Apidays New York 2024 - Accelerating FinTech Innovation by Vasa Krishnan, Fin...
Thursday, september 26, 2013
1. September 26, 2013
• Today:
• Make Up Tests?
• Grades shown Today, questions tomorrow
• Warm- Up: Equations
• The real number system: Integers &
Absolute Value
2. • Warm-Up, Equations:
• 1. A. 2 + 5 = 7; Solve for 5; 5 = B. Solve for 2; 2 =
• C. 14 - 8 = 6; Solve for -8; -8 = D. Solve for 14; 14 =
• E. 12/4 = 3; Solve for 4 F. Solve for 12; 12 =
• G. (5)(3) = 15; Solve for 5; 5 = H. Solve for 3; 3 =
The rules are the same for variables, because variables
are just numbers we haven't figured out yet.
2. Write 4 different equations so that the answer to each one
is x = 5. The first equation should have a sum, the second a
difference, the third a product and the fourth a quotient.
x + 5 = 7; x = 7 – 5; x = 2
5. Natural Numbers: {1, 2, 3, …}
Whole Numbers:
{0, 1, 2, 3, …}
Integers: {…, -2, -1, 0, 1, 2, …}
Rational Numbers:
Numbers expressed in the form
a/b, where a and b are integers
and b ≠ 0
The Real Numbers
Includes all fractions,
Mixed numbers, Ratios,
Proportions, & decimals.
Irrational Numbers:
These numbers don't end and
they don't repeat. Ex. √2, Pi,
6. Real Numbers
Practice: For each of the numbers, write
Natural, Whole, Integer, Rational, or Real according
to its type. Most will have more than one
classification.
1. 0.25 2. 8.25252525 3. -1/2 4. 5 5. 0 6. -5
7. √5 8. 200 ft. below sea level
7. Integer Rules:
1. For Multiplying : If there is an odd number of
negative values, the product will be negative. If the
number of values is even, the product will be negative:
3(-5) 3(-5)(-2) 3(-5)(-2)(-3)
2. For Dividing : Since we can only divide two numbers
at a time, the rule is easy: If one of the numbers is
negative, the quotient is negative. If both are
positive, or both are negative, the quotient is positive.
-28 ÷ (-4) 35 ÷ -7 -25 ÷ -5
-4ab•(-6) (-4)(-2)(8r)
8. Integer Rules:
3. For Adding & Subtracting
3a. If the signs are the same, add the values and use
the given sign.
3b. If the signs are different, subtract the values and
use the sign of the larger number.
3c. Sometimes you must reduce the number of signs
before you add or subtract. 1. -2 - (-19)
ALWAYS leave the sign of the first number alone. The
first number shows the starting point of the calculation
Our focus is always on the next two signs. If they are the
same, change them both to a single plus + sign. If they
are different, change them to a single negative sign.
1. 2 + 5 2. -2 - 5
9. Integer Rules:
At this point, the original expression -2 - (-19)
Now looks like: -2 +19
Only now can we go back to our original rules, 3a. And 3b.
(Signs are the same or signs are different)
1. -7 + - 5 2. - 8 + - 10 + 8 3. - 4 + (- 5) + 6
4. - 2 - (- 8) 5. - 4 - (- 10) 6. 4 - 22
1. - 12 2. - 10 3. - 3
4. 6 5. 6 6. - 18
11. Absolute Value: |x|
• An important concept in Algebra; one that you
should try to master.
Absolute Value measures the distance a number is
from zero.
The following are illustrations of what absolute
value means using the numbers 3 and -3:
Since Absolute Value is a measure of distance, the
result can never be negative. (There can be no negative
distances)
12. • If a number is positive (or zero), the absolute value function
does nothing to it: |4| = 4
• If a number is negative, the absolute value function makes it
positive: |-4| = 4
Find the value of the following: |5 + (-2)|
Did you get 7? Unfortunately, that's wrong.
If there is arithmetic to do inside the absolute value
sign, you must do it before taking the absolute value
sign. The correct answer is: |5 + (-2)| = |3| = 3
The following has no solution: | x |= -5 Why?
Absolute Value: |x|
16. 6 + 9 + 2 = 17; -3 + (-5) + (-4) = -12; 17 - 12 = 5
Use Method B: 5 + (-4) + (-6) + 8 + (-3) + 3 =
1 -6 = -5 + 8 = 13 - 3 = -0 + 3 = 13
1. There are 2 ways to combine long strings of integers.
A. Add all the positive numbers, then add all the
negative numbers. Then combine the two numbers using
rule for opposite signs.
B. Combine all numbers in order from left to right.
Use method A for these numbers:
-3 + (-5) + 6 + 9 + 2 + (-4)
