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The Positive, the Negative,
and the Ugly
Positive numbers are greater than zero. Negative numbers
are less than zero. Zero itself is neither positive nor negative.
On a number line, positive numbers appear to the right of zero and
negative numbers appear to the left.
..., –5, –4, –3, –2, –1, 0, 1, 2, 3, 4, 5,
...
In equations and expressions, positive numbers look like
normal numbers (for example, 7), while negative numbers have a negative
sign in front of them (–7).
Negative numbers can be confusing. It’s like you’re suddenly
told to read from right to left, but all of your instincts tell
you to do the opposite. Why are we telling you this? To convince
you to concentrate. The SAT Math preys on the careless, and negative
numbers are one of the weapons it uses most often.
Negative Numbers and Operations
Negative numbers act differently from positive numbers
when you add, subtract, multiply, or divide them.
Adding and Subtracting Signed Numbers
Adding a negative number is like subtracting a
positive number...
...while subtracting a negative number is the same as adding a
positive number.
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Multiplying and Dividing Negative Numbers
When negative numbers are involved in multiplication and
division, they affect whether the outcome is positive or negative.
You should know these rules cold.
| Multiplying with Negative Numbers | Dividing with Negative Numbers | ||||
|---|---|---|---|---|---|
| Positive × Positive = Positive | Positive ÷ Positive = Positive | ||||
| Negative × Negative = Positive | Negative ÷ Negative = Positive | ||||
| Positive × Negative = Negative | Positive ÷ Negative = Negative |
Here’s a helpful trick when dealing with a series of multiplied
or divided positive and negative numbers: If there’s an even number
of negative numbers in the series, the outcome will be positive.
If there’s an odd number, the outcome will be negative.
The Ugly: Negative Numbers and Parentheses
When negative signs and parentheses collide, well, the
heading says it all. The principle is simple: A negative sign outside
parentheses is distributed across the parentheses. Take the question 3 + 4
– (3 + 1 – 8) = ?. Solve this problem by following PEMDAS
and first working out the parentheses:
3 + 4 – (4 – 8) = 3 + 4 – (–4) = 3 + 4 + 4 = 11
When you start dealing with algebra, however, you won’t
always have like terms and you won’t be able to work out what’s
in the parentheses. You’ll instead have to get rid of the parentheses
by distributing the negative sign across it. Why can you do this?
Because 3 + 4 – (3 + 1 – 8) = ? secretly has multiplication in it.
It can also be written as 3 + 4 + (–1)(3 + 1 – 8) = ?. So every number
within the parentheses should be multiplied by –1.
But remember that multiplication with a negative number changes
the sign of the product. So the simplified expression is 3 + 4 – 3 – 1 + 8 = 11. Whenever you see a negative sign before
parentheses, take a deep breath and be careful of your signs.
Absolute Value
Negative numbers got you down? Absolute value can help.
The absolute value of a number is the distance on a number line
between that number and zero. Or, you could think of it as the positive
“version” of every number. The absolute value of a positive number
is that same number, and the absolute value of a negative number
is the number without a negative sign.
The absolute value of x is written this
way: |x|.
It is also possible to have expressions within absolute
value brackets:
You can’t just make that –7 positive because
it’s sitting between absolute value brackets. You have to work out
the math first:
Now you can get rid of the brackets and the negative sign
from that 4.
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