Chapter 1: Arithmetic & Prealgebra Section 1.1: Place Value, Estimation, Rounding Decimals & Order of Operations Applied Mathematics, 2nd Ed, Copyright 2018, Matovina & Yates

Applied Mathematics, 2nd Ed, Copyright 2018, Matovina & Yates Place Values Applied Mathematics, 2nd Ed, Copyright 2018, Matovina & Yates

Applied Mathematics, 2nd Ed, Copyright 2018, Matovina & Yates Place Values In a base 10 system, each place value is another power of 10. 100 = 1 (ones place) 101 = 10 (tens place) 102 = 10 (hundreds place) 103 = 10 (thousands place) The digit stated in the place indicated the quantity for that place. We have exactly ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 Applied Mathematics, 2nd Ed, Copyright 2018, Matovina & Yates

Applied Mathematics, 2nd Ed, Copyright 2018, Matovina & Yates Place Values Expanded form of a number is writing that number as an addition statement, showing the individual place values. Example: Write 7204 in expanded form. 7204 = 7000 + 200 + 0 + 4 = 7×1000 + 2×100 + 0×10 + 4×1 = 7×103 + 2×102 + 0×101 + 4×100 Applied Mathematics, 2nd Ed, Copyright 2018, Matovina & Yates

Applied Mathematics, 2nd Ed, Copyright 2018, Matovina & Yates Binary Numbers In a base two system (binary), each place value is another power of 2. 20 = 1 (ones place) 21 = 10 (twos place) 22 = 10 (fours place) 23 = 10 (eights place) The digit in a stated place indicates the quantity for that place. Every number is composed of some combination of 1s and or 0s. Applied Mathematics, 2nd Ed, Copyright 2018, Matovina & Yates

Applied Mathematics, 2nd Ed, Copyright 2018, Matovina & Yates Binary Numbers Example: Determine the decimal equivalent for the binary number 100101. 100101 = 1×25 + 0×24 + 0×23 + 1×22 + 0×21 + 1×20 = 1×32 + 0×16 + 0×8 + 1×4 + 0×2 + 1×1 = 32 + 0 + 0 + 4 + 0 + 1 = 37 Applied Mathematics, 2nd Ed, Copyright 2018, Matovina & Yates

Applied Mathematics, 2nd Ed, Copyright 2018, Matovina & Yates Binary Numbers When converting a decimal number to a binary number, remember, there is either a 1 or a 0 for each place value, and the binary number is some combination of 1s and 0s. A 1 in a given place value indicates to count that value, and a 0 tells us to skip it. Remember the powers of 2: 20 = 1 21 = 2 22 = 4 23 = 8 24 = 16 25 = 32 Applied Mathematics, 2nd Ed, Copyright 2018, Matovina & Yates

Applied Mathematics, 2nd Ed, Copyright 2018, Matovina & Yates Binary Numbers Find the largest power of 2 in the decimal number. This tells us how many digits are in the binary number. Indicate the inclusion in the binary number by putting a 1 in the corresponding place value for the binary number, and subtract that amount from the decimal number. Continue until we reach the desired sum. Applied Mathematics, 2nd Ed, Copyright 2018, Matovina & Yates

Applied Mathematics, 2nd Ed, Copyright 2018, Matovina & Yates Binary Numbers Example: Write the decimal number 91 as a binary number. Start with the basic framework for a binary number. ___ ___ ___ ___ ___ ___ ___ 64s 32s 16s 8s 4s 2s 1s Since we have one 64 in 91, put a 1 in the 64s place. How much is left? 91 – 64 = 27. There are no 32s in 27, so put a 0 in the 32s place. There’s one 16 in 27, so put a 1 in that place. 27– 16 = 11. 11 has an 8, no 4, a 2, and a 1. So finish out the number with 1 0 1 1. 1 1 1 1 1 Cleaning it up… 9110 = 10110112 Applied Mathematics, 2nd Ed, Copyright 2018, Matovina & Yates

Applied Mathematics, 2nd Ed, Copyright 2018, Matovina & Yates Hexadecimal Numbers In a hexadecimal place value system, each place value corresponds to a power of 16. 160 = 1 161 = 16 162 = 256 A hexadecimal system has sixteen unique, single-character digits. The first ten are easy: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 We cannot use “10” as a single-character digit, so we use the capital letter A to represent 10. Similarly, B represents 11, C is 12, D is 13, E is 14, and F is 15. To keep things simple (and short!), we will stick with two-digit hex numbers. Applied Mathematics, 2nd Ed, Copyright 2018, Matovina & Yates

Applied Mathematics, 2nd Ed, Copyright 2018, Matovina & Yates Hexadecimal Numbers Example: Convert the hex number D4 to a decimal number. The D is represents how many 16s are in the number, and the 4 represents how many 1s are in the number. Since D = 13, we have: D4 = D×16 + 4×1 = 13×16 + 4×1 = 208 + 4 = 212 Applied Mathematics, 2nd Ed, Copyright 2018, Matovina & Yates

Applied Mathematics, 2nd Ed, Copyright 2018, Matovina & Yates Hexadecimal Numbers To convert a decimal number (less than 256) to a hexadecimal number, we need to find out how many 16s are in the number. To do that, we divide by 16. For our two digit hex number, the quotient will be the digit in the 16s place, and the remainder will be the digit in the 1s place. Example: Convert the decimal number 91 to a hex number. Using long division, 91 divided by 16 is 5 with a remainder of 11. Using B instead of 11… 9110 = 5B16 16s 1s Applied Mathematics, 2nd Ed, Copyright 2018, Matovina & Yates

Estimation vs. Rounding Estimation is a process in which we determine an approximate value of a calculation. Some specific methodologies for estimations exist, but, unless directed to use a specific method, the degree of precision is up to the individual. Estimations can be done to the desired place value. 194.8 is approximately 195 or 200, depending on the desire of the one doing the estimation. Rounding is a process in which we follow a directive to make a quantity easier to visualize. In rounding, however, we MUST be given a specific place value to which to round. If no place value is indicated, it is improper to round. Applied Mathematics, 2nd Ed, Copyright 2018, Matovina & Yates

A Brief Review of Rounding The traditional rounding procedure: First, determine the round-off digit, which is the digit in the specified place value column. If the first digit to the right of the round-off digit is less than 5, do not change the round-off digit, but delete all the remaining digits to its right. If we are rounding to a whole number, such as tens or hundreds, all the digits between the round-off digit and the decimal point should become zeros, and no digits will appear after the decimal point. If the first digit to the right of the round-off digit is 5 or more, increase the round-off digit by 1, and delete all the remaining digits to its right. Again, if we are rounding to a non- decimal number, such as tens or hundreds, all the digits between the round-off digit and the decimal point should become zeros, and no digits will appear after the decimal point. For decimals, double-check to make sure the right-most digit of the decimal falls in the place value column to which we were directed to round, and there are no other digits to its right.

A Brief Review of Rounding Examples: Round 2578.3491 to the nearest Hundred 2600 Tenth 2578.3 Ten Thousand

Leading Digit Estimation Leading Digit Estimation is a technique in where we actually arrive at our estimated answer by rounding each number in the problem based on the first digit in each number. Although general estimations can vary from person to person, estimations done using Leading Digit Estimation should be the same, regardless of who has done them. Applied Mathematics, 2nd Ed, Copyright 2018, Matovina & Yates

Leading Digit Estimation Example: Use Leading Digit Estimation to estimate the sum: 2319 + 345 + 12 + 421 + 5698 Looking at the leading digit of each number… Round 2319 to 2000 Round 345 to 300 Round 12 to 10 Round 421 to 400 Round 5698 to 6000   2000 + 300 + 10 + 400 + 6000 = 8710 Applied Mathematics, 2nd Ed, Copyright 2018, Matovina & Yates

Arithmetic with Decimals Adding, subtracting, multiplying and dividing with decimals is almost exactly like doing so with whole numbers. The only extra step is the proper placement of the decimal point. With addition and subtraction, we like to set up the problem vertically and align the decimal points in the numbers. We also may need to include extra zeros in subtraction problems to ensure the decimals have the same place value agreement. Applied Mathematics, 2nd Ed, Copyright 2018, Matovina & Yates

Applied Mathematics, 2nd Ed, Copyright 2018, Matovina & Yates Order of Operations All operations contained within parentheses or other grouping symbols, such as brackets [ ], or braces { }, should be done first. Secondly, simplify all expressions containing exponents. Multiplication and division are done next, as we come to them going from left to right. Addition and subtraction are done last, again, as we come to them going from left to right. To help remember this order, many students like to memorize a cute little acronym like PEMDAS (Parentheses, Exponents, Multiplication, Division, Addition, Subtraction). Be careful! If you do not realize multiplication and division are done as we come to them going from the left to the right, you may fall into the trap of thinking multiplication always precedes division – it does not. The same hold true for addition and subtraction. Applied Mathematics, 2nd Ed, Copyright 2018, Matovina & Yates

Applied Mathematics, 2nd Ed, Copyright 2018, Matovina & Yates Order of Operations Example: Simplify 5 × (2 + 3)4 - (6 - 4) + 1 5 × (5)4 - (2) + 1 5 × 625 - 2 + 1 3125 - 2 + 1 3123 + 1 3124 Applied Mathematics, 2nd Ed, Copyright 2018, Matovina & Yates

Applied Mathematics, 2nd Ed, Copyright 2018, Matovina & Yates Calculators Make sure you have a scientific calculator and not a standard calculator. A standard calculator does not perform multi-step computations in accordance with the order of operations! If you are not sure if your calculator is a scientific or standard calculator, type in 5 + 2 × 3. If you get the correct answer of 11, you have a scientific calculator. If you get the incorrect answer of 21, you have a standard calculator. Applied Mathematics, 2nd Ed, Copyright 2018, Matovina & Yates