## MCAT Physics and Math Review

## Chapter 10: Mathematics

### Conclusion

In this chapter, we reviewed many of the skills that are necessary for successful performance on the MCAT science sections. We began by examining relevant arithmetic calculations for Test Day, including scientific notation and significant figures. We continued our review by examining logarithms and exponents before discussing the most common trigonometric functions and their values. We finished our math review by working on problem-solving skills that will be valuable in your studying and during the MCAT itself. In the next two chapters, we’ll review Test Day skills in experimental design and data analysis so that we’re ready to answer all of our Test Day questions.

### Concept Summary

*Arithmetic and Significant Figures*

· **Scientific notation** is a method of writing numbers in a way that improves the ease of calculations and the comparability of significant digits.

o Scientific notation takes the format [significand] × 10^{[exponent]}.

o The **significand** must be greater than or equal to 1 and less than 10.

o The **exponent** must be an integer.

· **Significant figures** include all nonzero digits and any trailing zeroes in a number with a decimal point.

o Measurements are an exception, in that the last digit provided is not significant.

o In addition and subtraction, reduce the answer to have the same number of decimal places as the number with the fewest number of decimal places.

o In multiplication and division, reduce the answer to have the same number of significant digits as the number with the fewest number of significant digits.

o The entire number should be maintained throughout calculations to minimize rounding error.

· Estimation of multiplication and division should be done logically.

o In multiplication, if one number is rounded up, the other should be rounded down in proportion.

o In division, if one number is rounded up, the other should also be rounded up in proportion.

*Exponents and Logarithms*

· **Exponents** are a notation for repeated multiplication. They may be manipulated mathematically, especially when the bases are the same.

· **Logarithms** are the inverse of exponents and are subject to similar mathematical manipulations.

· **Natural logarithms**, which use base ** e** (

**Euler’s number**) can be converted into

**common logarithms**, which use base 10.

*Trigonometry*

· Trigonometric relationships can be calculated based on the lengths of the sides of right triangles.

· **Sine** is the ratio of the length of the side opposite an angle to the length of the hypotenuse.

· **Cosine** is the ratio of the length of the side adjacent to an angle to the length of the hypotenuse.

· **Tangent** is the ratio of the side opposite an angle to the side adjacent to it.

· **Inverse trigonometric functions** use the calculated value from a ratio of side lengths to calculate the angle of interest.

*Problem-Solving*

· In** direct relationships**, as one variable increases, the other increases in proportion.

· In** inverse relationships**, as one variable increases, the other decreases in proportion.

· Conversions between metric prefixes require multiplication or division by corresponding powers of ten.

· Conversions between units of different scales require multiplication or division, and may require addition or subtraction.

· Unit analysis (**dimensional analysis**) can determine the appropriate computation based on given information.

· Algebraic systems may be solved by substitution, setting equations equal, or elimination. The general ideas are the same in each—solve for one variable, and then substitute the variable into an equation to solve for the other—although the specific methods are different.

### Answers to Concept Checks

· **10.1**

1. First, determine which digits are significant, as these will be preserved in scientific notation. Then, move the decimal point until the significand is greater than or equal to 1 and less than 10. Finally, determine what power of 10 is necessary for multiplication to restore the original number.

2. 34,600.; 0.0003201; 1.10; 525,800

3. In multiplication, adjust the two decimals in opposite directions. In division, adjust the two decimals in the same direction.

· **10.2**

1.

· (*a* + *b*)^{2} = *a*^{2} + 2*ab* + *b*^{2}

·

· log*a* (*a*) = 1

·

2. is between and so the value is between 19 and 20. We can also simplify this radical:

3. log 7,426,135,420 ≈ log (7.4 × 10^{9}) ≈ 9 + 0.74 = 9.74 (actual = 9.87). Note that—even with an absurdly large number—we can still get relatively accurate estimations by following basic logarithm rules.

· **10.3**

1. The value of a trigonometric function calculated from the dimensions of the resultant vector is used in the inverse tangent function to calculate the resultant vector angle. Inverse trigonometric ratios, in general, can be used to calculate angles.

2. The sine of an angle is equal to the ratio of the side opposite the angle to the hypotenuse. Cosine is the ratio of the side adjacent to the angle to the hypotenuse. Tangent is the ratio of the side opposite the angle to the side adjacent to the angle.

3. False. While calculating the values of sine, cosine, and tangent is more complicated in a triangle that does not contain a right angle, all possible angles do still have characteristic trigonometric values.

· **10.4**

1. Conversion between metric prefixes is accomplished by multiplication or division by the relevant power of ten.

2. In direct relationships, as one quantity increases, the other also increases in proportion. In inverse relationships, as one quantity increases, the other decreases in proportion.

3. In substitution, solve one equation for one variable in terms of the other; then, substitute this expression into the other equation. In setting equations equal (a modified version of substitution), solve both equations for the same variable and set them equal to each other. In elimination, multiply or divide one (or both) equations so that the coefficient in front of one of the variables is the same in both equations; then, add or subtract the equations to eliminate one of the variables.

### Equations to Remember

(10.1) **Zero exponent identity**: *X*^{0} = 1

(10.2) **Multiplying like bases with exponents**: *XA* × *XB* = *X*^{(}*A*^{ +}*B*^{)}

(10.3) **Dividing like bases with exponents**:

(10.4) **Raising an exponent to another exponent**: (*XA*)*B* = *X*^{(}*A*^{ ×}*B*^{)}

(10.5) **Raising fractions to exponents**:

(10.6) **Raising bases to negative exponents**:

(10.7) **Raising bases to fractional exponents**:

(10.8) **Square root approximations**:

(10.9) **Logarithm of 1 identity**: log*A* 1 = 0

(10.10) **Logarithm of base identity**: log*A* *A* = 1

(10.11) **Logarithm of product**: log *A* × *B* = log *A* + log*B*

(10.12) **Logarithm of quotient**:

(10.13) **Logarithm of exponent-containing expression**: log *A*^{B} = *B*log *A*

(10.14) **Logarithm of inverse**:

(10.15) **Conversion of natural to common logarithm**:

(10.16) **Scientific notation logarithm approximation**: log (*n* × 10*m*) ≈ *m* + 0.*n*

(10.17) **Definition of sine**:

(10.18) **Definition of cosine**:

(10.19) **Definition of tangent**:

(10.20) **Temperature conversions**:

### Shared Concepts

· **General Chemistry Chapter 5**

o Chemical Kinetics

· **General Chemistry Chapter 7**

o Thermochemistry

· **General Chemistry Chapter 10**

o Acids and Bases

· **Physics and Math Chapter 7**

o Waves and Sound

· **Physics and Math Chapter 9**

o Atomic and Nuclear Phenomena

· **Physics and Math Chapter 12**

o Data-Based and Statistical Reasoning