Negative numbers are numbers that exist below zero. Negative numbers usually have a (-) symbol in front of them to show that they are negative. Positive numbers can sometimes have the (+) symbol, but not always.

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Estimation

Estimating by Rounding

Estimation by rounding is one of the easier ways to estimate the answer to a problem. It is typically used when dealing with decimals or large numbers. To do this type of estimation, round each number in the problem and use the rounded values instead to get an answer. The answer isn't accurate, but if done correctly it should be close to the real answer.

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Rounding

Rounding is a way to simplify a number. Numbers, such as decimals, can be entirely too complex for the task at hand and need to be simplified. For instance, a gas station can charge 1.299 dollars per gallon of gasoline. The problem with this is that you can't really have 0.299 dollars. The closest is to round the decimal. It rounds to 0.30, and so a gas station will charge 1.30 per gallon if you only buy one gallon of gasoline.

Checking your Answer to a Problem

Rounding is a way to make numbers easier to work with. Rounding may not give the correct answer to a problem, but it will give CLOSE to the correct answer. It is a quick and easy way to check your answer.

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Place values

Whole Numbers

10,987,654,321

Each digit in the number above has a certain place value name associated with it. From left to right they are:

Ten Billions (the first 1)
Billions (the 0)
Hundred Millions (the 9)
Ten Millions (the 8)
Millions (the 7)
Hundred Thousands (the 6)
Ten Thousands (the 5)
Thousands (the 4)
Hundreds (the 3)
Tens (the 2)
Ones/Units (the 1)

Smaller Numbers

1.234567890

Each decimal place also has a place value name. From left to right, they are:

Ones/Units (the 1)
Tenths (the 2)
Hundredths (the 3)
Thousandths (the 4)
Ten Thousandths (the 5)
Hundred Thousandths (the 6)
Millionths (the 7)
Ten Millionths (the 8)
Hundred Millionths (the 9)

Notice that these are the same as the whole number place value names, but they all contain a ths at the end of the words. This ths shows that these numbers are very small.

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Calculator Usage

Using a calculator takes a bit practice. It may seem easy to use, but every calculator is different, and it is easy to make a mistake on a calculator if you are used to another one!

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Games

[Broken Calculator Game] - Can you get to a set of numbers using a broken calculator?

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Inequality

An inequality is a mathematical sentence that does not use an equal sign.

Inequality Symbols

≈	Approximately equal to or about equal to
≠	Not equal to
>	Greater than
≥	Greater than or equal to
<	Less than
≤	Less than or equal to

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Powers of Ten

The powers of ten are any number that is ten with an exponent. For example, 101 is the first power of ten. It is 1 with one zero after it, or 10. 102 is 1 with two zeroes after it, or 100. The exponent is the key here as it tells how many times you must move the decimal to the right to get an answer. Negative numbers, of course, reverse the direction to move the decimal.

Names of the Powers of Ten

The powers of ten:

Power of 10	Word Name	Written as a Decimal
101	Ten	10
102	Hundred	100
103	Thousand	1,000
104	Ten Thousand	10,000
105	Hundred Thousand	100,000
106	Million	1,000,000
107	Ten Million	10,000,000
108	Hundred Million	100,000,000
109	Billion	1,000,000,000
1012	Trillion	1,000,000,000,000
1015	Quadrillion	1,000,000,000,000,000
1018	Quintillion	1,000,000,000,000,000,000
1021	Sextillion	1,000,000,000,000,000,000,000
1024	Septillion	1,000,000,000,000,000,000,000,000
1027	Octillion	1,000,000,000,000,000,000,000,000,000
1030	Nonillion	1,000,000,000,000,000,000,000,000,000,000
10100	[Googol]	10,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000

Multiplying by Powers of Ten

Dividing by Powers of Ten

Exponents

An exponent is a number that indicates how many times to multiply a number by itself.

29, for example, means 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2. The answer to this works out to be 512

A number with an exponent has two parts. In the example above, the 2 is called the base and the 9 is known as the power or exponent

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Scientific Notation

Scientific Notation is a way of writing numbers that are very large and very small. Numbers like the weight of planets or the speed of light or the size of an atom may be represented in this form, and it is important to understand how to read these numbers.

Definition

6.70 x 108, which is the speed of light, is a number in scientific notation. What this means is that if you move the decimal point 8 times to the right, you will get the correct number. The eight comes from the exponent.