Signed Numbers, Absolute Value, Coordinate Geometry Flashcards
What is a number line?
A number line is a line segment on which integers are listed in numerical order.
*** Integers include all positive and negative whole numbers, as well as zero.
Define:
A signed number.
A signed number is a number with a plus or minus sign associated with it.
*** Special rules are needed for working with them.
*** When the sign is omitted, it is understood to be plus.
Define:
The absolute value of a number.
The absolute value of a number is the distance between that number and zero on the number line.
|a| = a
if a is greater than or equal to zero.
|a| = -a
if a is less than zero.
Is the absolute value of a sum of two numbers always equal to the sum of the absolute values?
|a + b| ? |a| + |b|
No, not always. The absolute value of a sum of two numbers is less than or equal to the sum of the absolute values.
|a + b| ≤ |a| + |b|
- Example:*
- I5 + 3I = I5I + I3I
I8I = 8 ; I5I + I3I = 5 + 3 = 8
- I5 + (-3)I I5I + I-3I
I2I = 2 ; I5I + I-3I = 5 + 3 = 8
Is the absolute value of a product of two numbers always equal to the product of the absolute values?
|a x b| ? |a| x |b|
Yes, the absolute value of a product of two numbers always equals to the product of the absolute values.
|a x b| = |a| x |b|
Example:
I5 x (-3)I = I-15I = 15
I5I x I-3I = 5 x 3 = 15
Fill in the blank:
Opposite numbers have …….. absolute values.
IaI ? I-aI
Opposite numbers have equal absolute values.
IaI = I-aI
Example:
I3I = I-3I = 3
Define:
The distance between two real numbers on the number line.
d (a,b)
The distance between two real numbers a and b on the number line is defined as the absolute value of b - a.
- d (a,b) =* Ib - aI
- Example:*
d (5, -3) = I-3 - 5I = 8
Fill in the blank:
The absolute value is always greater than or equal….?
The absolute value of a number is always greater than or equal zero.
The sum of two positive numbers is positive. What sign is the sum of two negative numbers?
- 5 + ( -3) = ?
When you add two negative numbers, add their absolute values. The sign of the sum is negative.
- Example:*
- 5 + (-3) = -8
|-5| + |-3| = 5 + 3 = 8
How do you add two numbers with different signs?
When you add two numbers with different signs, subtract the absolute values (the smaller from the larger) and keep the sign of the number with the greater absolute value.
- Example:*
- 5 + 3 = -2
|-5| - |3| = 5 - 3 = 2
Since -5 has the greater absolute value than 3, the result is negative.
What number is the negative of a negative number?
-(-a) = ?
The negative of a negative number is the opposite positive number.
- (-a) = + a
How do you subtract two numbers with the same sign?
3 - 7 = ?
To subtract two numbers with the same sign, rewrite the problem as follows:
(a - b) = a + (-b)
Follow the rules for adding signed numbers.
Example:
3 - 7 = 3 + (-7) = -4
I-7I - I3I = 7 - 3 = 4
*** The result is negative since the negative number in this case has a greater absolute value.
How do you subtract two numbers with different signs?
3 - (-7) = ?
Subtract an integer by adding its opposite. The negative of a negative number is the opposite positive number.
- (-7) = +7
3 + 7 = 10
How do you multiply or divide two numbers with the same sign?
What sign is the product or the quotient of two same sign numbers?
- Find the product or the quotient of the absolute values of the two numbers
- Two numbers with the same sign always give you a positive result
Example:
5 x 3 = 15
(-5) x (-3) = 15
|5| = |-5| = 5 ; |3| = |-3| = 3
How do you multiply or divide two numbers with different signs?
What sign is the product or the quotient of two numbers with different signs?
- Find the product or the quotient of the absolute values of the two numbers
- Two numbers with different signs always produce a negative result
Example:
(-5) x 3 = -15
|-5| = 5 ; |3| = 3
How do you add more than two signed numbers?
-4 + 6 + 5 + (-8) = ?
- 4 + 6 + 5 + (-8) = -1
- Work from left to right and add numbers two at a time, following the rules
-4 + 6 = 2
2 + 5 = 7
7 + (-8) = -1 or
- Add all positive numbers, then add all negative numbers, following the rules. Find the sum of your answers
6 + 5 = 11
- 4 + (-8) = -12
11 + (-12) = -1
When you multiply more than two signed numbers, when is their product positive?
When you multiply signed numbers, find the product of the absolute values of all factors.
The product is positive when:
- all factors are positive
- there is an even number of negative factors.
Example:
(-5) x (2) x (-1) x (2) = 20
When you multiply more than two signed numbers, when is the product negative?
Find the product of the absolute values of all factors.
The product is negative when there is an odd number of negative factors.
Example:
(-5) x (-2) x (-1) x (2) = -20
What is the definition of a coordinate plane?
A coordinate plane is a plane formed by the intersection of a vertical (y-axis) and a horizontal (x-axis) number lines.
What is a coordinate plane used for?
A coordinate plane is used to graph ordered pairs, straight lines and functions.
What is the definition of an ordered pair?
An ordered pair is two numbers written in a specific order.
It is used to locate a point on a coordinate plane.
Usually denoted (x,y) where x and y are the coordinates of the point.
Example:
A (3, -8)
B (-6, -2)
How do you plot a point on a coordinate plane?
To plot a point, you need to know the x and y coordinates of that point (the ordered pair).
- a point with (0,0) coordinates is the origin
- a point with (0,y) is the point on y-axis
- a point with (x,0) is the point on x-axis
There are four quadrants in a coordinate plane. They are labelled with Roman numerals I, II, III, IV, starting at the positive x-axis and going around anti-clockwise.
In which quadrant are both x and y coordinates of a point
- positive?
- negative?
- The x and y coordinates of a point are both positive in the first quadrant (QI)
- The x and y coordinates of a point are both negative in the third quadrant (QIII)
- x* > 0; y > 0 - first quadrant
- x* y
There are four quadrants in a coordinate plane. They are labelled with Roman numerals I, II, III, IV, starting at the positive x-axis and going around anti-clockwise.
In which quadrant(s) do the x and y coordinates of a point have different signs?
The x and y coordinates of a point have different signs in the second and the fourth quadrants (Q II and Q IV).
- x* y > 0 - second quadrant
- x* > 0; y
What type of graph represents a linear equation?
A linear equation is an equation whose graph is a straight line.
Write a standard form of a linear equation.
ax + by = c
where a, b and c are integers and both a and b cannot equal zero.
What is y-intercept?
y-intercept of a line is the point where this line crosses the y-axis.
What equation represents slope-intercept form of a linear equation?
y = mx + b
- m* stands for the slope,
- b* is y-intercept.
How can you prove that b is the y-intercept of a line in the equation below?
y = mx + b
By definitions, y-intercept of a line is the point where this line crosses y-axis. Therefore, the x coordinate of that point is zero. Plug in zero in the equation below and you’ll see that y equals b.
y = mx + b
y = m * 0 + b ⇒ y = b
Define:
The slope of a line.
The slope (m) of a line measures the steepness of that line.
- y = mx + b*
- m* is a ratio that shows the change in value of y to the change of value of x.
- m* =
When is the slope of a line:
- negative?
- positive?
- When the slope of a line is negative, the line goes down left to right.
- When the slope is positive, the line rises up left to right.
Fill in the blanks:
If two lines have the same graph,
- their _____ are ______
and
- their _____ are ______ .
If two lines have the same graph, their slopes are equal and their y-intercepts are equal.
