Arithmetic

Lowest Common Multiple

Find the lowest number the numbers divide into From above, use all factors from both numbers Good diagram here

$$ LCM = 2 * 2 * 2 * 2 * 3 * 3 * 11 = 1584 $$ Check: $$ 1584 / 144 = 11 $$ $$ 1584 / 66 = 24 $$

Fractions

$$ 1 \frac{1}{4} $$ This fraction is in its lowest terms because common factors include just unity Ratios occur in fractions with equal denominators. A percentage has 100 as denominator Multiply a fraction by 100 to turn it into a percentage $$ \frac{1}{4} * 100 = 25\% $$

28. Division of Integers

• Proper fraction; numerator less than denominator (1 /4)
• Improper; top heavy
• Mixed: $$ 1 \frac{1}{4} $$

Multiplying Fractions

$$ \frac{2}{4} * \frac{3}{8} = \frac{2 * 3}{4 * 8} = \frac{6}{32} $$

34. Equivalent Fractions

Multiply top and bottom by the same number, get the same $$ \frac{4}{5} \equiv \frac{12}{15}$$ Reverse the process; divide top and bottom by the same to reduce the fraction to lowest terms

37. Dividing Fractions

Flip the divider, multiply instead. $$ \frac{2}{4} \div \frac{3}{8} \equiv \frac{2}{4} * \frac{8}{3} $$

41. Adding Fractions

Find common denominator, using LCM, or just by figuring out a low number. Then can add $$ \frac{2}{4} - \frac{3}{8} \equiv \frac{4}{8} - \frac{3}{8} \equiv \frac{1}{8} $$

58. Significant Figures

0.1345 to 2 Significant figures = 0.13
1355 to 2 Significant figures = 1400

61. Decimal Places

0.0035 to 2 decimal places = 0.0035
0.2356 to 2 decimal places = 0.24A

Decimals as Fractions

$$ 1.224 = \frac{1224}{1000} \equiv \frac{153}{125} $$

70. Unending decimals

Put a dot over first and last number that repeats $$ 0.183183 \equiv 0.\dot{1}8\dot{3} $$ Any unnending repeating form can be a fraction. Convert to a fraction: $$ 1000x = 183.\dot{1}8\dot{3} $$ $$ x = 0.\dot{1}8\dot{3} $$ $$ 1000x - x = 183.\dot{1}8\dot{3} - 0.\dot{1}8\dot{3} $$ $$ 999x = 183 $$ $$ x = 183 / 999 $$

72. Rational, Irrational and Real Numbers

Fractions are rationals. Irrationals don’t work as fractions. e.g. $$ \sqrt{2} \equiv 0.14121... $$ $$ \Pi \equiv 3.14159... $$ Irrational and Rational numbers encompass all Real numbers.

78. Powers

$$ 2^3 \equiv 2 * 2 * 2 $$ $$ 3^1 \equiv 3 $$ $$ 3^0 \equiv 1 $$

Multiplication of same base

Same base to power, can add: $$ 2^3 * 2^6 \equiv 2^9 $$

Different bases to not work: $$ 2^4 * 3^5 \neq 2^9 $$

… But different bases with same power: $$ 2^4 * 3^4 \equiv 6^4 $$ i.e. multiply base, keep power

Division

$$ 5^6 \div 5^2 \equiv 5^4 $$ Same as mult; need the same base

Negative Powers

$$ 6^{-2} \equiv \frac{1}{6^2} $$ Because: $$ 6^{-2} = 6^0 - 6^{-2} = 6^0 \div 6^2 = \frac{1}{6^2} $$

Power products

Multiply the powers $$ (5^2)^{3} \equiv 5^6 $$

93. Fractional Powers and Roots

Odd roots are unique $$ 6^{1/3} \equiv Unique \equiv 1.817 $$ Even roots are not unique $$ 6^{1/4} \equiv 2\ OR -2 $$ Odd roots of negative are also negative $$ -32^{1/5} \equiv -2 $$ Even root of negative number cannot be found (yet) $$ -3^{1/4} $$

Integer powers of 10

$$ 0.01204 * 10^4 \equiv 120.4 $$ $$ 1.2 * 10^{-2} \equiv 0.012 $$

97. Precedence of powers

Evaluate powers before precedence rules

99. Standard Form

Express any integer as a number between 1 and 10 (mantissa) $$ 57.3 = 5.73 * 10^1 $$ Multiply/divide numbers by evaluating mantissas and adding or subtracting powers $$ 0.84 * 23000 = (8.4 * 10^-1) * (2.3 * 10^4) = 19.32*10^3 $$ For add/subtract, make the powers the same: $$ 4.72 * 10^3 + 3.648 * 10^4 \equiv 4.72 * 10^3 + 36.48 * 10^3 $$ $$ (4.72 + 36.48) * 10^3 $$

Preferred Standard Form

$$ 5.2746 * 10^4 $$ Powers of 3 for ‘preferred’; i.e. up to 3 digits before the point $$ 52.746 * 10^3 $$ ENG key on the Casio calculator
SCI mode is for working in standard form

Example in standard

$$ (4.72 * 10^2) * (8.36 * 10^5) = 39.4592 * 10^7 = 3.94592 * 10^8 $$

And in preferred standard:

$$ 394.592 * 10^6 $$

106. Error check

Convert to standard form, estimate: $$ 800.120 * 0.007953 \approx 8 * 10^2 * 8 * 10^-3 = 64 * 10^-1 = 6.4 $$

107. Accuracy

The smallest significant figures is all you can expect from an answer based on measured values. $$ 19.1 * 0.0053 \div 13.345 = 0.00758561 $$ in 2 figures: 0.0076

112. Denary (base 10)

$$ 10^3 10^2 10^1 10^0 . 10^{-1} 10^{-2} 10^{-3} $$

113. Binary (base 2)

$$ 2^3 2^2 2^1 2^0 . 2^{-1} 2^{-2} 2^{-3} $$

114 Octal (base 8)

$$ 8^3 8^2 8^1 8^0 . 8^{-1} 8^{-2} 8^{-3} $$

116 Duodecimal (base 12)

Extra symbols X and // ? How to do this in latex? $$ 12^3 12^2 12^1 12^0 . 12^{-1} 12^{-2} 12^{-3} $$

117 Hexadecimal (base 16)

Extra symbols ABCDEF $$ 16^3 16^2 16^1 16^0 . 16^{-1} 16^{-2} 16^{-3} $$

120 Alternative method

Take each digit, multiply by the base, add to the next digit and multiply again, etc. Just add the final number for the decimal point. For the bit after the point, multiply by the minus power of decimal digits. See 120 in stroud for a good diagram.

130 Change from Denary to a new base

Succesively divide by the base of the new number, and keep the remainder, then read the remainders in reverse. 12 in Octal $$ 12 / 8 = 4, 1 / 8 = 1 (14) $$

133 Change Denary decimal

This time multiply the decimal part by th new base, then discard the bit before the decimal. In base 8 $$ 0.526 * 8 = 4.208, .208 * 8 = 1.664.... $$ The answer is 0.41..

135 Convert both parts

We do the whole number part (130) and then the decimal part (133) and then squash them together

137 Octals for binary and hex

Convert to Octal first, then convert each digit to 3-entry binary digits: 5 = 101. That’s the binary equivalent. Then regroup the binary digits working outward from the center point into groups of 4 to get the hex digits. The same process works backwards from octal to hex/binary (147)