By the end of this lesson, you should be able to:

2. Identify and work with signed numbers.

3. Define the sets of integers, real numbers, rational numbers, irrational numbers, imaginary numbers, complex numbers, cardinal numbers, ordinal numbers, and transfinite numbers.

4. Describe and employ the rules of standard and scientific notation.

5. Convert numbers from one base to another.

6. Use addition, subtraction, multiplication, and division with correct application of the commutative, associative, and distributive laws.

This lesson looks at the types of numbers, notation, operations, and some common procedures.

Types of numbers

Numerals or numbers?

A numeral is a symbol, but it is not a number.

Which of the following numbers is larger:

9

3

The number 9 is larger, but the numeral 3 is taller.

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Without a negative sign, a number is assumed to be positive. Numbers that have positive or negative signs are called signed numbers.

Which is greater, +3 or -8?
Because it is "farther to the right on the number line," +3 is greater. However, the magnitude of -8 is greater.
Sometimes we wish to refer to the distance between number and zero, regardless of the sign of that number. This is referred to as the "absolute value" of the number, and is indicated with two vertical lines.

|7| = 7

|-7| = 7

-|7| = -7

-|-7| = -7

In some technical problems, a number with a plus sign can indicate a tension, an uphill slope, or another specified meaning.

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Integers are numbers with no fractional or decimal part.

Examples are 5, 0, and -343.

Positive integers are called natural numbers.

The set of integers is made up of the set of whole numbers {0, 1, 2, ...} and the set of negative whole numbers {-1, -2, ...}

Together, the set of integers and the set of non-integer fractions make up the set of real numbers.

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Real numbers are those on the number line.

Examples are 2, 1/3, pi, and -7.5.

The set of real numbers is composed of the sets of rational and irrational numbers.

{Reals}={{Rational}+{Irrational}

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Rational numbers are the real numbers that can be expressed as the quotient of two integers.

Examples include 1/3, 7, 3.7, and -5.

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Irrational numbers are the real numbers that cannot be expressed as the quotient of two integers.

Examples include non-terminating, non-repeating decimals, such as pi and the square root of 2.

Imaginary numbers cannot be found on the number line. The include numbers such as i, the square root of -1.

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Complex numbers include the set of real numbers and the set of imaginary numbers.

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Cardinal numbers refer to the count of the elements in a set.

For example, the cardinal number of the set of standard current english letters is 26.

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Ordinal numbers represent an order.

Examples include first, third, and tenth.

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There are other types of numbers based on different mathematical theories. One example of these are the Transfinite Numbers.
Transfinite numbers are not finite. They are different types of infinity. Not all transfinite numbers are equal, however. Here are some odd concepts to ponder, though probably of little use:

Aleph-null is the cardinal number of the set of Integers.

(There are as many positive integers as there are both positive and negative integers, and that is Aleph-null. Aleph-null plus aleph-null equals aleph-null.)

C is the cardinal number of the set of Real numbers. (There are more real numbers than integers, so C is greater than aleph-null.)

Notation

Standard

Standard notation refers to the way we normally write numbers. Here are some conventions in common use:
 

For positive number, the plus sign can be omitted.

 

Decimal points are usually omitted for integers.

 

Decimals that are between -1 and 1 may or may not have a leading zero before the decimal point, depending on the convention observed.

 

Each 3 digits to the left of the decimal point may have a comma, a space, or neither, as in:

• 53,000
• 53 000
• 53000

Some writing style guides specify that natural numbers of ten or less be spelled out with words rather than numerals; typically, where a number is the first term in a sentence it is also spelled out.

 

Modern conventions use K to indicate thousand (e.g., $53K), but there is disagreement as to whether M indicates million or MM indicates million. 

 

Numeral placement is critical, because a superscript usually indicates an exponent, and a subscript usually indicates a base or an index.

 

When writing numerals, some similarities between characters have caused problems. This typically happens with the following:

• the numeral 0 and the letter O;
• the numerals 1 and 7 and the letters I and l;
• the numeral 4 and the numeral 9;
• the numeral 5 and the letter S; and
• the numeral 2 and the letter Z.

The key to avoiding this confusion lies in careful lettering and writing habits. Sometimes, a slash is used in the numerals 0 and 7 and the letter Z. 
 

Parentheses are used to group mathematical terms. Because operations are performed in different orders, the convention of performing the operations for terms within parentheses avoids confusion.

• (3+5)*(8-5) = 24
• 3+(5*(8-5)) = 18

Percentages are indicated with a percent sign % and are identical to hundredths of a given value.

25% of 80 = .25 * 80 = 20

Some people make mistakes when dealing with percentages, and it is usually because they incorrectly answer the question: "Percent of what?"

For example, if production decreases 10% in January, and 10% in February, but then increases 20% in March, is the production level at the end of March the same as it was in the beginning of January?

The answer is "no." Let's say we begin with a level of 1000 kilograms on Jan. 1. At the end of January, the production is:

1000 kg - (10% * 1000 kg), or

1000 kg -100 kg, or 

900 kg.

But an additional 10% reduction in February produces the following production level:

900 kg - (10 % * 900 kg), or

900 kg - 90 kg, or

810 kg.

This is because February's 10% reduction means 10% less than the figure at the end of January, not at the beginning of January.

A 20% increase in March is not an additional 200 kg, but is computed based on the level at the end of February:

810 kg + (20% * 810), or

810 kg + 162 kg, or

972 kg

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Here's another type of percentage problem: You begin with a sheet of paper that measures 8.5 x 11 inches. Keeping the same width-to-length ratio, what is the size of the paper if it is reduced 50%?

The question you need to ask is, "Do you mean a 50% reduction in area, or a linear reduction?"

If you divide the width and the length in half, you get a piece that is 4.25 x 5.5, which corresponds to a 50% linear reduction, but a 75% reduction in area.

To solve the problem if the area is to be reduced by 50%, you may proceed as follows:

Area = 8.5" * 11" = 93.5 sq in

Area - (50% Area) = 93.5 sq in - (93.5 sq in * 50%)

50% Area = 46.75 sq in

But what are the width an length of the piece of paper with the same aspect ratio as an 8.5 * 11? Let's use a little algebra to find out. We'll call a new unit, U.

8.5 U * 11 U = 46.75 sq in

93.5 U² = 46.75 sq in

U² = 46.75 sq in / 93.5

U = square root of (46.75 sq in / 93.5)

U = + (6.837397... / 9.66954...) in

U = .7071 in

8.5U = 6.0104"

11U = 7.7782"

Check:

• (6.0104" * 7.7782")/50% = 93.5002 sq in

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Scientific notation can be used to easily refer to numbers of great or small magnitude. It consists of a decimal number multiplied by 10 raised to a specified power. The decimal is 1 or greater, but less than 10. The power of ten indicates the "number of places the decimal point is to be moved." For example:

734.89 = 7.3489*10²

0.000357 = 3.57*10^-4

Many calculators allow the user to specify scientific notation, or to input numbers in this form. Sometimes a key labeled EE (as in "Enter the Exponent of 10)" is used for this purpose.

For practice with scientific notation, look at the following:

http://work.ccv.vsc.edu/pubs/mathchps/scinote.pdf

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There are other notation formats, such as the use of Roman Numerals. 

Number Bases

Our system of numbers uses separate digits to indicate the ones (or units), tens, hundreds, thousands, etc. There are ten possible numerals for each digit, 0 through 9, thus this is called a "base-10" system. How much is 100 - 10? In our base-10 (i.e., decimal) system, the answer is 90.

But there are other bases in common use. Computers store most of the information they process as binary (not decimal) digits. Each binary digit, or bit can have the value of 0 or 1 in this base-2 system. The base-2 equivalents of 1, 2, 3, 4, 5 are 1, 10, 11, 100, 101. In base 2, the first digit on the right tells us how many units there are; the next digit tells us the number of twos, then the number of fours, then the number of eights, etc. Sometimes a chunk of binary digits is grouped together, and some of these are bytes.

Another common base system is base-16, or hexadecimal. Here is the base-16 equivalent to counting from 1 to 20: 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F, 10, 11, 12, 13, 14. The last in this series, 14, indicates that there are 4 units and 1 sixteen, which add together to equal 20. Typically, two-digit hex couplets are listed to indicate computer codes.

Try filling in the following table, without a calculator, assuming the numbers in each row are equivalent: