Exponents are used in all branches of science to describe extremely large and extremely small numbers. Because the Universe is so big, and the "Big Bang" proponents are going to use exponential notation, we need to be very accustomed to exponents to understand the arguments on both sides and to be able to respond well.
An exponent is a shortcut way of describing a longer process; you've seen shortcuts before:
Addition is the shortcut for repeated incrementations (think "counting on your fingers"):
10 + 5 is easier to convey than (starting at 10) 11, 12, 13, 14, 15
Addition says: “We have 10; add 5 to it.”
Multiplication is the shortcut for repeated additions:
10 × 5 is easier to convey than 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5
Multiplication says: “We add five, ten times” or “Ten times Five".
Exponentiation is the shortcut for repeated multiplications:
105 is easier to convey than 10 × 10 × 10 × 10 × 10
Exponentiation says: “Multiply 10 by itself, five times” or “Ten raised to the fifth power” or “Ten to the fifth.”
Ready to go on? Lesson 2
Ways of writing numbers
Comparing rows
The Zero exponent
Negative Exponents
| Table 1. | |||||
| Placement | Factors | Exponent | |||
|---|---|---|---|---|---|
| A. | 1,000 | = | 1 × 10 × 10 × 10 | = | 103 |
| B. | 100 | = | 1 × 10 × 10 | = | 102 |
| C. | 10 | = | 1 × 10 | = | 101 |
| D. | 1 | = | 1 | = | 100 |
| E. | 0.1 | = | 1 / 10 | = | 10-1 |
| F. | 0.01 | = | 1 /(10 × 10) | = | 10-2 |
| G. | 0.001 | = | 1 /(10 × 10 × 10) | = | 10-3 |
Table 1 shows three ways of noting products of ten; note that all three ways represent the same value.
The first column (labeled "Placement") contains numbers written the way we normally write numbers, but notice all the numbers in that column have only a single one and all the other digits are zeros. The column is called "Placement" because where the one is placed is important. The first number listed is 1,000, and for now, we'll just look at that row.
In the "Factors" column, we see another way to write 1,000: 10 × 10 × 10. That equals 1,000.
In the "Exponent" column, we see the way to write 1,000 in exponential notation. It shows us a '10' (called the base) and a '3' (called the exponent). which, when written together like this mean "multiply 10 by itself 3 times". Notice the 3 is raised up, and is smaller in size than the 1 or the 0. Writing a character raised up and small is called "superscript".
Next, we'll compare Row A. with Row B.
Ready to move on? Click here to go to Topic 2: Comparing Rows
Comparing Row A. with Row B., we see one thousand and one hundred, respectively, each written three ways.
Notice the 'one' in one hundred is further to the right - closer to the decimal point than the 'one' in one thousand; moving from Row A. to Row B., the 'one' has moved to the right.
In the "Factors" column, the 10 × 10 × 10 of Row A. is replaced by 10 × 10 of Row B. By moving down a row, the value in each column is smaller by a factor of 10.
Notice the exponents on the terms in the "Exponent" column.
Moving from Row A. to Row B., we see the exponent changes from 3 to 2 .
These are all ways of showing that the number in the second row is one thousand divided by ten. To move down a row, we divide by ten.
Moving from Row B. to Row C., the numbers are all divided by 10, and we show it in different ways: 100 becomes 10, 10 × 10 becomes just 10 and in the "Exponent" column the exponent on 10 changes from 2 to 1 .
All these are ways of describing ten.
So to move from the second row to the third row, we've divided by ten. And all this is spelled out carefully for a reason: the next row.
Click here to look at the Zero Exponent.
Now, we have to remember the patterns established because you will have to infer what should be:
So look what happens in the Position column: the value is divided by ten for every row we go down. In the Exponent column, the exponent decreases by one.
So what should happen when the exponent is one, and we move down a row?
Next, click here to look at negative exponents.
Negative numbers are the opposites of positive numbers, and so negative exponents are the opposites of positive exponents. But it may seem odd that you can multiply a number by itself a negative number of times.
For this to make sense, please resort to the table and remember the patterns established.
You've seen exponents before, using SI prefixes with the Metric System.
| Decimal | Exponent | Name | SI Prefix |
|---|---|---|---|
| 1,000,000,000,000 | 1012 | trillion | tera (T) |
| 1,000,000,000 | 109 | billion | giga (G) |
| 1,000,000 | 106 | million | mega (M) |
| 1,000 | 103 | thousand | kilo (k) |
| 1 | 100 | one | |
| 0.001 | 10-3 | thousandth | milli (m) |
| 0.000,001 | 10-6 | millionth | micro (µ) |
| 0.000,000,001 | 10-9 | billionth | nano (n) |
| 0.000,000,000,001 | 10-12 | trillionth | pico (p) |
We'll need exponents as we look at the Fine-Tuning Problem.