Arithmetic for the Practical Man

Concrete and Abstract number If the number is applied to specific counted or measured, things, objects, quantities the number is said to be concrete or sometimes a denominate number.

When number do not apply to those thing they are called abstract numbers.

Glossary

• Factor - 2 x 3 = 6 (Here 2 and 3 are factors of 6)
• Mantissa - 2.091514977 In this case, the characteristic is 2, and the mantissa is 0.091514977.
• Factorial - A number with (!) after it means it is factorial, eg: (3! = 3 x 2 x 1 = 6)

Keywords

• sum is simply sum
• subtraction
  • Minuend
  • Subtrahend
  • Remainders
• multiplication
  • Multiplicand
  • Multiplier
  • Product
• division
  • Divisor
  • Dividend
  • Quotient
• fractions
  • Numerator
  • Denominator
• root
  • Taken or Extracted (the calculation was made) The general operation of a root extraction is called evolution
• ratio
• series

Fractions

Decimal fractions are the fractions in which the denominator must be 10 or a multiple of 10 like 100, 1000, 10000.

Some division rules

1. If the last figure of any number is even, the number is divisible by 2
2. If the lasf figure of any number is 5 or 0, the number is divisible by 5
3. If the number consisting of the last two figures of any given number is divisible by 4 or by 25, the entire number is divisible by 4 or 25
4. If the number represented by the last three figures of a given number is divisible by 8, the given number is divisible by 8
5. If the sum of the separated figures of any number is divisible by 9, the number is divisible by 9
6. If the sum of the separated figures of any number is divisible by 3, the number is divisible by 3
7. If any number is divisible by 3 it is divisible by 6

Logarithms

$\log_{6}36 = 2$

[!info] “How many sixes need to be multiplied together to get 36?”

If any power of 2 is multiplied by any power of 2 number, the result is a power of two whose then the sum of the exponents corresponding to the multiplicand and the multiplier.

Finding the antilog

1. log₂ 16.
2. The logarithm base (2) and the logarithm value (16) indicate that 2 raised to what power equals 16.
3. In this case, 2 raised to the power of 4 equals 16, as 2^4 = 16.
4. Therefore, the antilog base 2 of log base 2 of 16 (log₂ 16) is 4.

Log can be used to simplify equations, you resolve the equation using the log of the values and then calculate the antilog to get the result.

Multiplication with logs Find the logarithm of each of the factors and add these logarithms. Find the antilog.

Division with logs Find the logarithm of each of the factors and subtract these logarithms. Find the antilog.

Power with logs Find the log of a number, and multiply this log by the exponent. Find the antilog.

Roots with logs Find the log of the number, and divide this log by the root index. The antilog is the result.

Series

1. First term
2. Last term1
3. Number of terms
4. Common difference
5. Sum of the series

[!info] If any of three are know the remaining two can be calculated.

Arithmetical Progression

• The number which form the series are called terms of the series.
• The method which each term is formed from the preceding term is called law of the series.
• The number added to each term of the series is called common difference.
• If each term in a series is formed by adding a certain number, to the preceding term, the terms of the series said to be in arithmetical progression.

Rules

1. The last term of an arithmetical progression is equal to the first term (65) plus the product of the common difference (2) by the number of terms (27) -1.

    [5, 7, 9, 11, 13, ...].length = 27
   
    5 + (26 x 2)
    5 + 52
    57
   

2. The first term of a progression is equal to the last term minus the product of the common difference by the number of terms -1.

   if we let a represent the first term, l represent the last term, n the number of terms, and d the common difference, we can express those two rules as follow:

   $l = a + [(n - 1) \enspace . \enspace d]$ $a = l - [(n - 1) \enspace . \enspace d]$

3. The common difference of an arithmetic progression is found by dividing the difference of the extremes by the number of terms less 1.

4. To find the number of terms in an arithmetical progression, divide the difference of the extremes by the common difference of the progression and to the quotient add 1. $d = (l-a) \div (n-1)$ $n = [(l-a) \div d] + 1$

5. The sum of the terms of an arithmetical progression is the product of half the number of terms by the sum of the extremes. $S = \frac{1}{2} \times (a + l)$

Geometrical Progression

• The number added to each term of the series is called common ratio.
• If each term in a series is formed by multiplying a certain number, to the preceding term, the terms of the series said to be in geometrical progression.

1. Initial term: In a geometric progression, the first number is called the initial term.
2. Common ratio: The ratio between a term in the sequence and the term before it is called the “common ratio.” 
3. The behavior of a geometric sequence depends on the value of the common ratio. If the common ratio is:
   • Positive, the terms will all be the same sign as the initial term.
   • Negative, the terms will alternate between positive and negative.
   • Greater than 1, there will be exponential growth towards positive or negative infinity (depending on the sign of the initial term). 
   • 1, the progression is a constant sequence.
   • Between -1 and 1 but not zero, there will be exponential decay towards zero.
   • -1, the progression is an alternating sequence.
   • Less than -1, for the absolute values there is exponential growth towards (unsigned) infinity, due to the alternating sign.

Angles

Angle Measures

• 60 seconds (´´) = 1 minute (´)
• 60 minutes = 1 degree (°)
• 90 degrees = 1 quadrant (quad.)
• 4 quadrants = 1 circle (⊙)

Latitude and Longitude

Those concepts uses imaginaries line between the globe: ![[Pasted image 20230617051422.png]]

It’s very helpful because you can use it to calculate time and distance as well, for example, one nautic mile is ~1 degree.

Percentage

Rules

1. To express a decimal fraction in percentage, shift the decimal point two places to the right. Example: .273 = $\frac{273}{1000}$ = $\frac{27.3}{100}$ = 27.3%;
2. To express a common fraction or ratio in percentage, multiply the numerator by 100 (add two zeros) and divide by the denominator. Example: $\frac{53}{79}$ 5300 $\div$ 79 = 67.1%

Binary

Counting by eight, four and binary:

10 = 2 100 = two x two = ($2^{2}$) 1000 = two x two x two = ($2^{3}$) 10_000_000_000 = ($2^{10}$)

Converting a denary to binary

eg 14 = 1110

2 / 14 = 7 (remainder 0) 2 / 07 = 3 (remainder 1) 2 / 03 = 1 (remainder 1) 2 / 01 = 1 (remainder 1)

[!info] The result here is inverted 0111 -> 1110

Converting a binary to denary Starting from the left side:

eg 1110 = 14

0 = 0 1 = 2 1 = 4 1 = 8

Table reference [[#Logarithms]]

14 = 2 + 4 + 8

Addition of binary 11 10 110 100 1100 = 11011

1. The addition of the right-hand column gives simply 1, write the 1;
2. The next column is 1+1+1 11, write the 1 and carry the 1;
3. The next column is 1 (carry) + 1 + 1 + 1 100, write the 0 and carry the 10;
4. The next column is 10 (carry) + 1 11, write the 1 and carry the 1;
5. The last entry is 1 (carry) with write the 1.

Subtraction with binary 100 1 11

1. We cannot subtract 0/1, borrow the left side
   1. Borrow 0 is not useful, borrow again 1
   2. Now we have 10
   3. 10 - 1 = 1