers can be expressed as numbers with repeating decimals. Some numbers (whose quotients are powers of 2 and 5) have terminating decimals ... or you can also think of them as numbers with repeating zero decimals.

> 1/64: % = evalf(%,70);

> 1/625: % = evalf(%,70);

> 1/2500: % = evalf(%,70);

Other rational numbers have infinitely repeating periods of digits. Can you state how many digits are in the repeating period of eac/

> 2/3: % = evalf(%,70);

> 1/14: % = evalf(%,70);

> 10/13: % = evalf(%,70);

> 5/39: % = evalf(%,70);

> 8/19: % = evalf(%,70);

Converting from Rational to Repeating Decimal


It is not obvious, but EVERY number with a repeating sequence of digits must be a rational number. That means it can be expressed as a fraction.

Example 1 :

> secret := evalf( 95/21, 60);

We notice that there is a period of six repeating digits. So if we "slide" the number over by a facor of 10^6, the periods will still be lined up. Then if we subtract the original from this one, all of the infinite part of the decimal will cancel - leaving only an integer part.

> A := round(secret*10^6 - secret);

This integer is equal to (10^6 - 1) times the original number S. We can now solve for S, as the quotient of two integers.

> S*10^n - S = S*(10^n - 1);
a = S*(10^n - 1); solve(%,S);

Lets execute this idea ....to find the original fraction.

> A/(10^6 - 1);

> `Check : `,% = evalf( %, 40);

Lets try another example.
Example 2 :


This one has a period of only two digits. However its a little more tricky.

> secret := evalf( 19/22, 60);

> A := round(secret*10^4 - secret*10^2);

> S = A/(10^4 - 10^2);

5. Real Numbers

Real numbers include both rational and irrational numbers. Irrational numbers can not be expressed as fractions. Here are some examples. You can try looking for repeating decimals if you wish, but don't waste too much time. ;->

> Pi: % = evalf(%,1000);

> sqrt(2): % = evalf(%,500);

> sqrt(Pi): % = evalf(%,500);

> exp(1): % = evalf(%,500);

> sqrt( 2 + sqrt( 1 + sqrt( 3))); % = evalf(%,500);

> 2^(1/3); % = evalf(%,500);

> 1/(Pi + sqrt(5) + exp(1)); % = evalf(%,500);

6. Properties of Numbers

There are a number of properties of real numbers which underly all that we do in math.

1. Identity Properties

For the operations of addition and multiplication there are special numbers which are identities - they don't change other numbers. The identity for addition is 0 and for multiplication is 1.

> a + 0; 0 + a; 0 + 12324;

> a*1; 1*a;; 1* 12324;

2. Inverse Properties

In regard to both addition and multiplication, each number has an inverse - a number that cancels out the original number and transforms it into the inverse. The additive inverse for a is -a, and the multiplicative inverse for a (non-zero) is 1/a.

> a + (-a); 3 + (-3); (324234/2306491) + (-324234/2306491);

> a*(1/a); 234 * (1/234); (345/671) * (671/345);

3. Associative Properties

> 3 + ( 4 + 5); (3 + 4) + 5;

> 50723423 + ( 1812388 + 99993211 );
( 50723423 + 1812388 ) + 99993211 ;

> 3 + ( 4 + 5); (3 + 4) + 5;

> 50723423 * ( 1812388 * 99993211 );
( 50723423 * 1812388 ) *99993211 ;

4. Commutative Properties

> a + b = b + a;

> 79234091912091147546 + 4856765464338;
4856765464338 + 79234091912091147546;

> a * b = b* a;

> 1299934 * 3692010; 3692010*1299934;

5. Distributive Property

This is the only property which involves both operations at the same time. All of the previous rules involved addition or multiplication but not both.

> a*(b+c) = a*b + a*c;

> 7105871 * ( 10344392 + 47183017439 );
( 7105871 * 10344392) + (7105871 * 47183017439 );

> (10000)*(333*x + 777777);

>

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