Chapter 2 - Numbers

Systems of equations are made up of equations, equations are made up of expressions, and expressions are made up of numbers. So at the very bottom of the mathematics food chain are numbers. In this chapter we look at the types of numbers, operations on numbers and uses of numbers. This chapter contains the following sections:

section 2.1 - In this section we talk about the types of numbers, ranging from the natural numbers to the complex numbers. We place each type on the number line.
section 2.2 - We explain the addition and subtraction of positive and negative numbers using arrows placed end-to-end on the number line.
section 2.3 - We explain the multiplication of positive and negative numbers in terms of stretching and reversing of arrows on the number line.
section 2.4 - We explain the division of positive and negative numbers.
section 2.5 - We introduce scientific notation as a way of representing very large or very small numbers.
section 2.6 - We talk about approximate numbers, rounding and significant figures.

2.1 - The Types of Numbers

Let's survey the types of numbers. We will look at natural and whole numbers, integers, rational numbers, irrational numbers, real numbers, imaginary numbers and complex numbers.

The Natural and Whole Numbers

We start with the natural numbers. These are the numbers 1, 2, 3, … (The … symbol means that the sequence goes on forever.) They are used for counting. If we include zero then we get the whole numbers, 0, 1, 2, 3, ….

The natural and whole numbers are usually considered to be exact (e.g. there are 4 tires on a car, 8 legs on a spider). But sometimes they are approximate (e.g. there were 1000 people in the crowd).

Here are the whole numbers, shown on the number line:

The Integers

Now for each of the numbers 1, 2, 3, … let’s create its opposite or negative and put it on the opposite side of the number line, like this:

We say that 1 and −1 are opposites, 2 and −2 are opposites, etc. and we also say that −1 is the opposite of 1 and that 1 is the opposite of −1.

Negative numbers are used to describe debts as opposed to assets, temperatures below zero as opposed to temperatures above zero, heights below sea level as opposed to heights above sea level, and so on.

The set of numbers …, −3, −2, −1, 0, 1, 2, 3, … (the whole numbers and their opposites) is called the integers.

The Rational Numbers

Next are the rational numbers. They are a generalization of common fractions so let’s review those first. Remember that the fraction notation a/b means that we break something into b equal pieces and that we have a of those pieces. For example if we break a pie into 4 pieces and we have 1 piece then we have 1/4 of the pie:

A rational number is defined to be any number that can be expressed as the quotient or ratio of two integers. We use the same fraction notation to express rational numbers:

Integer a is called the numerator and integer b is called the denominator. The denominator is not allowed to be zero.

Notice that if a and b are both natural numbers (1, 2, 3, etc.) then we get a common fraction. Thus the rational numbers include the common fractions.

Notice also that 3/1 = 3 and −5/1 = −5, so the rational numbers include all the integers.

Rational numbers can also be written in decimal notation instead of fraction notation. For example:

1/4 = 0.25

The decimal notation 0.25 means literally “25/100” and 25/100 and 1/4 are equivalent fractions.

Notice that some rational numbers have no exact decimal equivalent. For example 1/3 is approximately equal to 33/100 but not exactly:

1/3 ≈ 0.33

(The symbol ≈ means “is approximately equal to”.) Rational numbers are usually considered to be exact. For this reason the Algebra Coach program will not convert fractions to decimals when it is running in exact mode.

The Irrational Numbers

The irrational numbers are those that cannot be expressed as a ratio of two integers. Examples are

as well as the square roots of many other numbers, and special numbers like e and π. It turns out that there are as many irrational numbers as rational. Irrational numbers have no exact decimal equivalents. To write any irrational number in decimal notation would require an infinite number of decimal digits. Thus these are only approximations:

≈ 1.732, e ≈ 2.718 and π ≈ 3.14,

For this reason the Algebra Coach program will not convert irrational numbers to decimals when it is operating in exact mode.

The Real Numbers

The rational numbers and the irrational numbers together make up the real numbers. The real numbers are said to be dense. They include every single number that is on the number line.

The number line is useful for understanding the order of numbers. Smaller numbers are farther to the left and larger numbers are farther to the right.

We use the symbol < to mean “is less than” and the symbol > to mean “is greater than”. Here are some examples of the use of these symbols:

5 < 8 because 5 is to the left of 8.
−5 < 2 because −5 is to the left of 2. In fact any negative number is less than any positive number.
5 < 8 and 8 > 5 are two ways of stating the same fact.
3.14 < π < 3.15 I call this a “less than sandwich”. It means that 3.14 < π and also π < 3.15. In other words the number π is somewhere between 3.14 and 3.15.

Real numbers often result from making measurements and measurements are always approximate. For example, with one piece of equipment the length of an object might be measured to be 5.28 (in some units). This doesn’t mean the length is exactly 5.28. It just means that it is closer to 5.28 than it is to 5.27 or 5.29. With a more accurate (and usually much more expensive piece of equipment) the length might be measured to be 5.283, which just means it is closer to 5.283 than it is to 5.282 or 5.284, and so on. If an expression contains an approximate number then that whole expression is also approximate.

Click here for more information on accuracy and significant figures.

The Imaginary Numbers and the Complex Numbers

If real numbers include every single number on the number line, then what other numbers could there be? To answer that question, consider how we built up the number system so far:

We started with the whole numbers (numbers like 3)
We wanted opposites for these numbers so we created the integers (numbers like −3)
We wanted to divide these numbers but needed the rationals to describe some of the results (numbers like 3/4)
We wanted to take the square root of these numbers but needed the irrationals to describe some of the results (e.g. )

We now want to take the square root of negative numbers but we will need a new type of number to describe the result. We define the square root of a negative number to be an imaginary number. How much further can this process of creating new types of numbers go? The answer is one more step. We can add a real number to an imaginary number. The result is called a complex number. This is the end of the line because it turns out that every possible operation with every possible complex number results only in other complex numbers. We say that complex numbers make the numbers complete.

Where do imaginary and complex numbers go on the number line? The answer is they don’t. This picture shows the complex plane. It contains the number line (which is now called the real axis) and a new axis called the imaginary axis, perpendicular to it. Real numbers lie on the real axis, imaginary numbers lie on the imaginary axis and complex numbers generally lie off the real axis, either above it or below it.

The Algebra Coach program can run in real mode or complex mode. In real mode it will not carry out any operation that leads to a non-real number (such as taking the square root of a negative number).

Click here for more information on complex numbers.

2.2 - Addition and Subtraction of Numbers

We will find it very useful to represent a number using an arrow on the number line. The tail of the arrow will be at the origin (i.e. at zero) and the head of the arrow will be at the location of the number that is being represented. Here is how we will represent the numbers −4, −1, 2 and 3:

Normally we would draw the arrows directly on the number line. But in this case we wouldn’t be able to distinguish between so many arrows, so we have drawn them slightly above each other.

Addition

To add two numbers on the number line you put their arrows end-to-end (because that’s what adding means) and look where the final arrow ends up.

For example in the picture below the arrows of length 2 and 3 represent the numbers 2 and 3. To add 2 + 3 we put the arrow for 3 at the end of the arrow for 2 and see that we end up at 5, so 2 + 3 = 5.

The next example shows that the order of the two numbers being added does not matter.
3 + 2 also equals 5. This is called the commutative property of addition.

The commutative law of addition. This law states that for any numbers a and b,

a + b = b + a

In other words the result is the same no matter which number is considered to be the “first number” and which number is considered to be the “added on” number.

The associative law of addition. If we need to add three or more numbers together then we do this by adding two at a time. The associative law of addition says that the result is the same no matter which two we choose to start with. It states that for any numbers a, b and c,

(a + b) + c = a + (b + c)

The next example shows how to add negative numbers, which are represented by arrows pointing to the left. The rule for adding arrows end-to-end remains the same. This example shows that (−1) + (−4) = −5.

The next example shows the addition of a positive number and a negative number. The rule for adding arrows end-to-end remains the same. This example shows that 2 + (−5) = −3.

Zero. The number 0 is called the additive identity because adding it to any number a just gives back a :

a + 0 = a

For every number a there is a number −a, called the additive inverse. Added together they give zero:

a + (−a) = 0

Subtraction

There are at least two different ways of understanding subtraction.

One way is to replace the subtraction by the addition of an opposite. We will understand why this works when we look at multiplication in the next section. Here are several examples showing the method:

replace this subtraction by this addition of an opposite

2 − 5 = − 3 2 + (−5) = − 3

−5 − 3 = − 8 −5 + (−3) = − 8

−8 − (−2) = − 6 −8 + 2 = − 6

Another way to view subtraction is to use the idea that we subtract to find out how different two numbers are. (This is why the result is called the difference!) Thus the difference between 5 and 2 is 3. The trouble with this method is that we have to be careful which number we subtract from which.

2.3 - Multiplication of Numbers

First a word on notation:

a · b means multiply a times b. So does a b and so does a*b.
For example 3 · 2 = 6. The numbers 3 and 2 are called factors and the number 6 is called the product.
− ( ) means to take the opposite or negative of whatever is in the brackets.
For example − (3) = −3. Literally the negative of the number 3 is the number −3.
Also − (−3) = 3. Literally the negative of the number −3 is the number 3.

Multiplication originated as a way to describe repeated addition. For example adding 6 to itself 4 times is the same thing as 6 multiplied by 4:

6 + 6 + 6 + 6 = 4 · 6

A more advanced way to interpret multiplication is as stretching. Thus the number 4 · 6 is 4 times as long as the number 6:

Here are some important laws of multiplication.

The commutative law of multiplication. This law states that for any numbers a and b,

a · b = b · a

In other words the result is the same no matter which number is considered to be multiplied onto which.

The associative law of multiplication. If we need to multiply three or more numbers together then we do this by multiplying two at a time. The associative law says that the result is the same no matter which two we multiply first. It states that for any numbers a, b and c,

(a · b) · c = a · (b · c)

Zero Times Anything is Zero

No matter how complicated the factor a is,

0 · a = 0

The Multiplicative Identity

The number 1 is called the multiplicative identity because multiplying any number a by 1 just gives back a :

1 · a = a

A Negative Number Times a Positive Number is Negative

Probably the hardest thing about multiplication is understanding how to deal with negative numbers. We mentioned that multiplication originated as a way to do repeated addition. Then, as the following picture shows, it must be true that 3 · (−2) = (−2) + (−2) + (−2) = − 6 :

Notice that, as the next picture shows, it is also true that (−3) · 2 = − 6 :

Finally it is also true that − (3 · 2) = −6. Now put together all these ways of writing −6:

If you compare these three forms you see that you can move the − sign in front of either factor or in front of the entire product. This is true in general:

Moving the − sign around in a product.
For any numbers a and b,

a · (− b) = (− a) · b = − (a · b)

The Product of Two Negative Numbers is Positive

To derive this rule look at this example:

The first step was to move the − sign from the second factor to the first (as explained in the previous section).

The second step was to use the fact that the opposite or negative of − 3 is 3.

The last step was to just multiply 3 times 2. We conclude that the product of two negative numbers is a positive number.

A Product with Several Negative Factors

We can generalize the above results:

The sign of a product having several negative factors.
If a product has several negative factors then count how many negative factors there are. If there are an odd number then the product is negative. If there are an even number then the product is positive.

Here are some examples:

Definition: Even and odd numbers. The even numbers are the whole numbers 0, 2, 4, 6, … and the odd numbers are all the other whole numbers, namely 1, 3, 5, …. The even numbers are evenly divisible by 2 and the odd numbers are not.

The many roles of the − sign

Have you ever wondered why some calculators have more than one − key? The reason is that the − sign plays several roles in algebra, as this table shows. Each expression evaluates to − 5.

The expression What the − sign means in this expression

− 5 This is just a number to the left of 0 on the number line.

0 + (− 5) This is an addition. We are adding −5 onto zero.

0 − 5 This is a subtraction. We are subtracting 5 from zero.

−1 · 5 This is a multiplication. We are multiplying 5 by −1.

− ( 5 ) This means “the opposite of 5”.

In many algebra problems it is important for you to be able to switch the − sign from one role to another.

2.4 - Division of Numbers

There are several notations for division:

They all mean to divide the number a by the number b. The number a is called the numerator (or sometimes dividend), the number b is called the denominator (or sometimes divisor), and the ratio a/b is called the quotient. The notation is exactly the same as the notation for fractions because when a and b are whole numbers then the quotient that results from dividing a by b is just the fraction a/b. Click here to review fractions. In this section we study division when a and b are any real numbers in general.

Division defined. The result of dividing a real number a by a real number b is that real number c such that a = b · c. In other words, division is defined in terms of multiplication, which we studied in the previous section.

For example:

Division (of a non-zero number) by zero is undefined. The reason is this. Suppose that it was possible that a / 0 (where a is non-zero) could equal some number c. Then by the definition of division this would mean that a = 0 · c. But this is impossible because zero times anything is zero.

Zero divided by zero is indeterminate. (Indeterminate means that 0 / 0 could equal any number c.) The reason is that by the definition of division this means that 0 = 0 · c. But this is true for any number c whatsoever.

Note that this does not mean that 0 / n is undefined or indeterminate. On the contrary, 0 / n = 0 for any non-zero n.

Reciprocals. Two numbers whose product is 1 are called reciprocals.

For example:

4/5 and 5/4 are reciprocals because

8 and 1/8 are reciprocals because

Just as subtraction can be replaced by the addition of a negative so division can be replaced by multiplication by a reciprocal:

The division of a by b is equivalent to the multiplication of a by the reciprocal of b :

Here are some examples:

replace this division by this multiplication
by the reciprocal

Division with negative numbers

Because division is defined in terms of multiplication, the rule for finding the sign for division is the same as the rule for finding the sign for multiplication: Count the number of negative factors. (It doesn't matter whether the factors are in the numerator or the denominator.) If the number is even then the result is positive. If the number is odd then the result is negative. The minus sign is usually put in front of the quotient. In this example there are 5 negative factors so the result is negative:

2.5 - Scientific Notation

In scientific applications very large or very small numbers are encountered all the time. An example of a very large number is the speed of light (300000000 meters per second) and an example of a very small number is the charge of an electron (0.00000000000000000016 coulombs). To make these numbers easier to read they are usually expressed in scientific notation, like this:

300000000 = 3 × 10⁸

0.00000000000000000016 = 1.6 × 10^− 19

The rule for writing a number in scientific notation is to write it the product of two numbers. The first number, called the mantissa, must have its absolute value between 1 and 10, and the second number must be written as 10 raised to an integer power. (If the mantissa is exactly 1 then it may be omitted.) Calculators often replace the symbols “× 10 ” by the symbol E to save space on the display screen, like this:

300000000 = 3 E 8

0.00000000000000000016 = 1.6 E − 19

Notes:

A common mistake is the think that 3 E 8 means 3⁸. This is not correct. 3 E 8 means 3 × 10⁸.
The following forms all have the same value but only the first one is scientific notation. To change from one form to the next we multiply the mantissa by 10 and subtract 1 from the power of 10 (effectively dividing the second number by 10). The result is that the value of the product remains unchanged:

3 × 10⁸ = 30 × 10⁷ = 300 × 10⁶ = 3000 × 10⁵
A notation similar to scientific notation is engineering notation. In this notation the mantissa is multiplied or divided by enough powers of 10 so that the power of 10 in the second number can be a multiple of 3. (Engineers like to count in thousands, millions, billions, etc.) For example 3 × 10⁸ will be written either as 0.3 × 10⁹ or as 300 × 10⁶.
The Algebra Coach can accept any of these forms as input:

300000000 or 3 E 8 or 3 * 10 ^ 8

Multiplying or dividing numbers in scientific notation

To multiply or divide numbers that are expressed in scientific notation, simply multiply or divide the mantissas to produce the new mantissa and combine the powers of 10 together using the properties of exponents to produce the new power of 10. If the new mantissa is not between 1 and 10 then adjust it and the power of 10 as in the second note above.

Here are some examples:

(1.2 × 10⁸) (2.3 × 10¹⁰)
= 1.2 × 2.3 × 10⁸ × 10¹⁰

= 2.76 × 10¹⁸

(6 × 10⁸) (7 × 10¹⁰)
= 42 × 10¹⁸ ← mantissa is too big

= 4.2 × 10¹⁹ so divide it by 10 and add 1 to the power of 10

= 0.5 × 10^− 2 ← mantissa is too small

= 5 × 10^− 3 so multiply it by 10 and subtract 1 from the power of 10

Adding and subtracting numbers in scientific notation

In order to add or subtract numbers expressed in scientific notation the power of 10 factor must be the same for all of the numbers. To accomplish this use the above method of multiplying or dividing the mantissa by a power of 10 and adjusting the power of 10 factor accordingly. Here are some examples:

8.4 × 10¹⁰ + 5.3 × 10¹⁰ ← powers of 10 are the same so just go ahead and add mantissas
= 13.7 × 10¹⁰ ← mantissa is too big so adjust it

= 1.37 × 10¹¹

8 × 10⁶ + 5.1 × 10⁷ ← the powers of 10 are different; adjust the first number
= 0.8 × 10⁷ + 5.1 × 10⁷ ← powers of 10 the same; now add mantissas

= 5.9 × 10⁷

2.6 - Approximate Numbers and Significant Figures

An exact number is one that has no uncertainty. An example is the number of tires on a car (exactly 4) or the number of days in a week (exactly 7). An approximate number is one that does have uncertainty. A number can be approximate for one of two reasons:

The number can be the result of a measurement. For example a certain instrument capable of measuring to the nearest 0.1 cm may measure the length of a certain bolt to be 8.6 cm. A better quality instrument capable of measuring to the nearest 0.001 cm may give the length of the same bolt to be 8.617 cm. This new number is less approximate but is still not exact.
Certain numbers simply cannot be written exactly in decimal form. Many fractions and all irrational numbers fall into this category. For example the fraction 1/3 is approximately but not exactly equal to 0.333 and the irrational number is approximately but not exactly equal to 1.73.

When we state that the measured length of the bolt is 8.6 cm then we actually mean that the value is closer to 8.6 cm than it is to 8.5 cm or 8.7 cm. The true length could be anywhere in the gray area shown here:

And when we state that the more accurate instrument gave the length of the bolt to be approximately 8.617 cm then we mean that the value is closer to 8.617 cm than it is to 8.616 cm or 8.618 cm. The true length could still be anywhere in the gray area shown here:

If someone told us that they used this same instrument and got a reading of 8.61712345 would we believe it? No way! Adding just one atom to the end of the bolt would cause the last digit to change! The extra digits are meaningless and are said to be insignificant. To claim that they are correct is nonsense.

Significant Digits or Figures

Definitions:

In an approximate number the leftmost digit is said to be the most significant digit and the rightmost digit is the least significant digit. All the digits in the number are significant digits (also known as significant figures or sig. figs.) with one exception: if the digit is a zero that is used just to locate the decimal point then it is not significant.

The accuracy of an approximate number is given by the number of significant digits in it.

The precision of an approximate number is given by the position of the rightmost significant digit.

Examples:

The approximate number 8.617 has 4 significant digits. The digit 8 is the most significant digit and the digit 7 is the least significant digit.
The number 1.23, the number 0.000123 and the number 123000000 all have an accuracy of 3 sig. figs. All the zeros are used simply to locate the decimal point.
The number 1.23 has a precision of 0.01, the number 0.000123 has a precision of 0.000001 and the number 123000000 has a precision of 1000000.
The number 1.023, the number 0.01023 and the number 1002000 all have 4 sig. figs. (The zeros shown in red are used simply to locate the decimal point and don't count as sig. figs.)

Notes: The precision of a measuring instrument is the difference between the two closest readings that the instrument can differentiate. For example the above instruments had a precision of 0.1 cm and 0.001 cm.

Precision and accuracy are not the same thing. Accuracy has to do with the quality (and cost!) of the measurement. For example if your instrument has a precision of 1 centimeter then that may not be very accurate if that instrument is designed to measure distances between objects on your desk but it would be very accurate if it was designed to measure the distances between the planets.

Rounding

We saw above that if an instrument capable of measuring to the nearest 0.1 gave a measured value of 8.6 then the true value could be anywhere in the gray area:

Rounding is exactly the same idea but reversed. Rounding to the nearest 0.1 means to replace any number in the gray area (from 8.55 to 8.65) by 8.6. In general, rounding is done like this:

Rounding: When rounding to a certain place value then all digits to the right of that place are dropped. If the first dropped digit is 0, 1, 2, 3, or 4 then the least significant digit kept is not changed. (This is called rounding down.) If the first dropped digit is 5, 6, 7, 8 or 9 then the least significant digit kept is increased by 1. (This is called rounding up.)

You can round to either a given decimal place or to a given number of sig. figs. Here are some examples of rounding to 2 decimal places (the dropped digits are shown in red):

the rounding the rule used

4.384 → 4.38 first dropped digit is a 4 so round down

4.3851 → 4.39 first dropped digit is a 5 so round up

0.00043851 → 0.00 first dropped digit is a 0 so round down

Here are some examples of rounding to 3 sig. figs (the dropped digits are shown in red):

the rounding the rule used

4.384 → 4.38 first dropped digit is a 4 so round down

43851 → 43900 first dropped digit is a 5 so round up

0.00043851 → 0.000439 first dropped digit is a 5 so round up

replace this subtraction	by this addition of an opposite
2 − 5 = − 3	2 + (−5) = − 3
−5 − 3 = − 8	−5 + (−3) = − 8
−8 − (−2) = − 6	−8 + 2 = − 6

The expression	What the − sign means in this expression
− 5	This is just a number to the left of 0 on the number line.
0 + (− 5)	This is an addition. We are adding −5 onto zero.
0 − 5	This is a subtraction. We are subtracting 5 from zero.
−1 · 5	This is a multiplication. We are multiplying 5 by −1.
− ( 5 )	This means “the opposite of 5”.

replace this division	by this multiplication by the reciprocal

the rounding	the rule used
4.384 → 4.38	first dropped digit is a 4 so round down
4.3851 → 4.39	first dropped digit is a 5 so round up
0.00043851 → 0.00	first dropped digit is a 0 so round down