Basic Math and Pre-Algebra
PART 1. The World of Numbers
CHAPTER 1. Our Number System
Suppose you needed to talk about the distance from Earth to Mars (which keeps changing because both planets are moving, but you can give an approximate distance). You can say that Earth and Mars are at least 34,796,800 miles apart and probably not more than 249,169,848 miles apart, so on average, about 86,991,966.9 miles. If you read that last sentence and quickly lost track of what the numbers were and replaced their names with a mental “oh, big number,” you’re not alone.
Whether they’re written as a string of digits like 34,796,800 or in words like two hundred forty-nine million, one hundred sixty-nine thousand, eight hundred forty-eight, our brains have trouble really making sense of numbers that large. (Whether you think the numbers or the words are easier to understand is a personal matter. Our brains are not all the same.) Scientists and others who work with very large or very small numbers on a regular basis have a method for writing such numbers, called scientific notation.
Scientific notation is a system of expressing numbers as a number between one and ten, times a power of ten. The first number is always at least 1 and less than 10. Ten and any number bigger than ten can be written as a smaller number times a power of ten.
Let’s look at that with a few smallish numbers first. A single digit number like 8 would be 8 x 100. Ten to the zero power is 1, so 8 x 100 is 8 x 1 or 8. The number 20 would be 2 x 101. 101 is 10, so 2 x 101 is 2 x 10, or 20. For a larger number like 6,000,000 you would think of it as 6 x 1,000,000, or 6 x 106.
Scientific notation is a method for expressing very large or very small numbers as the product of a number between 1 and 10 and a power of 10.
To write a large number in scientific notation, copy the digits and place a decimal point after the first digit. This creates the number between 1 and 10. Count the number of places between where you just put the decimal point and where it actually belongs. This is the exponent on the ten. Once you write the number as a number between 1 and 10 times a power of 10, you can drop any trailing zeros, zeros at the end of the number.
Here’s how to write 83,900 in scientific notation:
1. Write the digits without a comma.
2. Insert a decimal point after the first digit.
3. Count the places from where the decimal is now to where it was originally.
4. Write as a number between 1 and 10 multiplied by a power of 10.
8.3900 x 104
5. Drop trailing zeros.
8.39 x 104
The number 83,900 can be written as 8.39 x 104.
To change a number that is written in scientific notation to standard notation, copy the digits of the number between 1 and 10 and move the decimal point to the right as many places as the exponent on the 10. You can add zeros if you run out of digits. The number 3.817 x 108 becomes or 381,700,000.
16. Write 59,400 in scientific notation.
17. Write 23,000,000 in scientific notation.
18. Write 5.8 x 109 in standard notation.
19. Write 2.492 x 1015 in standard notation.
20. Which is bigger: 1.2 x 1023 or 9.8 x 1022?