ACT Math 1: Pre-Algebra — Numbers and Operations

Counting numbers

The positive integers beginning with 1: $1, 2, 3, 4, \ldots$. They are also called natural numbers.

Whole numbers

The counting numbers together with zero: $0, 1, 2, 3, \ldots$.

How do counting numbers differ from whole numbers?

Counting numbers start at $1$, whereas whole numbers include $0$ as well as all counting numbers.

What do the origin and coordinate represent on a number line?

The origin is the point labeled $0$. The coordinate is the number associated with a particular point on the number line.

How does position on a number line relate to numerical value?

Numbers increase from left to right and decrease from right to left. Points are equally spaced according to the chosen unit.

Place-value system

A number system in which the value represented by a digit depends on both the digit itself and its position in the number.

Why do $537$ and $735$ have different values even though they contain the same digits?

The digits occupy different places. Since each place has a different value, rearranging digits changes the total number.

What are the values of the ones, tens, and hundreds places in base ten?

The ones place has value $1$, the tens place has value $10$, and the hundreds place has value $100$. Each place is $10$ times the value of the place immediately to its right.

How can $138$ be decomposed using place value?

$138 = 1(100)+3(10)+8(1)=100+30+8$.

How do base-10 blocks model whole numbers?

A one-block represents $1$, a tens rod represents $10$ ones, and a hundreds square represents $100$ ones, or $10$ tens. The blocks show how a number is composed by place value.

What number is represented by 2 hundreds blocks, 1 tens rod, and 5 ones blocks?

$2(100)+1(10)+5(1)=215$.

Place-value period

A group of three place values in a whole number. Beginning at the right, the periods are ones, thousands, millions, billions, trillions, and so on.

What is the relationship between adjacent place values in the base-ten system?

Each place is worth $10$ times the place immediately to its right. For example, one hundred is $10$ times one ten.

What place value does each digit occupy in $5,278,194$?

From left to right, the digits are in the millions, hundred-thousands, ten-thousands, thousands, hundreds, tens, and ones places.

What is the value of the digit $7$ in $63,407,218$?

The $7$ is in the thousands place, so its value is $7,000$.

How should a zero in a whole number be interpreted in place value?

A zero indicates that its place contributes no quantity, but it may still be essential as a placeholder to preserve the positions of digits in larger places.

How do you name a whole number in words using periods?

Starting with the leftmost period, name the number in each period followed by its period name, separating periods with commas. Do not say the name of the ones period, and do not insert “and” between whole-number periods.

How is $37,519,248$ written in words?

Thirty-seven million, five hundred nineteen thousand, two hundred forty-eight.

How do you convert a whole number written in words into digits?

Identify each named period, assign up to three digit positions within each period, and fill in the digits. Use zeros as placeholders whenever a period or place is not represented.

Write “nine billion, two hundred forty-six million, seventy-three thousand, one hundred eighty-nine” in standard form.

$9,246,073,189$. The zero in the hundred-thousands place preserves the three-place thousands period.

What is the standard-form meaning of “$77$ billion”?

$77,000,000,000$. All unnamed lower periods are filled with zeros.

Rounding

The process of replacing a number with a nearby value at a specified place to make it simpler while retaining an appropriate level of accuracy.

How do you round a whole number to a specified place value?

Look at the digit immediately to the right of the target place. If it is less than $5$, leave the target digit unchanged; if it is at least $5$, increase the target digit by $1$. Replace every digit to the right with zero.

Why does $75$ round to $80$ when rounded to the nearest ten?

$75$ is exactly halfway between $70$ and $80$. The stated convention is to round halfway cases up to the higher value.

Round $23,658$ to the nearest hundred.

The tens digit is $5$, so round the hundreds digit up: $23,700$.

Round $147,032$ to the nearest thousand.

The hundreds digit is $0$, which is less than $5$, so the result is $147,000$.

What happens when rounding requires increasing a target digit of $9$?

The $9$ becomes $0$, and $1$ is carried to the digit immediately to its left. This regrouping can continue through consecutive $9$s, as in $29,504$ rounded to the nearest thousand, which is $30,000$.

When should a rounded value be used instead of an exact whole number?

Use rounding when an approximate value is sufficient or when a simpler scale improves communication, such as reporting a population to the nearest million.

Integer

Any positive whole number, negative whole number, or zero: $\ldots,-3,-2,-1,0,1,2,3,\ldots$.

Negative number

A number less than zero, written with a minus sign. It can represent quantities such as temperatures below zero, overdrafts, elevations below sea level, or depths.

How are positive and negative numbers located on a number line?

Positive numbers lie to the right of zero, and negative numbers lie to the left. The number line extends indefinitely in both directions.

What does an omitted sign mean in front of a number?

A number without a sign is understood to be positive; for example, $5$ means $+5$.

Is zero positive or negative?

Zero is neither positive nor negative. It is the boundary between the positive and negative numbers.

How can an elevation of $-1,302$ feet be interpreted?

It represents a location $1,302$ feet below sea level, where sea level is assigned the value $0$.

How are integers ordered on a number line?

A number farther to the right is greater than a number farther to the left. Thus every positive integer is greater than zero and every negative integer, while zero is greater than every negative integer.

Opposites

Two numbers are opposites if they are the same distance from zero but lie on opposite sides of the number line. For example, $+4$ and $-4$ are opposites.

Absolute value

The absolute value of a number is its distance from zero on the number line, so it is never negative. It is written with vertical bars, such as $|-7|=7$ and $|5|=5$.

How do you simplify an expression containing absolute values?

Evaluate each absolute value as a nonnegative distance from zero, then perform the remaining operations. For example, $|-3|+|2|=3+2=5$.

How can integer word phrases be translated into numbers?

Use the reference point and direction indicated by the phrase: quantities above, gains, or increases are positive, while quantities below, losses, or decreases are negative. For example, $5$ degrees below zero is $-5^\circ$.